From 9cf76cbc101ef353eca77b563bf73e08e367b78d Mon Sep 17 00:00:00 2001
From: Stan Jenkins <mc.stanislaw@gmail.com>
Date: Mon, 8 Apr 2024 16:18:49 +0000
Subject: [PATCH] Documentation generated from Jenkins for docs-2-34

---
 docs/2_34/404.html                            |  172 ++-
 docs/2_34/cmdstan-guide-2_34.pdf              |  Bin 558861 -> 550389 bytes
 docs/2_34/cmdstan-guide/bib.html              |  172 ++-
 .../cmdstan-guide/command_line_options.html   |  224 +++-
 .../compiling_stan_programs.html              |  224 +++-
 docs/2_34/cmdstan-guide/diagnose_config.html  |  224 +++-
 docs/2_34/cmdstan-guide/diagnose_utility.html |  216 +++-
 .../2_34/cmdstan-guide/err_handling_apdx.html |  645 ----------
 .../cmdstan-guide/example_model_data.html     |  218 +++-
 docs/2_34/cmdstan-guide/external_code.html    |  216 +++-
 .../generate_quantities_config.html           |  352 ++++--
 .../generate_quantities_intro.html            |  979 ---------------
 docs/2_34/cmdstan-guide/index.html            |  218 +++-
 docs/2_34/cmdstan-guide/installation.html     |  224 +++-
 docs/2_34/cmdstan-guide/json_apdx.html        |  216 +++-
 .../cmdstan-guide/laplace_sample_config.html  |  224 +++-
 docs/2_34/cmdstan-guide/log_prob_config.html  |  218 +++-
 docs/2_34/cmdstan-guide/mcmc_config.html      |  431 +++++--
 .../cmdstan-guide/mcmc_sampling_intro.html    | 1094 ----------------
 .../cmdstan-guide/optimization_intro.html     |  961 --------------
 docs/2_34/cmdstan-guide/optimize_config.html  |  293 +++--
 docs/2_34/cmdstan-guide/parallelization.html  |  232 +++-
 .../2_34/cmdstan-guide/pathfinder_config.html |  339 +++--
 docs/2_34/cmdstan-guide/pathfinder_intro.html |  965 --------------
 docs/2_34/cmdstan-guide/print.html            |  216 +++-
 docs/2_34/cmdstan-guide/rdump_apdx.html       |  216 +++-
 docs/2_34/cmdstan-guide/stan_csv_apdx.html    |  218 +++-
 docs/2_34/cmdstan-guide/stanc.html            |  222 +++-
 docs/2_34/cmdstan-guide/stansummary.html      |  216 +++-
 .../cmdstan-guide/variational_config.html     |  329 +++--
 .../2_34/cmdstan-guide/variational_intro.html | 1027 ---------------
 docs/2_34/functions-reference-2_34.pdf        |  Bin 904075 -> 904346 bytes
 .../functions-reference/array_operations.html |  172 ++-
 .../binary_distributions.html                 |  172 ++-
 .../bounded_continuous_distributions.html     |  172 ++-
 .../bounded_discrete_distributions.html       |  172 ++-
 .../circular_distributions.html               |  172 ++-
 .../complex-valued_basic_functions.html       |  172 ++-
 .../complex_matrix_operations.html            |  172 ++-
 .../compound_arithmetic_and_assignment.html   |  172 ++-
 .../continuous_distributions_on_0_1.html      |  172 ++-
 ...conventions_for_probability_functions.html |  172 ++-
 .../correlation_matrix_distributions.html     |  172 ++-
 .../covariance_matrix_distributions.html      |  172 ++-
 .../deprecated_functions.html                 |  174 ++-
 .../distributions_over_unbounded_vectors.html |  172 ++-
 .../functions-reference/functions_index.html  |  172 ++-
 .../hidden_markov_models.html                 |  172 ++-
 .../higher-order_functions.html               |  172 ++-
 docs/2_34/functions-reference/index.html      |  174 ++-
 .../integer-valued_basic_functions.html       |  174 ++-
 .../mathematical_functions.html               |  172 ++-
 .../matrix_operations.html                    |  172 ++-
 .../functions-reference/mixed_operations.html |  172 ++-
 .../multivariate_discrete_distributions.html  |  172 ++-
 .../positive_continuous_distributions.html    |  172 ++-
 .../positive_lower-bounded_distributions.html |  172 ++-
 .../real-valued_basic_functions.html          |  172 ++-
 docs/2_34/functions-reference/references.html |  172 ++-
 .../removed_functions.html                    |  172 ++-
 .../simplex_distributions.html                |  172 ++-
 .../sparse_matrix_operations.html             |  172 ++-
 .../unbounded_continuous_distributions.html   |  172 ++-
 .../unbounded_discrete_distributions.html     |  172 ++-
 .../functions-reference/void_functions.html   |  172 ++-
 docs/2_34/index.html                          |  174 ++-
 docs/2_34/reference-manual-2_34.pdf           |  Bin 798928 -> 800192 bytes
 docs/2_34/reference-manual/analysis.html      |  172 ++-
 docs/2_34/reference-manual/blocks.html        |  176 ++-
 docs/2_34/reference-manual/comments.html      |  172 ++-
 docs/2_34/reference-manual/deprecations.html  |  174 ++-
 docs/2_34/reference-manual/diagnostics.html   |  174 ++-
 docs/2_34/reference-manual/encoding.html      |  172 ++-
 docs/2_34/reference-manual/execution.html     |  174 ++-
 docs/2_34/reference-manual/expressions.html   |  179 ++-
 docs/2_34/reference-manual/includes.html      |  186 ++-
 docs/2_34/reference-manual/index.html         |  174 ++-
 docs/2_34/reference-manual/laplace.html       |  172 ++-
 docs/2_34/reference-manual/licenses.html      |  181 ++-
 docs/2_34/reference-manual/mcmc.html          |  172 ++-
 docs/2_34/reference-manual/optimization.html  |  172 ++-
 docs/2_34/reference-manual/pathfinder.html    |  172 ++-
 docs/2_34/reference-manual/references.html    |  172 ++-
 docs/2_34/reference-manual/removals.html      |  172 ++-
 .../reference-manual/reproducibility.html     |  172 ++-
 docs/2_34/reference-manual/statements.html    |  349 ++++--
 docs/2_34/reference-manual/syntax.html        |  174 ++-
 docs/2_34/reference-manual/transforms.html    |  172 ++-
 docs/2_34/reference-manual/types.html         |  180 ++-
 .../2_34/reference-manual/user-functions.html |  173 ++-
 docs/2_34/reference-manual/variational.html   |  172 ++-
 docs/2_34/reference-manual/whitespace.html    |  172 ++-
 docs/2_34/search.json                         |  829 ++++++-------
 .../bootstrap/bootstrap-dark.min.css          |   12 +
 .../site_libs/bootstrap/bootstrap.min.css     |    6 +-
 .../quarto-syntax-highlighting-dark.css       |  185 +++
 docs/2_34/stan-users-guide-2_34.pdf           |  Bin 3083643 -> 3083836 bytes
 .../stan-users-guide/algebraic-equations.html |  172 ++-
 docs/2_34/stan-users-guide/bootstrap.html     |  172 ++-
 docs/2_34/stan-users-guide/clustering.html    |  172 ++-
 .../stan-users-guide/complex-numbers.html     |  172 ++-
 .../stan-users-guide/cross-validation.html    |  172 ++-
 .../stan-users-guide/custom-probability.html  |  172 ++-
 docs/2_34/stan-users-guide/dae.html           |  172 ++-
 .../stan-users-guide/decision-analysis.html   |  172 ++-
 .../stan-users-guide/efficiency-tuning.html   |  172 ++-
 .../stan-users-guide/finite-mixtures.html     |  172 ++-
 .../2_34/stan-users-guide/floating-point.html |  172 ++-
 .../2_34/stan-users-guide/for-bugs-users.html |  172 ++-
 .../stan-users-guide/gaussian-processes.html  |  172 ++-
 .../hyperspherical-models.html                |  172 ++-
 docs/2_34/stan-users-guide/index.html         |  174 ++-
 .../stan-users-guide/latent-discrete.html     |  174 ++-
 .../stan-users-guide/matrices-arrays.html     |  172 ++-
 .../stan-users-guide/measurement-error.html   |  172 ++-
 docs/2_34/stan-users-guide/missing-data.html  |  172 ++-
 .../2_34/stan-users-guide/multi-indexing.html |  172 ++-
 docs/2_34/stan-users-guide/odes.html          |  172 ++-
 .../one-dimensional-integrals.html            |  172 ++-
 .../stan-users-guide/parallelization.html     |  172 ++-
 .../posterior-prediction.html                 |  172 ++-
 .../posterior-predictive-checks.html          |  188 ++-
 .../stan-users-guide/poststratification.html  |  172 ++-
 .../problematic-posteriors.html               |  172 ++-
 .../proportionality-constants.html            |  172 ++-
 docs/2_34/stan-users-guide/references.html    |  172 ++-
 docs/2_34/stan-users-guide/regression.html    |  172 ++-
 .../stan-users-guide/reparameterization.html  |  172 ++-
 .../simulation-based-calibration.html         |  172 ++-
 docs/2_34/stan-users-guide/sparse-ragged.html |  172 ++-
 docs/2_34/stan-users-guide/style-guide.html   |  172 ++-
 docs/2_34/stan-users-guide/survival.html      |  172 ++-
 docs/2_34/stan-users-guide/time-series.html   |  172 ++-
 .../truncation-censoring.html                 |  172 ++-
 .../2_34/stan-users-guide/user-functions.html |  172 ++-
 docs/2_34/stan-users-guide/using-stanc.html   |  176 ++-
 docs/2_34/{ => theming}/quarto_styles.css     |   80 +-
 docs/404.html                                 |  172 ++-
 docs/cmdstan-guide/bib.html                   |  172 ++-
 docs/cmdstan-guide/command_line_options.html  |  224 +++-
 .../compiling_stan_programs.html              |  224 +++-
 docs/cmdstan-guide/diagnose_config.html       |  224 +++-
 docs/cmdstan-guide/diagnose_utility.html      |  216 +++-
 docs/cmdstan-guide/err_handling_apdx.html     |  645 ----------
 docs/cmdstan-guide/example_model_data.html    |  218 +++-
 docs/cmdstan-guide/external_code.html         |  216 +++-
 docs/cmdstan-guide/gc-intro.html              |    8 +-
 .../generate_quantities_config.html           |  352 ++++--
 .../generate_quantities_intro.html            |  988 +--------------
 docs/cmdstan-guide/index.html                 |  218 +++-
 docs/cmdstan-guide/installation.html          |  224 +++-
 docs/cmdstan-guide/json_apdx.html             |  216 +++-
 docs/cmdstan-guide/laplace_sample_config.html |  224 +++-
 docs/cmdstan-guide/log_prob_config.html       |  218 +++-
 docs/cmdstan-guide/mcmc-intro.html            |    8 +-
 docs/cmdstan-guide/mcmc_config.html           |  431 +++++--
 docs/cmdstan-guide/mcmc_sampling_intro.html   | 1103 +----------------
 docs/cmdstan-guide/optimization_intro.html    |  970 +--------------
 docs/cmdstan-guide/optimize_config.html       |  293 +++--
 docs/cmdstan-guide/parallelization.html       |  232 +++-
 docs/cmdstan-guide/pathfinder-intro.html      |    8 +-
 docs/cmdstan-guide/pathfinder_config.html     |  339 +++--
 docs/cmdstan-guide/pathfinder_intro.html      |  974 +--------------
 docs/cmdstan-guide/print.html                 |  216 +++-
 docs/cmdstan-guide/rdump_apdx.html            |  216 +++-
 docs/cmdstan-guide/stan_csv_apdx.html         |  218 +++-
 docs/cmdstan-guide/stanc.html                 |  222 +++-
 docs/cmdstan-guide/stansummary.html           |  216 +++-
 .../variational-inference-using-advi.html     |    8 +-
 docs/cmdstan-guide/variational_config.html    |  329 +++--
 docs/cmdstan-guide/variational_intro.html     | 1036 +---------------
 .../functions-reference/array_operations.html |  172 ++-
 .../binary_distributions.html                 |  172 ++-
 .../bounded_continuous_distributions.html     |  172 ++-
 .../bounded_discrete_distributions.html       |  172 ++-
 .../circular_distributions.html               |  172 ++-
 .../complex-valued_basic_functions.html       |  172 ++-
 .../complex_matrix_operations.html            |  172 ++-
 .../compound_arithmetic_and_assignment.html   |  172 ++-
 .../continuous_distributions_on_0_1.html      |  172 ++-
 ...conventions_for_probability_functions.html |  172 ++-
 .../correlation_matrix_distributions.html     |  172 ++-
 .../covariance_matrix_distributions.html      |  172 ++-
 .../deprecated_functions.html                 |  174 ++-
 .../distributions_over_unbounded_vectors.html |  172 ++-
 docs/functions-reference/functions_index.html |  172 ++-
 .../hidden_markov_models.html                 |  172 ++-
 .../higher-order_functions.html               |  172 ++-
 docs/functions-reference/index.html           |  174 ++-
 .../integer-valued_basic_functions.html       |  174 ++-
 .../mathematical_functions.html               |  172 ++-
 .../matrix_operations.html                    |  172 ++-
 .../functions-reference/mixed_operations.html |  172 ++-
 .../multivariate_discrete_distributions.html  |  172 ++-
 .../positive_continuous_distributions.html    |  172 ++-
 .../positive_lower-bounded_distributions.html |  172 ++-
 .../real-valued_basic_functions.html          |  172 ++-
 docs/functions-reference/references.html      |  172 ++-
 .../removed_functions.html                    |  172 ++-
 .../simplex_distributions.html                |  172 ++-
 .../sparse_matrix_operations.html             |  172 ++-
 .../unbounded_continuous_distributions.html   |  172 ++-
 .../unbounded_discrete_distributions.html     |  172 ++-
 docs/functions-reference/void_functions.html  |  172 ++-
 docs/index.html                               |  174 ++-
 docs/reference-manual/analysis.html           |  172 ++-
 docs/reference-manual/blocks.html             |  176 ++-
 docs/reference-manual/comments.html           |  172 ++-
 docs/reference-manual/deprecations.html       |  174 ++-
 docs/reference-manual/diagnostics.html        |  174 ++-
 docs/reference-manual/encoding.html           |  172 ++-
 docs/reference-manual/execution.html          |  174 ++-
 docs/reference-manual/expressions.html        |  179 ++-
 docs/reference-manual/includes.html           |  186 ++-
 docs/reference-manual/index.html              |  174 ++-
 docs/reference-manual/laplace.html            |  172 ++-
 docs/reference-manual/licenses.html           |  181 ++-
 docs/reference-manual/mcmc.html               |  172 ++-
 docs/reference-manual/optimization.html       |  172 ++-
 docs/reference-manual/pathfinder.html         |  172 ++-
 docs/reference-manual/references.html         |  172 ++-
 docs/reference-manual/removals.html           |  172 ++-
 docs/reference-manual/reproducibility.html    |  172 ++-
 docs/reference-manual/statements.html         |  349 ++++--
 docs/reference-manual/syntax.html             |  174 ++-
 docs/reference-manual/transforms.html         |  172 ++-
 docs/reference-manual/types.html              |  180 ++-
 docs/reference-manual/user-functions.html     |  173 ++-
 docs/reference-manual/variational.html        |  172 ++-
 docs/reference-manual/whitespace.html         |  172 ++-
 docs/search.json                              |  829 ++++++-------
 .../bootstrap/bootstrap-dark.min.css          |   12 +
 docs/site_libs/bootstrap/bootstrap.min.css    |    6 +-
 .../quarto-syntax-highlighting-dark.css       |  185 +++
 .../stan-users-guide/algebraic-equations.html |  172 ++-
 docs/stan-users-guide/bootstrap.html          |  172 ++-
 docs/stan-users-guide/clustering.html         |  172 ++-
 docs/stan-users-guide/complex-numbers.html    |  172 ++-
 docs/stan-users-guide/cross-validation.html   |  172 ++-
 docs/stan-users-guide/custom-probability.html |  172 ++-
 docs/stan-users-guide/dae.html                |  172 ++-
 docs/stan-users-guide/decision-analysis.html  |  172 ++-
 docs/stan-users-guide/efficiency-tuning.html  |  172 ++-
 docs/stan-users-guide/finite-mixtures.html    |  172 ++-
 docs/stan-users-guide/floating-point.html     |  172 ++-
 docs/stan-users-guide/for-bugs-users.html     |  172 ++-
 docs/stan-users-guide/gaussian-processes.html |  172 ++-
 .../hyperspherical-models.html                |  172 ++-
 docs/stan-users-guide/index.html              |  174 ++-
 docs/stan-users-guide/latent-discrete.html    |  174 ++-
 docs/stan-users-guide/matrices-arrays.html    |  172 ++-
 docs/stan-users-guide/measurement-error.html  |  172 ++-
 docs/stan-users-guide/missing-data.html       |  172 ++-
 docs/stan-users-guide/multi-indexing.html     |  172 ++-
 docs/stan-users-guide/odes.html               |  172 ++-
 .../one-dimensional-integrals.html            |  172 ++-
 docs/stan-users-guide/parallelization.html    |  172 ++-
 .../posterior-prediction.html                 |  172 ++-
 .../posterior-predictive-checks.html          |  188 ++-
 docs/stan-users-guide/poststratification.html |  172 ++-
 .../problematic-posteriors.html               |  172 ++-
 .../proportionality-constants.html            |  172 ++-
 docs/stan-users-guide/references.html         |  172 ++-
 docs/stan-users-guide/regression.html         |  172 ++-
 docs/stan-users-guide/reparameterization.html |  172 ++-
 .../simulation-based-calibration.html         |  172 ++-
 docs/stan-users-guide/sparse-ragged.html      |  172 ++-
 docs/stan-users-guide/style-guide.html        |  172 ++-
 docs/stan-users-guide/survival.html           |  172 ++-
 docs/stan-users-guide/time-series.html        |  172 ++-
 .../truncation-censoring.html                 |  172 ++-
 docs/stan-users-guide/user-functions.html     |  172 ++-
 docs/stan-users-guide/using-stanc.html        |  176 ++-
 docs/theming/quarto_styles.css                |  406 ++++++
 274 files changed, 40099 insertions(+), 19766 deletions(-)
 delete mode 100644 docs/2_34/cmdstan-guide/err_handling_apdx.html
 delete mode 100644 docs/2_34/cmdstan-guide/generate_quantities_intro.html
 delete mode 100644 docs/2_34/cmdstan-guide/mcmc_sampling_intro.html
 delete mode 100644 docs/2_34/cmdstan-guide/optimization_intro.html
 delete mode 100644 docs/2_34/cmdstan-guide/pathfinder_intro.html
 delete mode 100644 docs/2_34/cmdstan-guide/variational_intro.html
 create mode 100644 docs/2_34/site_libs/bootstrap/bootstrap-dark.min.css
 create mode 100644 docs/2_34/site_libs/quarto-html/quarto-syntax-highlighting-dark.css
 rename docs/2_34/{ => theming}/quarto_styles.css (79%)
 delete mode 100644 docs/cmdstan-guide/err_handling_apdx.html
 create mode 100644 docs/site_libs/bootstrap/bootstrap-dark.min.css
 create mode 100644 docs/site_libs/quarto-html/quarto-syntax-highlighting-dark.css
 create mode 100644 docs/theming/quarto_styles.css

diff --git a/docs/2_34/404.html b/docs/2_34/404.html
index b4c201dfd..1401c7548 100644
--- a/docs/2_34/404.html
+++ b/docs/2_34/404.html
@@ -70,10 +70,12 @@
 <script src="/docs/site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="/docs/site_libs/quarto-html/anchor.min.js"></script>
 <link href="/docs/site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="/docs/site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="/docs/site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="/docs/site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="/docs/site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="/docs/site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -102,7 +104,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="/docs/quarto_styles.css">
+<link rel="stylesheet" href="/docs/theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -217,6 +219,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -266,25 +269,6 @@ <h1 class="title">Page Not Found</h1>
 
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -305,6 +289,150 @@ <h1 class="title">Page Not Found</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/cmdstan-guide-2_34.pdf b/docs/2_34/cmdstan-guide-2_34.pdf
index 0bdc925ca805ce664bc8f0f89d52a469b1e17a17..1005961f4a6394025e4821919d1af54385062c8d 100644
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-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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 </li>
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@@ -311,7 +290,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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 </li>
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@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -563,25 +536,6 @@ <h2 class="anchored" data-anchor-id="error-messages-and-return-codes">Error mess
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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-
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@@ -602,6 +556,150 @@ <h2 class="anchored" data-anchor-id="error-messages-and-return-codes">Error mess
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  // Switch to dark mode if need be
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+    toggleColorMode(true);
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+  }
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@@ -971,8 +1069,8 @@ <h2 class="anchored" data-anchor-id="error-messages-and-return-codes">Error mess
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/generate_quantities_intro.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span>
+      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Parallelization</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/2_34/cmdstan-guide/compiling_stan_programs.html b/docs/2_34/cmdstan-guide/compiling_stan_programs.html
index 7cf03c839..495e4b249 100644
--- a/docs/2_34/cmdstan-guide/compiling_stan_programs.html
+++ b/docs/2_34/cmdstan-guide/compiling_stan_programs.html
@@ -64,7 +64,7 @@
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 <meta name="quarto:offset" content="../">
-<link href="../cmdstan-guide/mcmc_sampling_intro.html" rel="next">
+<link href="../cmdstan-guide/parallelization.html" rel="next">
 <link href="../cmdstan-guide/example_model_data.html" rel="prev">
 <link href="../img/logo_tm.png" rel="icon" type="image/png">
 <script src="../site_libs/quarto-html/quarto.js"></script>
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -261,7 +264,7 @@
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Quickstart Guide</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/compiling_stan_programs.html">Compiling a Stan Program</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/compiling_stan_programs.html">Compiling a Stan Program</a></li></ol></nav>
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@@ -286,7 +289,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -312,32 +315,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -345,7 +324,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -366,7 +345,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -378,13 +357,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -404,12 +383,6 @@
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  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -605,25 +578,6 @@ <h3 class="anchored" data-anchor-id="optimizing-by-ignoring-range-checks">Optimi
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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-
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 <script id="quarto-html-after-body" type="application/javascript">
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@@ -644,6 +598,150 @@ <h3 class="anchored" data-anchor-id="optimizing-by-ignoring-range-checks">Optimi
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
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+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
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+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
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+    }
+    // Hack to workaround the fact that safari doesn't
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-      <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="pagination-link" aria-label="MCMC Sampling">
-        <span class="nav-page-text">MCMC Sampling</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
+        <span class="nav-page-text">Parallelization</span> <i class="bi bi-arrow-right-short"></i>
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diff --git a/docs/2_34/cmdstan-guide/diagnose_config.html b/docs/2_34/cmdstan-guide/diagnose_config.html
index 110eb6e35..9cd0c2f9d 100644
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/diagnose_config.html">Diagnosing HMC by Comparison of Gradients</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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+ <span class="menu-text">Running CmdStan</span></a>
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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- <span class="menu-text">Parallelization</span></a>
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@@ -516,25 +489,6 @@ <h1>Diagnosing HMC by Comparison of Gradients</h1>
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@@ -929,8 +1027,8 @@ <h1>Diagnosing HMC by Comparison of Gradients</h1>
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-      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
-        <span class="nav-page-text">Parallelization</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../cmdstan-guide/stanc.html" class="pagination-link" aria-label="`stanc`: Translating Stan to C++">
+        <span class="nav-page-text"><code>stanc</code>: Translating Stan to C++</span> <i class="bi bi-arrow-right-short"></i>
       </a>
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diff --git a/docs/2_34/cmdstan-guide/diagnose_utility.html b/docs/2_34/cmdstan-guide/diagnose_utility.html
index 861ab386b..4048bcf1f 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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@@ -133,7 +135,7 @@
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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@@ -312,32 +315,8 @@
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- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
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diff --git a/docs/2_34/cmdstan-guide/example_model_data.html b/docs/2_34/cmdstan-guide/example_model_data.html
index 8115bd0d2..7bd8f42f8 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/example_model_data.html">Example Model and Data</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
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+ <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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           <li class="sidebar-item">
@@ -375,12 +354,6 @@
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- <span class="menu-text">Parallelization</span></a>
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 </li>
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   </li>
@@ -505,25 +478,6 @@ <h1>Example Model and Data</h1>
 <li id="fn1"><p>The model is available with the CmdStan distribution at the path <code>&lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan</code>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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@@ -544,6 +498,150 @@ <h1>Example Model and Data</h1>
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diff --git a/docs/2_34/cmdstan-guide/external_code.html b/docs/2_34/cmdstan-guide/external_code.html
index 6bdcca97b..1ef9aac18 100644
--- a/docs/2_34/cmdstan-guide/external_code.html
+++ b/docs/2_34/cmdstan-guide/external_code.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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@@ -282,32 +285,8 @@
 </li>
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- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
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-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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 </li>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -336,7 +315,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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 </li>
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@@ -348,13 +327,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -374,12 +353,6 @@
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  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
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- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
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@@ -614,25 +587,6 @@ <h2 class="anchored" data-anchor-id="special-functions-rngs-distributions-editin
 <li id="fn1"><p>Details of programming in the Stan Math style are omitted from this section, it is presented only as an example<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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diff --git a/docs/2_34/cmdstan-guide/generate_quantities_config.html b/docs/2_34/cmdstan-guide/generate_quantities_config.html
index 3567a4f6b..f7263340d 100644
--- a/docs/2_34/cmdstan-guide/generate_quantities_config.html
+++ b/docs/2_34/cmdstan-guide/generate_quantities_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
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+<title>Generating Quantities of Interest from a Fitted Model</title>
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+code.sourceCode > span { color: inherit; text-decoration: inherit; }
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+pre.sourceCode { margin: 0; }
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+div.sourceCode { overflow: auto; }
+}
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+}
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+@media screen {
+pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
+}
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/generate_quantities_config.html">Generating Quantities of Interest from a Fitted Model</a></li></ol></nav>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -249,32 +286,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
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- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
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-  <div class="sidebar-item-container"> 
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-</li>
-          <li class="sidebar-item">
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- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -341,12 +354,6 @@
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@@ -430,7 +437,11 @@
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-  <li><a href="#standalone-generate-quantities" id="toc-standalone-generate-quantities" class="nav-link active" data-scroll-target="#standalone-generate-quantities">Standalone Generate Quantities</a></li>
+  <li><a href="#gc-intro" id="toc-gc-intro" class="nav-link active" data-scroll-target="#gc-intro">Generating Quantities of Interest from a Fitted Model</a>
+  <ul class="collapse">
+  <li><a href="#example" id="toc-example" class="nav-link" data-scroll-target="#example">Example</a></li>
+  <li><a href="#errors" id="toc-errors" class="nav-link" data-scroll-target="#errors">Errors</a></li>
+  </ul></li>
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@@ -440,46 +451,101 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="standalone-generate-quantities" class="level1">
-<h1>Standalone Generate Quantities</h1>
-<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. For an overview of the uses of this feature, see the <a href="../cmdstan-guide/generate_quantities_intro.html">QuickStart Guide section</a> and the Stan User’s Guide section on <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
-<p>This method requires sub-argument <code>fitted_params</code> which takes as its value an existing Stan CSV file that contains a sample from an equivalent model, i.e., a model with the same parameters, transformed parameters, and model blocks, conditioned on the same data.</p>
+<section id="gc-intro" class="level1">
+<h1>Generating Quantities of Interest from a Fitted Model</h1>
+<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the <code>generate_quantities</code> method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block.</p>
+<p>This method requires sub-argument <code>fitted_params</code> which takes as its value an existing <a href="../cmdstan-guide/stan_csv_apdx.html">Stan CSV</a> file that contains a parameter values from an equivalent model, i.e., a model with the same parameters block, conditioned on the same data.</p>
+<p>The <a href="https://mc-stan.org/docs/reference-manual/blocks.html#generated-quantities">generated quantities block</a> computes <em>quantities of interest</em> (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:</p>
+<ul>
+<li>generate simulated data for model testing by forward sampling</li>
+<li>generate predictions for new data</li>
+<li>calculate posterior event probabilities, including multiple comparisons, sign tests, etc.</li>
+<li>calculate posterior expectations</li>
+<li>transform parameters for reporting</li>
+<li>apply full Bayesian decision theory</li>
+<li>calculate log likelihoods, deviances, etc. for model comparison</li>
+</ul>
+<p>For an overview of the uses of this feature, see the Stan User’s Guide section on <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
+<section id="example" class="level2">
+<h2 class="anchored" data-anchor-id="example">Example</h2>
+<p>To illustrate how this works we use the <code>generate_quantities</code> method to do posterior predictive checks using the estimate of <code>theta</code> given the example bernoulli model and data, following the <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#posterior-predictive-simulation-in-stan">posterior predictive simulation</a> procedure in the Stan User’s Guide.</p>
+<p>We write a program <code>bernoulli_ppc.stan</code> which contains the following generated quantities block, with comments to explain the procedure:</p>
+<div class="sourceCode" id="cb1"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
+<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
+<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
+<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
+<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
+<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
+<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>The rest of the program is the same as in <code>bernoulli.stan</code>.</p>
+<p>The <code>generate_method</code> requires the sub-argument <code>fitted_params</code> which takes as its value the name of a Stan CSV file. The per-draw parameter values from the <code>fitted_params</code> file will be used to run the generated quantities block.</p>
 <p>If we run the <code>bernoulli.stan</code> program for a single chain to generate a sample in file <code>bernoulli_fit.csv</code>:</p>
 <pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv</code></pre>
 <p>Then we can run the <code>bernoulli_ppc.stan</code> to carry out the posterior predictive checks:</p>
 <pre><code>&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \
                   data file=bernoulli.data.json \
                   output file=bernoulli_ppc.csv</code></pre>
-<p>The <code>fitted_params</code> file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:</p>
+<p>The output file <code>bernoulli_ppc.csv</code> contains only the values for the variables declared in the <code>generated quantities</code> block, i.e., <code>theta_rep</code> and the elements of <code>y_sim</code>:</p>
+<pre><code># model = bernoulli_ppc_model
+# method = generate_quantities
+#   generate_quantities
+#     fitted_params = bernoulli_fit.csv
+# id = 1 (Default)
+# data
+#   file = bernoulli.data.json
+# init = 2 (Default)
+# random
+#   seed = 2983956445 (Default)
+# output
+#   file = output.csv (Default)
+y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep
+1,1,1,0,0,0,1,1,0,1,0.6
+1,1,0,1,0,0,1,0,1,0,0.5
+1,0,1,1,1,1,1,1,0,1,0.8
+0,1,0,1,0,1,0,1,0,0,0.4
+1,0,0,0,0,0,0,0,0,0,0.1
+0,0,0,0,0,1,1,1,0,0,0.3
+0,0,1,0,1,0,0,0,0,0,0.2
+1,0,1,0,1,1,0,1,1,0,0.6
+...</code></pre>
+<p>Given the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:</p>
+<div class="sourceCode" id="cb5"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
+<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
+<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
+<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
+<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_cp = theta;</span>
+<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
+<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
+<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>Now the output is slightly more interpretable: <code>theta_cp</code> is the same as the <code>theta</code> used to generate the values <code>y_sim[1]</code> through <code>y_sim[1]</code>. Comparing columns <code>theta_cp</code> and <code>theta_rep</code> allows us to see how the uncertainty in our estimate of <code>theta</code> is carried forward into our predictions:</p>
+<pre><code>y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep
+0,1,1,0,1,0,0,1,1,0,0.545679,0.5
+1,1,1,1,1,1,0,1,1,0,0.527164,0.8
+1,1,1,1,0,1,1,1,1,0,0.529116,0.8
+1,0,1,1,1,1,0,0,1,0,0.478844,0.6
+0,1,0,0,0,0,1,0,1,0,0.238793,0.3
+0,0,0,0,0,1,1,0,0,0,0.258294,0.2
+1,1,1,0,0,0,0,0,0,0,0.258465,0.3</code></pre>
+</section>
+<section id="errors" class="level2">
+<h2 class="anchored" data-anchor-id="errors">Errors</h2>
+<p>The <code>fitted_params</code> file must be a <a href="../cmdstan-guide/stan_csv_apdx.html">Stan CSV</a> file; attempts to use a regular CSV file will result an error message of the form:</p>
 <pre><code>Error reading fitted param names from sample csv file &lt;filename.csv&gt;</code></pre>
 <p>The <code>fitted_params</code> file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:</p>
 <pre><code>Error reading fitted param names from sample csv file &lt;filename.csv&gt;</code></pre>
-<p>The parameter values of the <code>fitted_params</code> are on the constrained scale and must obey all constraints. For example, if we modify the contencts of the first reported draw in <code>bernoulli_fit.csv</code> so that the value of <code>theta</code> is outside the declared bounds <code>real&lt;lower=0, upper=1&gt;</code>, the program will return the following error message:</p>
-<pre><code>Exception: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] (in 'bernoulli_ppc.stan', line 5, column 2 to column 30)</code></pre>
+<p>The parameter values of the <code>fitted_params</code> are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in <code>bernoulli_fit.csv</code> so that the value of <code>theta</code> is outside the declared bounds <code>real&lt;lower=0, upper=1&gt;</code>, the program will return the following error message:</p>
+<pre><code>Exception: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \
+(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)</code></pre>
 
 
+</section>
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@@ -869,8 +1079,8 @@ <h1>Standalone Generate Quantities</h1>
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-<section id="gc-intro" class="level1">
-<h1>Generating Quantities of Interest from a Fitted Model</h1>
-<p>The <a href="https://mc-stan.org/docs/reference-manual/blocks.html#generated-quantities">generated quantities block</a> computes <em>quantities of interest</em> (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:</p>
-<ul>
-<li>generate simulated data for model testing by forward sampling</li>
-<li>generate predictions for new data</li>
-<li>calculate posterior event probabilities, including multiple comparisons, sign tests, etc.</li>
-<li>calculating posterior expectations</li>
-<li>transform parameters for reporting</li>
-<li>apply full Bayesian decision theory</li>
-<li>calculate log likelihoods, deviances, etc. for model comparison</li>
-</ul>
-<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the <code>generate_quantities</code> method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block. See the Stan User’s Guide section <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
-<p>To illustrate how this works we use the <code>generate_quantities</code> method to do posterior predictive checks using the estimate of <code>theta</code> given the example bernoulli model and data, following the <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#posterior-predictive-simulation-in-stan">posterior predictive simulation</a> procedure in the Stan User’s Guide.</p>
-<p>We write a program <code>bernoulli_ppc.stan</code> which contains the following generated quantities block, with comments to explain the procedure:</p>
-<div class="sourceCode" id="cb1"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
-<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep;</span>
-<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
-<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
-<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
-<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
-<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>  theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
-<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>The rest of the program is the same as in <code>bernoulli.stan</code>.</p>
-<p>The <code>generate_method</code> requires the sub-argument <code>fitted_params</code> which takes as its value the name of a Stan CSV file. The per-draw parameter estimates from the <code>fitted_params</code> file will be used to run the generated quantities block.</p>
-<p>If we run the <code>bernoulli.stan</code> program for a single chain to generate a sample in file <code>bernoulli_fit.csv</code>:</p>
-<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv</code></pre>
-<p>Then we can run the <code>bernoulli_ppc.stan</code> to carry out the posterior predictive checks:</p>
-<pre><code>&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \
-                  data file=bernoulli.data.json \
-                  output file=bernoulli_ppc.csv</code></pre>
-<p>The output file <code>bernoulli_ppc.csv</code> consists of just the values for the variables declared in the generated quantities block, i.e., <code>theta_rep</code> and the elements of <code>y_sim</code>:</p>
-<pre><code># model = bernoulli_ppc_model
-# method = generate_quantities
-#   generate_quantities
-#     fitted_params = bernoulli_fit.csv
-# id = 0 (Default)
-# data
-#   file = bernoulli.data.json
-# init = 2 (Default)
-# random
-#   seed = 2135140492 (Default)
-# output
-#   file = bernoulli_ppc.csv
-#   diagnostic_file =  (Default)
-#   refresh = 100 (Default)
-theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10
-0.2,0,0,1,0,0,0,0,0,1,0
-0.3,1,0,0,1,0,1,0,0,0,0
-0.8,1,0,1,1,1,1,1,1,1,0
-0.1,0,0,0,0,0,1,0,0,0,0
-0.3,0,0,0,0,0,0,1,1,1,0</code></pre>
-<p><em>Note</em>: the only relevant analysis of the resulting CSV output is computing per-column statistics; this can easily be done in Python, R, Excel or similar, or you can use the CmdStanPy and CmdStanR interfaces which provide a better user experience for this workflow.</p>
-<p>Given the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:</p>
-<div class="sourceCode" id="cb5"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
-<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_cp = theta;</span>
-<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep;</span>
-<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
-<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
-<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
-<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
-<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a>  theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
-<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>Now the output is slightly more interpretable: <code>theta_cp</code> is the same as the <code>theta</code> used to generate the values <code>y_sim[1]</code> through <code>y_sim[1]</code>. Comparing columns <code>theta_cp</code> and <code>theta_rep</code> allows us to see how the uncertainty in our estimate of <code>theta</code> is carried forward into our predictions:</p>
-<pre><code>theta_cp,theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10
-0.102391,0,0,0,0,0,0,0,0,0,0,0
-0.519567,0.2,0,1,0,0,1,0,0,0,0,0
-0.544634,0.6,1,0,0,0,0,1,1,1,1,1
-0.167651,0,0,0,0,0,0,0,0,0,0,0
-0.167651,0.1,1,0,0,0,0,0,0,0,0,0</code></pre>
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-  </div>
-  <div class="nav-page nav-page-next">
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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+<p>This document is a user’s guide for CmdStan, the command-line interface to the <a href="https://mc-stan.org/">Stan</a> statistical modeling language. CmdStan provides the programs and tools to compile Stan programs into C++ executables that can be run directly from the command line, together with a few utilities to check and summarize the resulting outputs.</p>
 <p>In CmdStan, statistical models written in the Stan probabilistic programming language are translated into a C++ program which is then compiled together with the CmdStan routines that provide the logic needed to manage all user inputs and program outputs and the Stan inference algorithms and math library. The resulting command line executable program can be used to</p>
 <ul>
 <li><p>do inference on data, producing an exact or approximate estimate of the posterior;</p></li>
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diff --git a/docs/2_34/cmdstan-guide/installation.html b/docs/2_34/cmdstan-guide/installation.html
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">CmdStan Installation</a></li></ol></nav>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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@@ -278,32 +281,8 @@
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           <li class="sidebar-item">
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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+ <span class="menu-text">Running CmdStan</span></a>
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+ <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -496,7 +469,7 @@ <h1>CmdStan Installation</h1>
 </ul>
 <section id="conda-install" class="level2">
 <h2 class="anchored" data-anchor-id="conda-install">Installation via <code>conda</code></h2>
-<p>With conda, you can install CmdStan from the <a href="https://conda-forge.org/">conda-forge channel</a>. This will install a pre-built version of CmdStan along with the required dependencies (i.e.&nbsp;a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command <code>cmdstan_model</code> to activate the CmdStan makefile from anywhere.</p>
+<p>With conda, you can install CmdStan from the <a href="https://conda-forge.org/">conda-forge channel</a>. This will install a pre-built version of CmdStan along with the required dependencies (i.e.&nbsp;a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command <code>cmdstan_model</code> to activate the CmdStan makefile from anywhere.</p>
 <p><em>Note</em>: This requires that conda has been installed already on your machine. You can either install <a href="https://docs.conda.io/en/latest/miniconda.html">miniconda</a>, a free, minimal installer for conda or you can get the full <a href="https://docs.continuum.io/anaconda/">Anaconda</a> system which provides graphical installer wizards for <a href="https://www.anaconda.com/products/distribution#macos">MacOS</a> and <a href="https://www.anaconda.com/products/distribution#windows">Windows</a> users.</p>
 <p>We recommend installing CmdStan in a new conda environment:</p>
 <pre><code> conda create -n stan -c conda-forge cmdstan</code></pre>
@@ -570,7 +543,7 @@ <h2 class="anchored" data-anchor-id="checking-the-stan-compiler">Checking the St
 &gt; make examples/bernoulli/bernoulli
 
 # fit to provided data (results of 10 trials, 2 out of 10 successes)
-&gt; ./examples/bernoulli/bernoulli sample\ 
+&gt; ./examples/bernoulli/bernoulli sample\
   data file=examples/bernoulli/bernoulli.data.json
 
 # default output written to file `output.csv`,
@@ -741,7 +714,7 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
     4. Build the diagnose utility bin/diagnose
     5. Build all libraries and object files compile and link an executable Stan program
 
-    Note: to build using multiple cores, use the -j option to make, e.g., 
+    Note: to build using multiple cores, use the -j option to make, e.g.,
     for 4 cores:
     &gt; make build -j4
 
@@ -798,25 +771,6 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
 <li id="fn1"><p>To open a Windows command shell, first open the <strong>Start Menu</strong>, (usually in the lower left of the screen), select option <strong>All Programs</strong>, then option <strong>Accessories</strong>, then program <strong>Command Prompt</strong>. Alternatively, enter <code>[Windows+r]</code> (both keys together on the keyboard), and enter <code>cmd</code> into the text field that pops up in the Run window, then press <code>[Return]</code> on the keyboard to run.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -837,6 +791,150 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+        el.classList.remove('notransition');
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
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+        }
+      });
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
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+    }
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+  const isFileUrl = () => { 
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+      return storageValue != null ? storageValue : localAlternateSentinel;
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+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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diff --git a/docs/2_34/cmdstan-guide/json_apdx.html b/docs/2_34/cmdstan-guide/json_apdx.html
index 12a1949c2..f6b4763d3 100644
--- a/docs/2_34/cmdstan-guide/json_apdx.html
+++ b/docs/2_34/cmdstan-guide/json_apdx.html
@@ -72,10 +72,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -286,7 +289,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -312,32 +315,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -345,7 +324,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -366,7 +345,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -378,13 +357,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -404,12 +383,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -687,25 +660,6 @@ <h3 class="anchored" data-anchor-id="empty-arrays-in-json">Empty arrays in JSON<
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -726,6 +680,150 @@ <h3 class="anchored" data-anchor-id="empty-arrays-in-json">Empty arrays in JSON<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+      const toggle = toggles[i];
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+        } else {
+          toggle.classList.remove("alternate");
+        }
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
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+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
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+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
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+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
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+    setStyleSentinel(toAlternate);
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+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/2_34/cmdstan-guide/laplace_sample_config.html b/docs/2_34/cmdstan-guide/laplace_sample_config.html
index 473c935c6..f42ba54f9 100644
--- a/docs/2_34/cmdstan-guide/laplace_sample_config.html
+++ b/docs/2_34/cmdstan-guide/laplace_sample_config.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/laplace_sample_config.html">Laplace sampling</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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-  <div class="sidebar-item-container"> 
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- <span class="menu-text">Optimization</span></a>
-  </div>
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-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
           <li class="sidebar-item">
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
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- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
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@@ -447,7 +420,7 @@ <h2 id="toc-title">On this page</h2>
 
 <section id="laplace-sampling" class="level1">
 <h1>Laplace sampling</h1>
-<p>The <code>laplace</code> method produces a sample from a normal approximation centered at the mode of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See <a href="https://avehtari.github.io/casestudies/Jacobian/jacobian.html">this case study</a> for a visual illustration.)</p>
+<p>The <code>laplace</code> method produces a sample from a normal approximation centered at the <a href="../cmdstan-guide/optimize_config.html">mode</a> of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See <a href="https://avehtari.github.io/casestudies/Jacobian/jacobian.html">this case study</a> for a visual illustration.)</p>
 <p>This is computationally inexpensive compared to exact Bayesian inference with MCMC. The goodness of this estimate depends on both the estimate of the mode and how much the true posterior in the unconstrained space resembles a Gaussian.</p>
 <section id="configuration" class="level2">
 <h2 class="anchored" data-anchor-id="configuration">Configuration</h2>
@@ -518,25 +491,6 @@ <h2 class="anchored" data-anchor-id="example">Example</h2>
 </section>
 
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+  const disableStylesheet = (stylesheets) => {
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
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+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
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+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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@@ -926,8 +1024,8 @@ <h2 class="anchored" data-anchor-id="example">Example</h2>
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Standalone Generate Quantities">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Standalone Generate Quantities</span>
+      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/2_34/cmdstan-guide/log_prob_config.html b/docs/2_34/cmdstan-guide/log_prob_config.html
index f0996f429..ed65b3aba 100644
--- a/docs/2_34/cmdstan-guide/log_prob_config.html
+++ b/docs/2_34/cmdstan-guide/log_prob_config.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -104,7 +106,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -232,7 +235,7 @@
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/log_prob_config.html">Extracting log probabilities and gradients for diagnostics</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/log_prob_config.html">Extracting log probabilities and gradients for diagnostics</a></li></ol></nav>
         <a class="flex-grow-1" role="button" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">      
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@@ -257,7 +260,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -283,32 +286,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -316,7 +295,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -337,7 +316,7 @@
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   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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 </li>
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@@ -349,13 +328,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -375,12 +354,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -542,25 +515,6 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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diff --git a/docs/2_34/cmdstan-guide/mcmc_config.html b/docs/2_34/cmdstan-guide/mcmc_config.html
index 114955e9b..18ceb1d81 100644
--- a/docs/2_34/cmdstan-guide/mcmc_config.html
+++ b/docs/2_34/cmdstan-guide/mcmc_config.html
@@ -7,7 +7,7 @@
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-<title>MCMC Sampling Configuration</title>
+<title>MCMC Sampling</title>
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 span.smallcaps{font-variant: small-caps;}
@@ -58,10 +58,12 @@
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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@@ -119,7 +121,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -234,6 +236,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -247,7 +250,7 @@
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
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-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/mcmc_config.html">MCMC Sampling using Hamiltonian Monte Carlo</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/mcmc_config.html">MCMC Sampling using Hamiltonian Monte Carlo</a></li></ol></nav>
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         </a>
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@@ -272,7 +275,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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@@ -298,32 +301,8 @@
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           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -331,7 +310,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
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+ <span class="menu-text">Running CmdStan</span></a>
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@@ -352,7 +331,7 @@
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   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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@@ -364,13 +343,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
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   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -390,12 +369,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
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-  <div class="sidebar-item-container"> 
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- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -481,6 +454,8 @@ <h2 id="toc-title">On this page</h2>
   <ul>
   <li><a href="#mcmc-config" id="toc-mcmc-config" class="nav-link active" data-scroll-target="#mcmc-config">MCMC Sampling using Hamiltonian Monte Carlo</a>
   <ul class="collapse">
+  <li><a href="#running-the-sampler" id="toc-running-the-sampler" class="nav-link" data-scroll-target="#running-the-sampler">Running the sampler</a></li>
+  <li><a href="#mcmc_output_csv" id="toc-mcmc_output_csv" class="nav-link" data-scroll-target="#mcmc_output_csv">Stan CSV output file</a></li>
   <li><a href="#iterations" id="toc-iterations" class="nav-link" data-scroll-target="#iterations">Iterations</a></li>
   <li><a href="#adaptation" id="toc-adaptation" class="nav-link" data-scroll-target="#adaptation">Adaptation</a>
   <ul class="collapse">
@@ -493,8 +468,12 @@ <h2 id="toc-title">On this page</h2>
   <li><a href="#hmc-samplers" id="toc-hmc-samplers" class="nav-link" data-scroll-target="#hmc-samplers">HMC samplers</a></li>
   </ul></li>
   <li><a href="#sampler-diag-file" id="toc-sampler-diag-file" class="nav-link" data-scroll-target="#sampler-diag-file">Sampler diagnostic file</a></li>
-  <li><a href="#sampler-num-chains" id="toc-sampler-num-chains" class="nav-link" data-scroll-target="#sampler-num-chains">Multiple chains in one executable</a></li>
-  <li><a href="#examples---older-parallelism" id="toc-examples---older-parallelism" class="nav-link" data-scroll-target="#examples---older-parallelism">Examples - older parallelism</a>
+  <li><a href="#multi-chain-sampling" id="toc-multi-chain-sampling" class="nav-link" data-scroll-target="#multi-chain-sampling">Running multiple chains</a>
+  <ul class="collapse">
+  <li><a href="#sampler-num-chains" id="toc-sampler-num-chains" class="nav-link" data-scroll-target="#sampler-num-chains">Using the num_chains argument to run multiple chains</a></li>
+  </ul></li>
+  <li><a href="#summarizing-sampler-outputs-with-stansummary" id="toc-summarizing-sampler-outputs-with-stansummary" class="nav-link" data-scroll-target="#summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></a></li>
+  <li><a href="#old-multichain" id="toc-old-multichain" class="nav-link" data-scroll-target="#old-multichain">Examples - older parallelism</a>
   <ul class="collapse">
   <li><a href="#running-multiple-chains-with-a-specified-rng-seed" id="toc-running-multiple-chains-with-a-specified-rng-seed" class="nav-link" data-scroll-target="#running-multiple-chains-with-a-specified-rng-seed">Running multiple chains with a specified RNG seed</a></li>
   <li><a href="#changing-the-default-warmup-and-sampling-iterations" id="toc-changing-the-default-warmup-and-sampling-iterations" class="nav-link" data-scroll-target="#changing-the-default-warmup-and-sampling-iterations">Changing the default warmup and sampling iterations</a></li>
@@ -520,8 +499,78 @@ <h2 id="toc-title">On this page</h2>
 <section id="mcmc-config" class="level1">
 <h1>MCMC Sampling using Hamiltonian Monte Carlo</h1>
 <p>The <code>sample</code> method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on <a href="https://mc-stan.org/docs/reference-manual/mcmc.html">MCMC Sampling</a>.</p>
-<p>The full set of configuration options available for the <code>sample</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default sample method configuration options:</p>
-<pre><code># method = sample (Default)
+<section id="running-the-sampler" class="level2">
+<h2 class="anchored" data-anchor-id="running-the-sampler">Running the sampler</h2>
+<p>To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument <code>sample</code> or <code>method=sample</code> together with the input data. The executable can be run from any directory.</p>
+<p>The full set of configuration options available for the <code>sample</code> method is available by using the <code>sample help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.</p>
+<p>Here, we run it in the directory which contains the Stan program and input data, <code>&lt;cmdstan-home&gt;/examples/bernoulli</code>:</p>
+<pre><code>&gt; cd examples/bernoulli
+&gt; ls
+  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan</code></pre>
+<p>To execute sampling of the model under Linux or Mac, use:</p>
+<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json</code></pre>
+<p>In Windows, the <code>./</code> prefix is not needed:</p>
+<pre><code>&gt; bernoulli.exe sample data file=bernoulli.data.json</code></pre>
+<p>The output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:</p>
+<pre><code>method = sample (Default)
+  sample
+    num_samples = 1000 (Default)
+    num_warmup = 1000 (Default)
+    save_warmup = 0 (Default)
+    thin = 1 (Default)
+    adapt
+      engaged = 1 (Default)
+      gamma = 0.050000000000000003 (Default)
+      delta = 0.80000000000000004 (Default)
+      kappa = 0.75 (Default)
+      t0 = 10 (Default)
+      init_buffer = 75 (Default)
+      term_buffer = 50 (Default)
+      window = 25 (Default)
+      save_metric = 0 (Default)
+    algorithm = hmc (Default)
+      hmc
+        engine = nuts (Default)
+          nuts
+            max_depth = 10 (Default)
+        metric = diag_e (Default)
+        metric_file =  (Default)
+        stepsize = 1 (Default)
+        stepsize_jitter = 0 (Default)
+    num_chains = 1 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 3252652196 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>After the configuration has been displayed, a short timing message is given.</p>
+<pre><code>Gradient evaluation took 1.2e-05 seconds
+1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
+Adjust your expectations accordingly!</code></pre>
+<p>Next, the sampler reports the iteration number, reporting the percentage complete.</p>
+<pre><code>Iteration:    1 / 2000 [  0%]  (Warmup)
+...
+Iteration: 2000 / 2000 [100%]  (Sampling)</code></pre>
+<p>Finally, the sampler reports timing information:</p>
+<pre><code> Elapsed Time: 0.007 seconds (Warm-up)
+               0.017 seconds (Sampling)
+               0.024 seconds (Total)</code></pre>
+</section>
+<section id="mcmc_output_csv" class="level2">
+<h2 class="anchored" data-anchor-id="mcmc_output_csv">Stan CSV output file</h2>
+<p>Each execution of the model results in draws from a single Markov chain being written to a file in <a href="../cmdstan-guide/stan_csv_apdx.html">comma-separated value (CSV) format</a>. The default name of the output file is <code>output.csv</code>.</p>
+<p>The first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (<code>#</code>)).</p>
+<p>When the example model <code>bernoulli.stan</code> is run via the command line with all default arguments, the following configuration is displayed:</p>
+<pre><code># stan_version_major = 2
+# stan_version_minor = 23
+# stan_version_patch = 0
+# model = bernoulli_model
+# method = sample (Default)
 #   sample
 #     num_samples = 1000 (Default)
 #     num_warmup = 1000 (Default)
@@ -546,7 +595,44 @@ <h1>MCMC Sampling using Hamiltonian Monte Carlo</h1>
 #         metric_file =  (Default)
 #         stepsize = 1.000000 (Default)
 #         stepsize_jitter = 0.000000 (Default)
-#     num_chains = 1 (Default)</code></pre>
+#     num_chains = 1 (Default)
+# output
+#   file = output.csv (Default)
+#   diagnostic_file =  (Default)
+#   refresh = 100 (Default)</code></pre>
+<p>This is followed by a CSV header indicating the names of the values sampled.</p>
+<pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
+<p>The first output columns report the HMC sampler information:</p>
+<ul>
+<li><code>lp__</code> - the total log probability density (up to an additive constant) at each sample</li>
+<li><code>accept_stat__</code> - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory</li>
+<li><code>stepsize__</code> - integrator step size</li>
+<li><code>treedepth__</code> - depth of tree used by NUTS (NUTS sampler)</li>
+<li><code>n_leapfrog__</code> - number of leapfrog calculations (NUTS sampler)</li>
+<li><code>divergent__</code> - has value <code>1</code> if trajectory diverged, otherwise <code>0</code>. (NUTS sampler)</li>
+<li><code>energy__</code> - value of the Hamiltonian</li>
+<li><code>int_time__</code> - total integration time (static HMC sampler)</li>
+</ul>
+<p>Because the above header is from the NUTS sampler, it has columns <code>treedepth__</code>, <code>n_leapfrog__</code>, and <code>divergent__</code> and doesn’t have column <code>int_time__</code>. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, <code>theta</code>.</p>
+<p>The header line is written to the output file before warmup begins. If option <code>save_warmup</code> is set to <code>1</code>, the warmup draws are output directly after the header. The total number of warmup draws saved is <code>num_warmup</code> divided by <code>thin</code>, rounded up (i.e., <code>ceiling</code>).</p>
+<p>Following the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:</p>
+<pre><code># Adaptation terminated
+# Step size = 0.884484
+# Diagonal elements of inverse mass matrix:
+# 0.535006</code></pre>
+<p>The default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter <code>theta</code>.</p>
+<p>Draws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.</p>
+<pre><code>-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853
+-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295
+-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299
+-6.88712,1,0.884484,1,1,0,7.02101,0.188229
+-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596
+...</code></pre>
+<p>The output ends with timing details:</p>
+<pre><code>#  Elapsed Time: 0.007 seconds (Warm-up)
+#                0.017 seconds (Sampling)
+#                0.024 seconds (Total)</code></pre>
+</section>
 <section id="iterations" class="level2">
 <h2 class="anchored" data-anchor-id="iterations">Iterations</h2>
 <p>At every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters <code>metric</code> and <code>stepsize</code> in order to efficiently sample from <em>typical set</em>, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.</p>
@@ -652,32 +738,86 @@ <h4 class="anchored" data-anchor-id="integration-time">Integration time</h4>
 <h2 class="anchored" data-anchor-id="sampler-diag-file">Sampler diagnostic file</h2>
 <p>The <code>output</code> keyword sub-argument <code>diagnostic_file=&lt;filepath&gt;</code> specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.</p>
 </section>
-<section id="sampler-num-chains" class="level2">
-<h2 class="anchored" data-anchor-id="sampler-num-chains">Multiple chains in one executable</h2>
-<p>As described in the <a href="../cmdstan-guide/mcmc_sampling_intro.html#multi-chain-sampling">quickstart section on parallelism</a>, the preferred way to run multiple chains is to use the <code>num_chains</code> argument.</p>
-<p>This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with <code>STAN_THREADS=true</code>.</p>
+<section id="multi-chain-sampling" class="level2">
+<h2 class="anchored" data-anchor-id="multi-chain-sampling">Running multiple chains</h2>
+<p>A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.</p>
+<p>The preferred way of using multiple chains is to run them all from the same executable using the <code>num_chains</code> argument. There is also the option to use the <a href="#old-multichain">Unix or DOS shell to run multiple executables</a>.</p>
+<section id="sampler-num-chains" class="level3">
+<h3 class="anchored" data-anchor-id="sampler-num-chains">Using the num_chains argument to run multiple chains</h3>
+<p>The <code>num_chains</code> argument can be used for all of Stan’s samplers with the exception of the <code>static</code> HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with <code>STAN_THREADS=true</code>, these will either run in parallel or one after the other.</p>
 <p>The <code>num_chains</code> argument changes the meanings of several other arguments when it is greater than <code>1</code> (the default). Many arguments are now interpreted as a “template” which is used for each chain.</p>
 <p>For example, when <code>num_chains=2</code>, the argument <code>output file=foo.csv</code> <em>no longer produces a file <code>foo.csv</code></em>, but instead produces two files, <code>foo_1.csv</code> and <code>foo_2.csv</code>. If you also supply <code>id=5</code>, the files produced will be <code>foo_5.csv</code> and <code>foo_6.csv</code> – <code>id=5</code> gives the id of the first chain, and the remaining chains are sequential from there.</p>
 <p>This also applies to input files, like those used for initialization. For example, if <code>num_chains=3</code> and <code>init=bar.json</code> will first look for <code>bar_1.json</code>. If it exists, it will use <code>bar_1.json</code> for the first chain, <code>bar_2.json</code> for the second, and so on. If <code>bar_1.json</code> does not exist, it falls back to looking for <code>bar.json</code>, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the <code>id</code> argument, which defaults to <code>1</code>.</p>
+<p>For example, this will run 4 chains:</p>
+<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv</code></pre>
+<p>This will produce samples in <code>output_1.csv</code>, <code>output_2.csv</code>, <code>output_3.csv</code>, <code>output_4.csv</code>. A suffix with the chain id is appended to the provided output filename (<code>output.csv</code> in the above command).</p>
+<p>If the model was not compiled with <code>STAN_THREADS=true</code>, the above command will run 4 chains sequentially.</p>
+<p>If the model was compiled with <code>STAN_THREADS=true</code>, the chains can run in parallel, with the <code>num_threads</code> argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the below command will run 4 chains in parallel, provided there are cores available:</p>
+<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4</code></pre>
+<p>If the model uses within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the <a href="https://www.intel.com/content/www/us/en/docs/onetbb/developer-guide-api-reference/2021-11/how-task-scheduler-works.html">Threading Building Blocks scheduler</a>.</p>
+<pre><code>./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16</code></pre>
 </section>
-<section id="examples---older-parallelism" class="level2">
-<h2 class="anchored" data-anchor-id="examples---older-parallelism">Examples - older parallelism</h2>
+</section>
+<section id="summarizing-sampler-outputs-with-stansummary" class="level2">
+<h2 class="anchored" data-anchor-id="summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></h2>
+<p>The <a href="../cmdstan-guide/stansummary.html"><code>stansummary</code> utility</a> processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file <code>stansummary</code> reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and <span class="math inline">\(\hat{R}\)</span> values.</p>
+<p>To run <code>stansummary</code> on the output files generated by the for loop above, by the above run of the <code>bernoulli</code> model on Mac or Linux:</p>
+<pre><code>&lt;cmdstan-home&gt;/bin/stansummary output_*.csv</code></pre>
+<p>On Windows, use backslashes to call the <code>stansummary.exe</code>.</p>
+<pre><code>&lt;cmdstan-home&gt;\bin\stansummary.exe output_*.csv</code></pre>
+<p>The stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable <code>theta</code>:</p>
+<pre><code>Inference for Stan model: bernoulli_model
+4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.
+
+Warmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total
+Sampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total
+
+                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat
+lp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00
+accept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00
+stepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13
+treedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00
+n_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00
+divergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan
+energy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00
+theta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00
+
+Samples were drawn using hmc with nuts.
+For each parameter, N_Eff is a crude measure of effective sample size,
+and R_hat is the potential scale reduction factor on split chains (at
+convergence, R_hat=1).</code></pre>
+<p>In this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for <code>theta</code> reflect the uncertainty in our estimate, due to the small amount of data, given the prior of <code>beta(1, 1)</code></p>
+</section>
+<section id="old-multichain" class="level2">
+<h2 class="anchored" data-anchor-id="old-multichain">Examples - older parallelism</h2>
 <p><strong>Note</strong>: Many of these examples can be simplified by using the <a href="#sampler-num-chains"><code>num_chains</code> argument</a>.</p>
-<p>The Quickstart Guide <a href="../cmdstan-guide/mcmc_sampling_intro.html">MCMC Sampling chapter</a> section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.</p>
-<p>This creates multiple copies of the model process which will all load the data.</p>
-<p>To run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:</p>
+<p>When the <code>num_chains</code> argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.</p>
+<p>To run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell <code>for</code> loop to set up index variables needed to identify each chain and its outputs.</p>
+<p>On MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:</p>
+<pre><code>for NAME [in LIST]; do COMMANDS; done</code></pre>
+<p>The list can be a simple sequence of numbers, or you can use the shell expansion syntax <code>{1..N}</code> which expands to the sequence from <span class="math inline">\(1\)</span> to <span class="math inline">\(N\)</span>, e.g.&nbsp;<code>{1..4}</code> expands to <code>1 2 3 4</code>. Note that the expression <code>{1..N}</code> cannot contain spaces.</p>
+<p>To run 4 chains for the example bernoulli model on MacOS or Linux:</p>
+<pre><code>&gt; for i in {1..4}
+    do
+      ./bernoulli sample data file=bernoulli.data.json \
+      output file=output_${i}.csv
+    done</code></pre>
+<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. To run chains in parallel, put an ampersand (<code>&amp;</code>) at the end of the nested sampler command:</p>
 <pre><code>&gt; for i in {1..4}
     do
-      ./bernoulli sample data file=my_model.data.json \
-                  output file=output_${i}.csv &amp;
+      ./bernoulli sample data file=bernoulli.data.json \
+      output file=output_${i}.csv &amp;
     done</code></pre>
-<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. The ampersand (<code>&amp;</code>) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
-<p>On Windows the corresponding loop is:</p>
+<p>This pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
+<p>On Windows, the DOS <a href="https://www.windows-commandline.com/windows-for-loop-examples/">for-loop syntax</a> is one of:</p>
+<pre><code>for %i in (SET) do COMMAND COMMAND-ARGUMENTS
+for /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS</code></pre>
+<p>To run 4 chains in parallel on Windows:</p>
 <pre><code>&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^
-                                    data file=my_model.data.json my_data ^
+                                    data file=bernoulli.data.json my_data ^
                                     output file=output_%i.csv</code></pre>
 <p>The caret (<code>^</code>) indicates a line continuation in DOS. The expression <code>%i</code> is the loop index.</p>
-<p>In the following examples, we focus on just the nested sampler command for Unix.</p>
+<p>In the following extended examples, we focus on just the nested sampler command for Unix.</p>
 <section id="running-multiple-chains-with-a-specified-rng-seed" class="level3">
 <h3 class="anchored" data-anchor-id="running-multiple-chains-with-a-specified-rng-seed">Running multiple chains with a specified RNG seed</h3>
 <p>For reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.</p>
@@ -835,25 +975,6 @@ <h3 class="anchored" data-anchor-id="everything-example">Everything example</h3>
 Betancourt, Michael. 2017. <span>“A Conceptual Introduction to <span>H</span>amiltonian <span>M</span>onte <span>C</span>arlo.”</span> <em>arXiv</em> 1701.02434. <a href="https://arxiv.org/abs/1701.02434">https://arxiv.org/abs/1701.02434</a>.
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-<section id="mcmc-intro" class="level1">
-<h1>MCMC Sampling</h1>
-<section id="running-the-sampler" class="level2">
-<h2 class="anchored" data-anchor-id="running-the-sampler">Running the sampler</h2>
-<p>To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument <code>sample</code> or <code>method=sample</code> together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, <code>&lt;cmdstan-home&gt;/examples/bernoulli</code>:</p>
-<pre><code>&gt; cd examples/bernoulli</code></pre>
-<p>To execute sampling of the model under Linux or Mac, use:</p>
-<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json</code></pre>
-<p>In Windows, the <code>./</code> prefix is not needed:</p>
-<pre><code>&gt; bernoulli.exe sample data file=bernoulli.data.json</code></pre>
-<p>The output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:</p>
-<pre><code>method = sample (Default)
-  sample
-    num_samples = 1000 (Default)
-    num_warmup = 1000 (Default)
-    save_warmup = 0 (Default)
-    thin = 1 (Default)
-    adapt
-      engaged = 1 (Default)
-      gamma = 0.050000000000000003 (Default)
-      delta = 0.80000000000000004 (Default)
-      kappa = 0.75 (Default)
-      t0 = 10 (Default)
-      init_buffer = 75 (Default)
-      term_buffer = 50 (Default)
-      window = 25 (Default)
-      save_metric = 0 (Default)
-    algorithm = hmc (Default)
-      hmc
-        engine = nuts (Default)
-          nuts
-            max_depth = 10 (Default)
-        metric = diag_e (Default)
-        metric_file =  (Default)
-        stepsize = 1 (Default)
-        stepsize_jitter = 0 (Default)
-    num_chains = 1 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 3252652196 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>After the configuration has been displayed, a short timing message is given.</p>
-<pre><code>Gradient evaluation took 1.2e-05 seconds
-1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
-Adjust your expectations accordingly!</code></pre>
-<p>Next, the sampler reports the iteration number, reporting the percentage complete.</p>
-<pre><code>Iteration:    1 / 2000 [  0%]  (Warmup)
-....
-Iteration: 2000 / 2000 [100%]  (Sampling)</code></pre>
-<p>Finally, the sampler reports timing information:</p>
-<pre><code> Elapsed Time: 0.007 seconds (Warm-up)
-               0.017 seconds (Sampling)
-               0.024 seconds (Total)</code></pre>
-</section>
-<section id="multi-chain-sampling" class="level2">
-<h2 class="anchored" data-anchor-id="multi-chain-sampling">Running multiple chains</h2>
-<p>A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.</p>
-<p>There are two different ways of running multiple chains, with the <code>num_chains</code> argument using a single executable and by using the Unix and DOS shell to run multiple executables.</p>
-<section id="using-the-num_chains-argument-to-run-multiple-chains" class="level3">
-<h3 class="anchored" data-anchor-id="using-the-num_chains-argument-to-run-multiple-chains">Using the num_chains argument to run multiple chains</h3>
-<p>The <code>num_chains</code> argument can be used for all of Stan’s samplers with the exception of the <code>static</code> HMC engine.</p>
-<p>Example that will run 4 chains:</p>
-<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv</code></pre>
-<p>If the model was not compiled with <code>STAN_THREADS=true</code>, the above command will run 4 chains sequentially and will produce the sample in <code>output_1.csv</code>, <code>output_2.csv</code>, <code>output_3.csv</code>, <code>output_4.csv</code>. A suffix with the chain id is appended to the provided output filename (<code>output.csv</code> in the above command).</p>
-<p>If the model was compiled with <code>STAN_THREADS=true</code>, the chains can run in parallel, with the <code>num_threads</code> argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the below command will run 4 chains in parallel, provided there are cores available:</p>
-<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4</code></pre>
-<p>If the model uses within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the <a href="https://www.intel.com/content/www/us/en/docs/onetbb/developer-guide-api-reference/2021-11/how-task-scheduler-works.html">Threading Building Blocks scheduler</a>.</p>
-<pre><code>./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16</code></pre>
-</section>
-<section id="using-shell-for-running-multiple-chains" class="level3">
-<h3 class="anchored" data-anchor-id="using-shell-for-running-multiple-chains">Using shell for running multiple chains</h3>
-<p>To run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell <code>for</code> loop to set up index variables needed to identify each chain and its outputs.</p>
-<p>On MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:</p>
-<pre><code>for NAME [in LIST]; do COMMANDS; done</code></pre>
-<p>The list can be a simple sequence of numbers, or you can use the shell expansion syntax <code>{1..N}</code> which expands to the sequence from <span class="math inline">\(1\)</span> to <span class="math inline">\(N\)</span>, e.g.&nbsp;<code>{1..4}</code> expands to <code>1 2 3 4</code>. Note that the expression <code>{1..N}</code> cannot contain spaces.</p>
-<p>To run 4 chains for the example bernoulli model on MacOS or Linux:</p>
-<pre><code>&gt; for i in {1..4}
-    do
-      ./bernoulli sample data file=bernoulli.data.json \
-      output file=output_${i}.csv
-    done</code></pre>
-<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. To run chains in parallel, put an ampersand (<code>&amp;</code>) at the end of the nested sampler command:</p>
-<pre><code>&gt; for i in {1..4}
-    do
-      ./bernoulli sample data file=bernoulli.data.json \
-      output file=output_${i}.csv &amp;
-    done</code></pre>
-<p>This pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
-<p>On Windows, the DOS <a href="https://www.windows-commandline.com/windows-for-loop-examples/">for-loop syntax</a> is one of:</p>
-<pre><code>for %i in (SET) do COMMAND COMMAND-ARGUMENTS
-for /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS</code></pre>
-<p>To run 4 chains in parallel on Windows:</p>
-<pre><code>&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^
-                                    data file=bernoulli.data.json my_data ^
-                                    output file=output_%i.csv</code></pre>
-<p>The caret (<code>^</code>) indicates a line continuation in DOS.</p>
-</section>
-</section>
-<section id="mcmc_output_csv" class="level2">
-<h2 class="anchored" data-anchor-id="mcmc_output_csv">Stan CSV output file</h2>
-<p>Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is <code>output.csv</code>.</p>
-<p>The first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (<code>#</code>)).</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = sample (Default)
-#   sample
-#     num_samples = 1000 (Default)
-#     num_warmup = 1000 (Default)
-...
-# output
-#   file = output.csv (Default)
-#   diagnostic_file =  (Default)
-#   refresh = 100 (Default)</code></pre>
-<p>This is followed by a CSV header indicating the names of the values sampled.</p>
-<pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
-<p>The first output columns report the HMC sampler information:</p>
-<ul>
-<li><code>lp__</code> - the total log probability density (up to an additive constant) at each sample</li>
-<li><code>accept_stat__</code> - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory</li>
-<li><code>stepsize__</code> - integrator step size</li>
-<li><code>treedepth__</code> - depth of tree used by NUTS (NUTS sampler)</li>
-<li><code>n_leapfrog__</code> - number of leapfrog calculations (NUTS sampler)</li>
-<li><code>divergent__</code> - has value <code>1</code> if trajectory diverged, otherwise <code>0</code>. (NUTS sampler)</li>
-<li><code>energy__</code> - value of the Hamiltonian</li>
-<li><code>int_time__</code> - total integration time (static HMC sampler)</li>
-</ul>
-<p>Because the above header is from the NUTS sampler, it has columns <code>treedepth__</code>, <code>n_leapfrog__</code>, and <code>divergent__</code> and doesn’t have column <code>int_time__</code>. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, <code>theta</code>.</p>
-<p>The header line is written to the output file before warmup begins. If option <code>save_warmup</code> is set to <code>1</code>, the warmup draws are output directly after the header. The total number of warmup draws saved is <code>num_warmup</code> divided by <code>thin</code>, rounded up (i.e., <code>ceiling</code>).</p>
-<p>Following the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:</p>
-<pre><code># Adaptation terminated
-# Step size = 0.884484
-# Diagonal elements of inverse mass matrix:
-# 0.535006</code></pre>
-<p>The default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter <code>theta</code>.</p>
-<p>Draws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.</p>
-<pre><code>-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853
--6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295
--7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299
--6.88712,1,0.884484,1,1,0,7.02101,0.188229
--7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596
-...</code></pre>
-<p>The output ends with timing details:</p>
-<pre><code>#  Elapsed Time: 0.007 seconds (Warm-up)
-#                0.017 seconds (Sampling)
-#                0.024 seconds (Total)</code></pre>
-</section>
-<section id="summarizing-sampler-outputs-with-stansummary" class="level2">
-<h2 class="anchored" data-anchor-id="summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></h2>
-<p>The <code>stansummary</code> utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file <code>stansummary</code> reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and <span class="math inline">\(\hat{R}\)</span> values.</p>
-<p>To run <code>stansummary</code> on the output files generated by the for loop above, by the above run of the <code>bernoulli</code> model on Mac or Linux:</p>
-<pre><code>&lt;cmdstan-home&gt;/bin/stansummary output_*.csv</code></pre>
-<p>On Windows, use backslashes to call the <code>stansummary.exe</code>.</p>
-<pre><code>&lt;cmdstan-home&gt;\bin\stansummary.exe output_*.csv</code></pre>
-<p>The stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable <code>theta</code>:</p>
-<pre><code>Inference for Stan model: bernoulli_model
-4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.
-
-Warmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total
-Sampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total
-
-                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat
-lp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00
-accept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00
-stepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13
-treedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00
-n_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00
-divergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan
-energy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00
-theta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00
-
-Samples were drawn using hmc with nuts.
-For each parameter, N_Eff is a crude measure of effective sample size,
-and R_hat is the potential scale reduction factor on split chains (at
-convergence, R_hat=1).</code></pre>
-<p>In this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for <code>theta</code> reflect the uncertainty in our estimate, due to the small amount of data, given the prior of <code>beta(1, 1)</code></p>
-
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-<section id="optimization" class="level1">
-<h1>Optimization</h1>
-<p>The CmdStan executable can run Stan’s optimization algorithms which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability.</p>
-<p>The executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli optimize data file=bernoulli.data.json</code></pre>
-<p>Executing this command prints both output to the console and to a csv file.</p>
-<p>The first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:</p>
-<pre><code> ./bernoulli optimize data file=bernoulli.data.json
-method = optimize
-  optimize
-    algorithm = lbfgs (Default)
-      lbfgs
-        init_alpha = 0.001 (Default)
-        tol_obj = 1e-12 (Default)
-        tol_rel_obj = 10000 (Default)
-        tol_grad = 1e-08 (Default)
-        tol_rel_grad = 10000000 (Default)
-        tol_param = 1e-08 (Default)
-        history_size = 5 (Default)
-    iter = 2000 (Default)
-    save_iterations = 0 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 87122538 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>The second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The <code>alpha</code> value is for step size used. In the final state the change in parameters was roughly <span class="math inline">\(0.002\)</span> and the length of the gradient roughly 3e-05 (<span class="math inline">\(0.00003\)</span>).</p>
-<pre><code>Initial log joint probability = -6.85653
-    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
-       5      -5.00402    0.00184936   3.35074e-05           1           1        8   
-Optimization terminated normally: 
-  Convergence detected: relative gradient magnitude is below tolerance</code></pre>
-<p>The output from optimization is written into the file <code>output.csv</code> by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = optimize
-#   optimize
-#     algorithm = lbfgs (Default)
-...</code></pre>
-<p>Following the config information, are two lines of output: the CSV headers and the recorded values:</p>
-<pre><code>lp__,theta
--5.00402,0.200003</code></pre>
-<p>Note that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with <code>lp__</code> and the model parameter <code>theta</code>. The maximum log probability is -5.0 and the posterior mode for <code>theta</code> is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.</p>
-<p>All of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument <code>save_iterations</code> specifies whether or not to save the intermediate iterations to the output file. Allowed values are <span class="math inline">\(0\)</span> or <span class="math inline">\(1\)</span>, corresponding to <code>False</code> and <code>True</code> respectively. The default value is <span class="math inline">\(0\)</span>, i.e., intermediate iterations are not saved to the output file. Running the optimizer with <code>save_iterations=1</code> writes both the initial log joint probability and values for all iterations to the output CSV file.</p>
-<p>Running the example model with option <code>save_iterations=1</code>, i.e., the command</p>
-<pre><code>&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json</code></pre>
-<p>produces CSV file output rows:</p>
-<pre><code>lp__,theta
--6.85653,0.493689
--6.10128,0.420936
--5.02953,0.22956
--5.00517,0.206107
--5.00403,0.200299
--5.00402,0.200003</code></pre>
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diff --git a/docs/2_34/cmdstan-guide/optimize_config.html b/docs/2_34/cmdstan-guide/optimize_config.html
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@@ -7,7 +7,7 @@
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-<title>Optimization Configuration</title>
+<title>Optimization</title>
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+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/optimize_config.html">Optimization</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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+ <span class="menu-text">Optimization</span></a>
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-  <li><a href="#maximum-likelihood-estimation" id="toc-maximum-likelihood-estimation" class="nav-link active" data-scroll-target="#maximum-likelihood-estimation">Maximum Likelihood Estimation</a>
+  <li><a href="#optimization" id="toc-optimization" class="nav-link active" data-scroll-target="#optimization">Optimization</a>
   <ul class="collapse">
   <li><a href="#jacobian-adjustments" id="toc-jacobian-adjustments" class="nav-link" data-scroll-target="#jacobian-adjustments">Jacobian adjustments</a></li>
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@@ -475,11 +448,49 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="maximum-likelihood-estimation" class="level1">
-<h1>Maximum Likelihood Estimation</h1>
-<p>The <code>optimize</code> method finds the mode of the posterior distribution, assuming that there is one. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability. For optimization, the mode is calculated without the Jacobian adjustment for constrained variables, which shifts the mode due to the change of variables. Thus modes correspond to modes of the model as written.</p>
-<p>The full set of configuration options available for the <code>optimize</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=optimize</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
-<pre><code># model = bernoulli_model
+<section id="optimization" class="level1">
+<h1>Optimization</h1>
+<p>The CmdStan executable can run Stan’s optimization algorithms, which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global optimum as opposed to a local optimum of log probability.</p>
+<p>The full set of configuration options available for the <code>optimize</code> method is available by using the <code>optimize help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the optimizer console output and in the output CSV file’s comments.</p>
+<p>The executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli optimize data file=bernoulli.data.json</code></pre>
+<p>Executing this command prints both output to the console and to a CSV file.</p>
+<p>The first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:</p>
+<pre><code> ./bernoulli optimize data file=bernoulli.data.json
+method = optimize
+  optimize
+    algorithm = lbfgs (Default)
+      lbfgs
+        init_alpha = 0.001 (Default)
+        tol_obj = 1e-12 (Default)
+        tol_rel_obj = 10000 (Default)
+        tol_grad = 1e-08 (Default)
+        tol_rel_grad = 10000000 (Default)
+        tol_param = 1e-08 (Default)
+        history_size = 5 (Default)
+    iter = 2000 (Default)
+    save_iterations = 0 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 87122538 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>The second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The <code>alpha</code> value is for step size used. In the final state the change in parameters was roughly <span class="math inline">\(0.002\)</span> and the length of the gradient roughly 3e-05 (<span class="math inline">\(0.00003\)</span>).</p>
+<pre><code>Initial log joint probability = -6.85653
+    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes
+       5      -5.00402    0.00184936   3.35074e-05           1           1        8
+Optimization terminated normally:
+  Convergence detected: relative gradient magnitude is below tolerance</code></pre>
+<p>The output from optimization is written into the file <code>output.csv</code> by default. The output follows the <a href="../cmdstan-guide/stan_csv_apdx.html">same pattern</a> as the output for sampling, first dumping the entire set of parameters used as comment lines:</p>
+<pre><code># stan_version_major = 2
+# stan_version_minor = 23
+# stan_version_patch = 0
+# model = bernoulli_model
 # method = optimize
 #   optimize
 #     algorithm = lbfgs (Default)
@@ -493,10 +504,29 @@ <h1>Maximum Likelihood Estimation</h1>
 #         history_size = 5 (Default)
 #     jacobian = 0 (Default)
 #     iter = 2000 (Default)
-#     save_iterations = 0 (Default)</code></pre>
+#     save_iterations = 0 (Default)
+...</code></pre>
+<p>Following the config information are two lines of output, the CSV headers and the recorded values:</p>
+<pre><code>lp__,theta
+-5.00402,0.200003</code></pre>
+<p>Note that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with <code>lp__</code> and the model parameter <code>theta</code>. The maximum log probability is -5.0 and the posterior mode for <code>theta</code> is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.</p>
+<p>All of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument <code>save_iterations</code> specifies whether or not to save the intermediate iterations to the output file. Allowed values are <span class="math inline">\(0\)</span> or <span class="math inline">\(1\)</span>, corresponding to <code>False</code> and <code>True</code> respectively. The default value is <span class="math inline">\(0\)</span>, i.e., intermediate iterations are not saved to the output file. Running the optimizer with <code>save_iterations=1</code> writes both the initial log joint probability and values for all iterations to the output CSV file.</p>
+<p>Running the example model with option <code>save_iterations=1</code>, i.e., the command</p>
+<pre><code>&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json</code></pre>
+<p>produces CSV file output rows:</p>
+<pre><code>lp__,theta
+-6.85653,0.493689
+-6.10128,0.420936
+-5.02953,0.22956
+-5.00517,0.206107
+-5.00403,0.200299
+-5.00402,0.200003</code></pre>
 <section id="jacobian-adjustments" class="level2">
 <h2 class="anchored" data-anchor-id="jacobian-adjustments">Jacobian adjustments</h2>
-<p>The <code>jacobian</code> argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms. Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\)</span>, the value which maximizes the likelihood of the data given the parameters, (including prior terms). Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\,p(\theta)\)</span>. By default this value is <code>0</code> (false), do not include the Jacobian adjustment.</p>
+<p>The <code>jacobian</code> argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of <a href="https://mc-stan.org/docs/reference-manual/transforms.html">inverse parameter transforms</a>.</p>
+<p>Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\)</span>, the value which maximizes the likelihood of the data given the parameters, (including prior terms).</p>
+<p>Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\,p(\theta)\)</span>.</p>
+<p>By default this value is <code>0</code> (false), do not include the Jacobian adjustment.</p>
 </section>
 <section id="optimization-algorithms" class="level2">
 <h2 class="anchored" data-anchor-id="optimization-algorithms">Optimization algorithms</h2>
@@ -534,25 +564,6 @@ <h2 class="anchored" data-anchor-id="the-newton-optimizer">The Newton optimizer<
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diff --git a/docs/2_34/cmdstan-guide/parallelization.html b/docs/2_34/cmdstan-guide/parallelization.html
index e1ec1024e..6538b46b9 100644
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+++ b/docs/2_34/cmdstan-guide/parallelization.html
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-<link href="../cmdstan-guide/stanc.html" rel="next">
-<link href="../cmdstan-guide/diagnose_config.html" rel="prev">
+<link href="../cmdstan-guide/command_line_options.html" rel="next">
+<link href="../cmdstan-guide/compiling_stan_programs.html" rel="prev">
 <link href="../img/logo_tm.png" rel="icon" type="image/png">
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -70,7 +72,7 @@
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/parallelization.html">Parallelization</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/parallelization.html">Parallelization</a></li></ol></nav>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link active">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -303,7 +282,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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@@ -315,13 +294,13 @@
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   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link active">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -484,7 +457,7 @@ <h3 class="anchored" data-anchor-id="running">Running</h3>
 <p>Generally, this number should not exceed the number of available cores for best performance.</p>
 <p>Example:</p>
 <pre><code>./model sample data file=data.json num_threads=4 ...</code></pre>
-<p>When the model is compiled with <code>STAN_THREADS</code> we can sample with multiple chains with a single executable (see section <span id="multi-chain-sampling">running multiple chains</span> for cases when this is available). When running multiple chains <code>num_threads</code> is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:</p>
+<p>When the model is compiled with <code>STAN_THREADS</code> we can sample with multiple chains with a single executable (see section <a href="../cmdstan-guide/mcmc_config.html#multi-chain-sampling">running multiple chains</a> for cases when this is available). When running multiple chains <code>num_threads</code> is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:</p>
 <pre><code>./model sample num_chains=2 data file=data.json num_threads=8 ...</code></pre>
 </section>
 </section>
@@ -576,25 +549,6 @@ <h3 class="anchored" data-anchor-id="running-2">Running</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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@@ -615,6 +569,150 @@ <h3 class="anchored" data-anchor-id="running-2">Running</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
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+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+        }
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+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
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+        } else {
+          toggle.classList.remove("alternate");
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
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+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
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+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
@@ -984,13 +1082,13 @@ <h3 class="anchored" data-anchor-id="running-2">Running</h3>
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/diagnose_config.html" class="pagination-link" aria-label="Diagnosing HMC by Comparison of Gradients">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Diagnosing HMC by Comparison of Gradients</span>
+      <a href="../cmdstan-guide/compiling_stan_programs.html" class="pagination-link" aria-label="Compiling a Stan Program">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Compiling a Stan Program</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
-      <a href="../cmdstan-guide/stanc.html" class="pagination-link" aria-label="`stanc`: Translating Stan to C++">
-        <span class="nav-page-text"><code>stanc</code>: Translating Stan to C++</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../cmdstan-guide/command_line_options.html" class="pagination-link" aria-label="Command-Line Interface Overview">
+        <span class="nav-page-text">Command-Line Interface Overview</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/2_34/cmdstan-guide/pathfinder_config.html b/docs/2_34/cmdstan-guide/pathfinder_config.html
index 7c6136f7f..5cf450850 100644
--- a/docs/2_34/cmdstan-guide/pathfinder_config.html
+++ b/docs/2_34/cmdstan-guide/pathfinder_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>Pathfinder Configuration</title>
+<title>Pathfinder Method for Approximate Bayesian Inference</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -20,7 +20,27 @@
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   vertical-align: middle;
 }
-</style>
+/* CSS for citations */
+div.csl-bib-body { }
+div.csl-entry {
+  clear: both;
+  margin-bottom: 0em;
+}
+.hanging-indent div.csl-entry {
+  margin-left:2em;
+  text-indent:-2em;
+}
+div.csl-left-margin {
+  min-width:2em;
+  float:left;
+}
+div.csl-right-inline {
+  margin-left:2em;
+  padding-left:1em;
+}
+div.csl-indent {
+  margin-left: 2em;
+}</style>
 
 
 <script src="../site_libs/quarto-nav/quarto-nav.js"></script>
@@ -38,10 +58,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/pathfinder_config.html">Pathfinder Method for Approximate Bayesian Inference</a></li></ol></nav>
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@@ -477,9 +470,14 @@ <h2 id="toc-title">On this page</h2>
 
 <section id="pathfinder-config" class="level1">
 <h1>Pathfinder Method for Approximate Bayesian Inference</h1>
-<p>The Pathfinder algorithm is described in section <a href="../cmdstan-guide/pathfinder_intro.html">Pathfinder overview</a>.</p>
+<p>The CmdStan method <code>pathfinder</code> uses the Pathfinder algorithm of <span class="citation" data-cites="zhang_pathfinder:2022">Zhang et al. (<a href="#ref-zhang_pathfinder:2022" role="doc-biblioref">2022</a>)</span>, which is further described in the <a href="https://mc-stan.org/docs/reference-manual/pathfinder.html">Stan Reference Manual</a>.</p>
+<p>A single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes.</p>
 <p>The <code>pathfinder</code> method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument <code>num_paths</code> specifies the number of single-path Pathfinders, the default is <span class="math inline">\(4\)</span>. If <code>num_paths</code> is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.</p>
-<p>The full set of configuration options available for the <code>pathfinder</code> method is reported at the beginning of the pathfinder output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=pathfinder</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
+<p>The full set of configuration options available for the <code>pathfinder</code> method is available by using the <code>pathfinder help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.</p>
+<p>The following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli pathfinder data file=bernoulli.data.R</code></pre>
+<p>Executing this command prints both output to the console and csv files.</p>
+<p>The first part of the console output reports on the configuration used.</p>
 <pre><code>method = pathfinder
   pathfinder
     init_alpha = 0.001 (Default)
@@ -496,7 +494,39 @@ <h1>Pathfinder Method for Approximate Bayesian Inference</h1>
     save_single_paths = 0 (Default)
     max_lbfgs_iters = 1000 (Default)
     num_draws = 1000 (Default)
-    num_elbo_draws = 25 (Default)</code></pre>
+    num_elbo_draws = 25 (Default)
+id = 1 (Default)
+data
+  file = examples/bernoulli/bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 1995513073 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)
+  sig_figs = -1 (Default)
+  profile_file = profile.csv (Default)
+num_threads = 1 (Default)</code></pre>
+<p>The rest of the output describes the progression of the algorithm.</p>
+<p>By default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, then uses importance resampling on the set of returned draws to produce the specified number of draws.</p>
+<pre><code>Path [1] :Initial log joint density = -11.543343
+Path [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00
+Path [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)
+Path [2] :Initial log joint density = -7.443345
+Path [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00
+Path [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)
+Path [3] :Initial log joint density = -18.986308
+Path [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00
+Path [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)
+Path [4] :Initial log joint density = -8.304453
+Path [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00
+Path [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)
+Total log probability function evaluations:8404</code></pre>
 <section id="pathfinder-configuration" class="level2">
 <h2 class="anchored" data-anchor-id="pathfinder-configuration">Pathfinder Configuration</h2>
 <ul>
@@ -512,18 +542,43 @@ <h2 class="anchored" data-anchor-id="pathfinder-configuration">Pathfinder Config
 </section>
 <section id="l-bfgs-configuration" class="level2">
 <h2 class="anchored" data-anchor-id="l-bfgs-configuration">L-BFGS Configuration</h2>
-<p>Arguments <code>init_alpha</code> through <code>history_size</code> are the full set of arguments to the L-BFGS optimizer and have the same defaults for <a href="../cmdstan-guide/optimize_config.html">optimization</a>.</p>
+<p>Arguments <code>init_alpha</code> through <code>history_size</code> are the full set of arguments to the L-BFGS optimizer and have the same defaults for <a href="../cmdstan-guide/optimize_config.html#the-quasi-newton-optimizers">optimization</a>.</p>
 </section>
 <section id="pathfinder_csv" class="level2">
 <h2 class="anchored" data-anchor-id="pathfinder_csv">Multi-path Pathfinder CSV files</h2>
 <p>By default, the <code>pathfinder</code> method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a <a href="../cmdstan-guide/stan_csv_apdx.html">StanCSV file</a>.</p>
 <p>The CSV files have the following structure:</p>
-<ul>
-<li><p>The full set of configuration options available for the <code>pathfinder</code> method is reported at the beginning of the sampler output file as CSV comments.</p></li>
-<li><p>The CSV header row consists of columns <code>lp_approx__</code>, <code>lp__</code>, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.</p></li>
-<li><p>The data rows contain the draws from the single- or multi-path run.</p></li>
-<li><p>Final comments containing timing information.</p></li>
-</ul>
+<p>The initial CSV comment rows contain the complete set of CmdStan configuration options.</p>
+<pre><code>...
+# method = pathfinder
+#   pathfinder
+#     init_alpha = 0.001 (Default)
+#     tol_obj = 9.9999999999999998e-13 (Default)
+#     tol_rel_obj = 10000 (Default)
+#     tol_grad = 1e-08 (Default)
+#     tol_rel_grad = 10000000 (Default)
+#     tol_param = 1e-08 (Default)
+#     history_size = 5 (Default)
+#     num_psis_draws = 1000 (Default)
+#     num_paths = 4 (Default)
+#     psis_resample = 1 (Default)
+#     calculate_lp = 1 (Default)
+#     save_single_paths = 0 (Default)
+#     max_lbfgs_iters = 1000 (Default)
+#     num_draws = 1000 (Default)
+#     num_elbo_draws = 25 (Default)
+...</code></pre>
+<p>Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs <code>lp_approx__</code>, the log density in the approximating distribution, and <code>lp__</code>, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.</p>
+<pre><code>lp_approx__,lp__,theta
+-2.4973, -8.2951, 0.0811852
+-0.87445, -7.06526, 0.160207
+-0.812285, -7.07124, 0.35819
+...</code></pre>
+<p>The final lines are comment lines which give timing information.</p>
+<pre><code># Elapsed Time: 0.016000 seconds (Pathfinders)
+#               0.003000 seconds (PSIS)
+#               0.019000 seconds (Total)</code></pre>
+<p>Pathfinder provides option <code>save_single_paths</code> which will save output from the single-path Pathfinder runs.</p>
 </section>
 <section id="single-path-pathfinder-outputs." class="level2">
 <h2 class="anchored" data-anchor-id="single-path-pathfinder-outputs.">Single-path Pathfinder Outputs.</h2>
@@ -591,29 +646,15 @@ <h2 class="anchored" data-anchor-id="single-path-pathfinder-outputs.">Single-pat
 <p>Option <code>num_paths=1</code> runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments <code>num_paths=1 save_single_paths=1</code> creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.</p>
 
 
+
 </section>
 </section>
 
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+<a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a><div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography" id="quarto-bibliography"><h2 class="anchored quarto-appendix-heading">References</h2><div id="refs" class="references csl-bib-body hanging-indent" data-entry-spacing="0" role="list">
+<div id="ref-zhang_pathfinder:2022" class="csl-entry" role="listitem">
+Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
+</div>
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-<h1>Variational Inference using Pathfinder</h1>
-<p>The CmdStan method <code>pathfinder</code> uses the Pathfinder algorithm of <span class="citation" data-cites="zhang_pathfinder:2022">Zhang et al. (<a href="#ref-zhang_pathfinder:2022" role="doc-biblioref">2022</a>)</span>. Pathfinder is a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the negative inverse Hessian estimates produced by the L-BFGS optimizer. Pathfinder returns draws from the Gaussian approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior.</p>
-<p>Pathfinder differs from the ADVI method in that it uses quasi-Newton optimization on the log posterior instead of stochastic gradient descent (SGD) on the Monte Carlo computation of the evidence lower bound (ELBO). Pathfinder’s approach is both faster and more stable than that of ADVI. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors.</p>
-<p>A single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes. By default, the <code>pathfinder</code> method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws.</p>
-<p>The following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli pathfinder data file=bernoulli.data.R</code></pre>
-<p>Executing this command prints both output to the console and csv files.</p>
-<p>The first part of the console output reports on the configuration used.</p>
-<pre><code>method = pathfinder
-  pathfinder
-    init_alpha = 0.001 (Default)
-    tol_obj = 9.9999999999999998e-13 (Default)
-    tol_rel_obj = 10000 (Default)
-    tol_grad = 1e-08 (Default)
-    tol_rel_grad = 10000000 (Default)
-    tol_param = 1e-08 (Default)
-    history_size = 5 (Default)
-    num_psis_draws = 1000 (Default)
-    num_paths = 4 (Default)
-    psis_resample = 1 (Default)
-    calculate_lp = 1 (Default)
-    save_single_paths = 0 (Default)
-    max_lbfgs_iters = 1000 (Default)
-    num_draws = 1000 (Default)
-    num_elbo_draws = 25 (Default)
-id = 1 (Default)
-data
-  file = examples/bernoulli/bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 1995513073 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)
-  sig_figs = -1 (Default)
-  profile_file = profile.csv (Default)
-num_threads = 1 (Default)</code></pre>
-<p>The rest of the output describes the progression of the algorithm.</p>
-<p>By default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, the uses importance resampling on the set of returned draws to produce the specified number of draws.</p>
-<pre><code>Path [1] :Initial log joint density = -11.543343
-Path [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00
-Path [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)
-Path [2] :Initial log joint density = -7.443345
-Path [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00
-Path [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)
-Path [3] :Initial log joint density = -18.986308
-Path [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00
-Path [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)
-Path [4] :Initial log joint density = -8.304453
-Path [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00
-Path [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)
-Total log probability function evaluations:8404</code></pre>
-<p>Pathfinder outputs a <a href="../cmdstan-guide/pathfinder_config.html#pathfinder_csv">StanCSV file</a> file which contains the importance resampled draws from multi-path Pathfinder. The initial CSV comment rows contain the complete set of CmdStan configuration options. Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs <code>lp_approx__</code>, the log density in the approximating distribution, and <code>lp__</code>, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.</p>
-<pre><code>lp_approx__,lp__,theta
--2.4973, -8.2951, 0.0811852
--0.87445, -7.06526, 0.160207
--0.812285, -7.07124, 0.35819
-...</code></pre>
-<p>The final lines are comment lines which give timing information.</p>
-<pre><code># Elapsed Time: 0.016000 seconds (Pathfinders)
-#               0.003000 seconds (PSIS)
-#               0.019000 seconds (Total)</code></pre>
-<p>Pathfinder provides option <code>save_single_paths</code> which will save output from the single-path Pathfinder runs. See section <a href="../cmdstan-guide/pathfinder_config.html">Pathfinder Method</a> for details.</p>
-
-
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-<div id="ref-zhang_pathfinder:2022" class="csl-entry" role="listitem">
-Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -223,7 +226,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -282,7 +261,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -303,7 +282,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -315,13 +294,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -448,25 +421,6 @@ <h1><code>print</code> (deprecated): MCMC Output Analysis</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -487,6 +441,150 @@ <h1><code>print</code> (deprecated): MCMC Output Analysis</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
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+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
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+  }
+  const setStyleSentinel = (alternate) => {
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+      return storageValue != null ? storageValue : localAlternateSentinel;
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+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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diff --git a/docs/2_34/cmdstan-guide/rdump_apdx.html b/docs/2_34/cmdstan-guide/rdump_apdx.html
index e2bdbddc4..0504edc45 100644
--- a/docs/2_34/cmdstan-guide/rdump_apdx.html
+++ b/docs/2_34/cmdstan-guide/rdump_apdx.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
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@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -704,25 +677,6 @@ <h2 class="anchored" data-anchor-id="bnf-grammar-for-dump-data">BNF grammar for
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -743,6 +697,150 @@ <h2 class="anchored" data-anchor-id="bnf-grammar-for-dump-data">BNF grammar for
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+        } else {
+          toggle.classList.remove("alternate");
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
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+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+  const hasAlternateSentinel = () => {  
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+  const setStyleSentinel = (alternate) => {
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+    if (!isFileUrl()) {
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+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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diff --git a/docs/2_34/cmdstan-guide/stan_csv_apdx.html b/docs/2_34/cmdstan-guide/stan_csv_apdx.html
index 94d260a78..15a31cda0 100644
--- a/docs/2_34/cmdstan-guide/stan_csv_apdx.html
+++ b/docs/2_34/cmdstan-guide/stan_csv_apdx.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
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@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -570,7 +543,7 @@ <h3 class="anchored" data-anchor-id="sampler-stan-csv-output-file">Sampler Stan
 #   refresh = 100 (Default)</code></pre>
 <p>Note that when running multi-threaded programs which use <code>reduce_sum</code> for <a href="../cmdstan-guide/parallelization.html">high-level parallelization</a>, the number of threads used will also be included in this initial comment header.</p>
 <p><strong>Column headers</strong></p>
-<p>The CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the <a href="../cmdstan-guide/mcmc_sampling_intro.html#mcmc_output_csv">output file</a> section of the Quickstart Guide chapter on MCMC Sampling. The example model <code>bernoulli.stan</code> only contains one parameter <code>theta</code>, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:</p>
+<p>The CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the <a href="../cmdstan-guide/mcmc_config.html#mcmc_output_csv">output file</a> section of the chapter on MCMC Sampling. The example model <code>bernoulli.stan</code> only contains one parameter <code>theta</code>, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:</p>
 <pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
 <p>As a second example, we show the output of the <code>eight_schools.stan</code> model on run on example dataset. This model has 3 parameters: <code>mu</code>, <code>theta</code> a vector whose length is dependent on the input data, here <code>N = 8</code>, and <code>tau</code>. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:</p>
 <pre><code>mu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau</code></pre>
@@ -693,25 +666,6 @@ <h2 class="anchored" data-anchor-id="diagnose-method-outputs">Diagnose method ou
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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@@ -732,6 +686,150 @@ <h2 class="anchored" data-anchor-id="diagnose-method-outputs">Diagnose method ou
     }
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+  const disableStylesheet = (stylesheets) => {
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diff --git a/docs/2_34/cmdstan-guide/stanc.html b/docs/2_34/cmdstan-guide/stanc.html
index c40d78df8..c59f92f4f 100644
--- a/docs/2_34/cmdstan-guide/stanc.html
+++ b/docs/2_34/cmdstan-guide/stanc.html
@@ -31,17 +31,19 @@
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
 <link href="../cmdstan-guide/stansummary.html" rel="next">
-<link href="../cmdstan-guide/parallelization.html" rel="prev">
+<link href="../cmdstan-guide/diagnose_config.html" rel="prev">
 <link href="../img/logo_tm.png" rel="icon" type="image/png">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -223,7 +226,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -282,7 +261,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -303,7 +282,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -315,13 +294,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -478,25 +451,6 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -517,6 +471,150 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
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+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
@@ -886,8 +984,8 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Parallelization</span>
+      <a href="../cmdstan-guide/diagnose_config.html" class="pagination-link" aria-label="Diagnosing HMC by Comparison of Gradients">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Diagnosing HMC by Comparison of Gradients</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/2_34/cmdstan-guide/stansummary.html b/docs/2_34/cmdstan-guide/stansummary.html
index 63a4aa3e6..339920e0a 100644
--- a/docs/2_34/cmdstan-guide/stansummary.html
+++ b/docs/2_34/cmdstan-guide/stansummary.html
@@ -38,10 +38,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "copy-button": false,
@@ -99,7 +101,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -599,25 +572,6 @@ <h2 class="anchored" data-anchor-id="command-line-options">Command-line options<
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -638,6 +592,150 @@ <h2 class="anchored" data-anchor-id="command-line-options">Command-line options<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
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+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
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+  }
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diff --git a/docs/2_34/cmdstan-guide/variational_config.html b/docs/2_34/cmdstan-guide/variational_config.html
index 09f3a71b1..3ed610e23 100644
--- a/docs/2_34/cmdstan-guide/variational_config.html
+++ b/docs/2_34/cmdstan-guide/variational_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>ADVI Configuration</title>
+<title>ADVI for Variational Inference</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -58,10 +58,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -119,7 +121,7 @@
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@@ -479,7 +452,7 @@
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-  <li><a href="#variational-inference-algorithm-advi" id="toc-variational-inference-algorithm-advi" class="nav-link active" data-scroll-target="#variational-inference-algorithm-advi">Variational Inference Algorithm: ADVI</a>
+  <li><a href="#variational-inference-using-advi" id="toc-variational-inference-using-advi" class="nav-link active" data-scroll-target="#variational-inference-using-advi">Variational Inference using ADVI</a>
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@@ -494,33 +467,81 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="variational-inference-algorithm-advi" class="level1">
-<h1>Variational Inference Algorithm: ADVI</h1>
-<p>CmdStan can approximate the posterior distribution using variational inference. The approximation is a Gaussian in the unconstrained variable space.</p>
+<section id="variational-inference-using-advi" class="level1">
+<h1>Variational Inference using ADVI</h1>
 <p>Stan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) <span class="citation" data-cites="kucukelbir:2017">Kucukelbir et al. (<a href="#ref-kucukelbir:2017" role="doc-biblioref">2017</a>)</span>. ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using <a href="https://mc-stan.org/docs/reference-manual/variational.html#stochastic-gradient-ascent">stochastic gradient ascent</a>. The measures of convergence are similar to the relative tolerance scheme of Stan’s <a href="https://mc-stan.org/docs/reference-manual/optimization.html">optimization algorithms</a>.</p>
 <p>The algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.</p>
-<p>The full set of configuration options available for the <code>variational</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=variational</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
-<pre><code># method = variational
-#   variational
-#     algorithm = meanfield (Default)
-#       meanfield
-#     iter = 10000 (Default)
-#     grad_samples = 1 (Default)
-#     elbo_samples = 100 (Default)
-#     eta = 1 (Default)
-#     adapt
-#       engaged = 1 (Default)
-#       iter = 50 (Default)
-#     tol_rel_obj = 0.01 (Default)
-#     eval_elbo = 100 (Default)
-#     output_samples = 1000 (Default)</code></pre>
-<p>The console output includes a notice that this algorithm is considered to be experimental:</p>
-<pre><code>EXPERIMENTAL ALGORITHM:
+<p>The full set of configuration options available for the <code>variational</code> method is available by using the <code>variational help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.</p>
+<p>The following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
+<p>Executing this command prints both output to the console and to a csv file.</p>
+<p>The first part of the console output reports on the configuration used: the default option <code>algorithm=meanfield</code> and the default tolerances for monitoring the algorithm’s convergence.</p>
+<pre><code>method = variational
+  variational
+    algorithm = meanfield (Default)
+      meanfield
+    iter = 10000 (Default)
+    grad_samples = 1 (Default)
+    elbo_samples = 100 (Default)
+    eta = 1 (Default)
+    adapt
+      engaged = 1 (Default)
+      iter = 50 (Default)
+    tol_rel_obj = 0.01 (Default)
+    eval_elbo = 100 (Default)
+    output_samples = 1000 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 3323783840 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>After the configuration has been displayed, informational and timing messages are output:</p>
+<pre><code>------------------------------------------------------------
+EXPERIMENTAL ALGORITHM:
   This procedure has not been thoroughly tested and may be unstable
-  or buggy. The interface is subject to change.</code></pre>
+  or buggy. The interface is subject to change.
+------------------------------------------------------------
+
+Gradient evaluation took 2.1e-05 seconds
+1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
+Adjust your expectations accordingly!</code></pre>
+<p>The rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in <span class="math inline">\(\sim\)</span> 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.</p>
+<pre><code>Begin eta adaptation.
+Iteration:   1 / 250 [  0%]  (Adaptation)
+Iteration:  50 / 250 [ 20%]  (Adaptation)
+Iteration: 100 / 250 [ 40%]  (Adaptation)
+Iteration: 150 / 250 [ 60%]  (Adaptation)
+Iteration: 200 / 250 [ 80%]  (Adaptation)
+Success! Found best value [eta = 1] earlier than expected.
+
+Begin stochastic gradient ascent.
+  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes
+   100           -6.131             1.000            1.000
+   200           -6.458             0.525            1.000
+   300           -6.300             0.359            0.051
+   400           -6.137             0.276            0.051
+   500           -6.243             0.224            0.027
+   600           -6.305             0.188            0.027
+   700           -6.289             0.162            0.025
+   800           -6.402             0.144            0.025
+   900           -6.103             0.133            0.025
+  1000           -6.314             0.123            0.027
+  1100           -6.348             0.024            0.025
+  1200           -6.244             0.020            0.018
+  1300           -6.293             0.019            0.017
+  1400           -6.250             0.017            0.017
+  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED
+
+Drawing a sample of size 1000 from the approximate posterior...
+COMPLETED.</code></pre>
 <section id="variational-algorithms" class="level2">
 <h2 class="anchored" data-anchor-id="variational-algorithms">Variational algorithms</h2>
-<p>Stan implements two variational algorithms. The <code>algorithm</code> argument specifies the variational algorithm.</p>
+<p>Stan implements two variational algorithms. They differ in the approximating distribution used in the unconstrained variable space. By default, ADVI uses option <code>algorithm=meanfield</code>. The <code>algorithm</code> argument specifies the variational algorithm.</p>
 <ul>
 <li><p><code>algorithm=meanfield</code> - Use a fully factorized Gaussian for the approximation. This is the default algorithm.</p></li>
 <li><p><code>algorithm=fullrank</code> Use a Gaussian with a full-rank covariance matrix for the approximation.</p></li>
@@ -555,7 +576,8 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 </ul>
 <p>To illustrate, we call Stan’s variational inference on the example model and data:</p>
 <pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
-<p>By default, the output file is <code>output.csv</code>. Lines 1 - 28 contain configuration information:</p>
+<p>By default, the output file is <code>output.csv</code>.</p>
+<p>The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:</p>
 <pre><code># stan_version_major = 2
 # stan_version_minor = 23
 # stan_version_patch = 0
@@ -575,19 +597,21 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 #     eval_elbo = 100 (Default)
 #     output_samples = 1000 (Default)
 ...</code></pre>
-<p>The column header row is:</p>
+<p>Next, the column header row:</p>
 <pre><code>lp__,log_p__,log_g__,theta</code></pre>
-<p>The stepsize adaptation information is:</p>
+<p>Additional comments provide stepsize adaptation information:</p>
 <pre><code># Stepsize adaptation complete.
 # eta = 1</code></pre>
-<p>The reported mean variational approximations information is:</p>
+<p>Followed by the data rows. The first line is special — it is the mean of the variational approximation.</p>
 <pre><code>0,0,0,0.214911</code></pre>
 <p>That is, the estimate for <code>theta</code> given the data is <code>0.2</code>.</p>
+<p>The rest of the output contains <code>output_samples</code> number of samples drawn from the variational approximation.</p>
 <p>The following is a sample based on this approximation:</p>
 <pre><code>0,-14.0252,-5.21718,0.770397
 0,-7.05063,-0.10025,0.162061
 0,-6.75031,-0.0191099,0.241606
 ...</code></pre>
+<p>The header indicates the unnormalized log probability with <code>lp__</code>. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.</p>
 
 
 
@@ -599,25 +623,6 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
 </div>
 </div></section></div></main> <!-- /main -->
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@@ -1012,8 +1161,8 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
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+      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
+        <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span> <i class="bi bi-arrow-right-short"></i>
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diff --git a/docs/2_34/cmdstan-guide/variational_intro.html b/docs/2_34/cmdstan-guide/variational_intro.html
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-<section id="variational-inference-using-advi" class="level1">
-<h1>Variational Inference using ADVI</h1>
-<p>The CmdStan method <code>variational</code> uses the Automatic Differentiation Variational Inference (ADVI) algorithm of <span class="citation" data-cites="kucukelbir:2017">Kucukelbir et al. (<a href="#ref-kucukelbir:2017" role="doc-biblioref">2017</a>)</span> to provide an approximate posterior distribution of the model conditioned on the data. The approximating distribution it uses is a Gaussian in the unconstrained variable space, either a fully factorized Gaussian approximation, specified by argument <code>algorithm=meanfield</code> option, or a Gaussian approximation using a full-rank covariance matrix, specified by argument <code>algorithm=fullrank</code>. By default, ADVI uses option <code>algorithm=meanfield</code>.</p>
-<p>The following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
-<p>Executing this command prints both output to the console and to a csv file.</p>
-<p>The first part of the console output reports on the configuration used: the default option <code>algorithm=meanfield</code> and the default tolerances for monitoring the algorithm’s convergence.</p>
-<pre><code>method = variational
-  variational
-    algorithm = meanfield (Default)
-      meanfield
-    iter = 10000 (Default)
-    grad_samples = 1 (Default)
-    elbo_samples = 100 (Default)
-    eta = 1 (Default)
-    adapt
-      engaged = 1 (Default)
-      iter = 50 (Default)
-    tol_rel_obj = 0.01 (Default)
-    eval_elbo = 100 (Default)
-    output_samples = 1000 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 3323783840 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>After the configuration has been displayed, informational and timing messages are output:</p>
-<pre><code>------------------------------------------------------------
-EXPERIMENTAL ALGORITHM:
-  This procedure has not been thoroughly tested and may be unstable
-  or buggy. The interface is subject to change.
-------------------------------------------------------------
-
-Gradient evaluation took 2.1e-05 seconds
-1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
-Adjust your expectations accordingly!</code></pre>
-<p>The rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in <span class="math inline">\(\sim\)</span> 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.</p>
-<pre><code>Begin eta adaptation.
-Iteration:   1 / 250 [  0%]  (Adaptation)
-Iteration:  50 / 250 [ 20%]  (Adaptation)
-Iteration: 100 / 250 [ 40%]  (Adaptation)
-Iteration: 150 / 250 [ 60%]  (Adaptation)
-Iteration: 200 / 250 [ 80%]  (Adaptation)
-Success! Found best value [eta = 1] earlier than expected.
-
-Begin stochastic gradient ascent.
-  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes 
-   100           -6.131             1.000            1.000
-   200           -6.458             0.525            1.000
-   300           -6.300             0.359            0.051
-   400           -6.137             0.276            0.051
-   500           -6.243             0.224            0.027
-   600           -6.305             0.188            0.027
-   700           -6.289             0.162            0.025
-   800           -6.402             0.144            0.025
-   900           -6.103             0.133            0.025
-  1000           -6.314             0.123            0.027
-  1100           -6.348             0.024            0.025
-  1200           -6.244             0.020            0.018
-  1300           -6.293             0.019            0.017
-  1400           -6.250             0.017            0.017
-  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED
-
-Drawing a sample of size 1000 from the approximate posterior... 
-COMPLETED.</code></pre>
-<p>The output from variational is written into the file <code>output.csv</code> by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = variational
-#   variational
-#     algorithm = meanfield (Default)
-#       meanfield
-#     iter = 10000 (Default)
-#     grad_samples = 1 (Default)
-#     elbo_samples = 100 (Default)
-#     eta = 1 (Default)
-#     adapt
-#       engaged = 1 (Default)
-#       iter = 50 (Default)
-#     tol_rel_obj = 0.01 (Default)
-#     eval_elbo = 100 (Default)
-#     output_samples = 1000 (Default)
-...</code></pre>
-<p>Next is the column header line, followed more CSV comments reporting the adapted value for the stepsize, followed by the values. The first line is special: it is the mean of the variational approximation. The rest of the output contains <code>output_samples</code> number of samples drawn from the variational approximation.</p>
-<pre><code>lp__,log_p__,log_g__,theta
-# Stepsize adaptation complete.
-# eta = 1
-0,0,0,0.236261
-0,-6.82318,-0.0929121,0.300415
-0,-6.89701,-0.158687,0.321982
-0,-6.99391,-0.23916,0.343643
-0,-7.35801,-0.51787,0.401554
-0,-7.4668,-0.539473,0.123081
-...</code></pre>
-<p>The header indicates the unnormalized log probability with <code>lp__</code>. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.</p>
-
-
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-<div id="ref-kucukelbir:2017" class="csl-entry" role="listitem">
-Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -880,25 +883,6 @@ <h2 class="anchored" data-anchor-id="reversing-functions">Reversing functions</h
 <li id="fn1"><p>Dividing by <span class="math inline">\(N\)</span> rather than <span class="math inline">\((N-1)\)</span> produces a maximum likelihood estimate of variance, which is biased to underestimate variance.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+      }, 40);  
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+    a.classList.add('quarto-color-scheme-toggle');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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+    i.classList.add('bi');
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diff --git a/docs/2_34/functions-reference/binary_distributions.html b/docs/2_34/functions-reference/binary_distributions.html
index 11519b89c..be66aa289 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -703,25 +706,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan Functions</h3>
 </section>
 
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -742,6 +726,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan Functions</h3>
     }
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   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+  const enableStylesheet = (stylesheets) => {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
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+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
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+    }
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+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
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+        manageTransitions("body", true);
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+  }
+  const isFileUrl = () => { 
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+    i.classList.add('bi');
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diff --git a/docs/2_34/functions-reference/bounded_continuous_distributions.html b/docs/2_34/functions-reference/bounded_continuous_distributions.html
index 499c6f71b..00a1498b0 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
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 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -569,25 +572,6 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -608,6 +592,150 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/bounded_discrete_distributions.html b/docs/2_34/functions-reference/bounded_discrete_distributions.html
index 4c2f1b515..d418d5988 100644
--- a/docs/2_34/functions-reference/bounded_discrete_distributions.html
+++ b/docs/2_34/functions-reference/bounded_discrete_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1140,25 +1143,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1179,6 +1163,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/circular_distributions.html b/docs/2_34/functions-reference/circular_distributions.html
index d97541370..58464c502 100644
--- a/docs/2_34/functions-reference/circular_distributions.html
+++ b/docs/2_34/functions-reference/circular_distributions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -620,25 +623,6 @@ <h3 class="anchored" data-anchor-id="numerical-stability">Numerical stability</h
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -659,6 +643,150 @@ <h3 class="anchored" data-anchor-id="numerical-stability">Numerical stability</h
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
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diff --git a/docs/2_34/functions-reference/complex-valued_basic_functions.html b/docs/2_34/functions-reference/complex-valued_basic_functions.html
index 5814d44f6..35ec0abe9 100644
--- a/docs/2_34/functions-reference/complex-valued_basic_functions.html
+++ b/docs/2_34/functions-reference/complex-valued_basic_functions.html
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -916,25 +919,6 @@ <h2 class="anchored" data-anchor-id="complex-h-trig">Complex hyperbolic trigonom
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -955,6 +939,150 @@ <h2 class="anchored" data-anchor-id="complex-h-trig">Complex hyperbolic trigonom
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/complex_matrix_operations.html b/docs/2_34/functions-reference/complex_matrix_operations.html
index 921b2106b..01e52f20e 100644
--- a/docs/2_34/functions-reference/complex_matrix_operations.html
+++ b/docs/2_34/functions-reference/complex_matrix_operations.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1479,25 +1482,6 @@ <h2 class="anchored" data-anchor-id="reverse-functions-for-complex-matrices">Rev
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1518,6 +1502,150 @@ <h2 class="anchored" data-anchor-id="reverse-functions-for-complex-matrices">Rev
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/compound_arithmetic_and_assignment.html b/docs/2_34/functions-reference/compound_arithmetic_and_assignment.html
index 804d67c8c..0076bc5b8 100644
--- a/docs/2_34/functions-reference/compound_arithmetic_and_assignment.html
+++ b/docs/2_34/functions-reference/compound_arithmetic_and_assignment.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -549,25 +552,6 @@ <h2 class="anchored" data-anchor-id="compound-elementwise-division-and-assignmen
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -588,6 +572,150 @@ <h2 class="anchored" data-anchor-id="compound-elementwise-division-and-assignmen
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/continuous_distributions_on_0_1.html b/docs/2_34/functions-reference/continuous_distributions_on_0_1.html
index bd879dd66..06a043783 100644
--- a/docs/2_34/functions-reference/continuous_distributions_on_0_1.html
+++ b/docs/2_34/functions-reference/continuous_distributions_on_0_1.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -620,25 +623,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -659,6 +643,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/conventions_for_probability_functions.html b/docs/2_34/functions-reference/conventions_for_probability_functions.html
index 8f4c64d26..11186c168 100644
--- a/docs/2_34/functions-reference/conventions_for_probability_functions.html
+++ b/docs/2_34/functions-reference/conventions_for_probability_functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -751,25 +754,6 @@ <h4 class="anchored" data-anchor-id="return-type">Return type</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -790,6 +774,150 @@ <h4 class="anchored" data-anchor-id="return-type">Return type</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
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+    toggleColorMode(toAlternate);
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/2_34/functions-reference/correlation_matrix_distributions.html b/docs/2_34/functions-reference/correlation_matrix_distributions.html
index 009a7845a..3f52b0530 100644
--- a/docs/2_34/functions-reference/correlation_matrix_distributions.html
+++ b/docs/2_34/functions-reference/correlation_matrix_distributions.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -680,25 +683,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
 <li id="fn1"><p>It is possible to build up a valid <code>L</code> within Stan, but that would then require Jacobian adjustments to imply the intended posterior.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -719,6 +703,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
     }
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+  const disableStylesheet = (stylesheets) => {
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+  const enableStylesheet = (stylesheets) => {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
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+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
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+      const el = els[i];
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+      } else {
+        el.classList.add('notransition');
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+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/covariance_matrix_distributions.html b/docs/2_34/functions-reference/covariance_matrix_distributions.html
index e6fe7f383..1b5cbd87e 100644
--- a/docs/2_34/functions-reference/covariance_matrix_distributions.html
+++ b/docs/2_34/functions-reference/covariance_matrix_distributions.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -684,25 +687,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-3">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -723,6 +707,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-3">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
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+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
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+    } else {
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    // Read the current dark / light value 
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+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/deprecated_functions.html b/docs/2_34/functions-reference/deprecated_functions.html
index a048564b6..ab815ffc5 100644
--- a/docs/2_34/functions-reference/deprecated_functions.html
+++ b/docs/2_34/functions-reference/deprecated_functions.html
@@ -72,10 +72,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -562,7 +565,7 @@ <h2 id="toc-title">On this page</h2>
 <section id="deprecated-functions" class="level1">
 <h1>Deprecated Functions</h1>
 <p>This appendix lists currently deprecated functionality along with how to replace it.</p>
-<p>Starting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of <code>get_lp</code> or <code>multiply_log</code>) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Functions]. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> these on the behalf of the user for the entire deprecation window and at least one version following the removal.</p>
+<p>Starting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of <code>get_lp</code> or <code>multiply_log</code>) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in <a href="../functions-reference/removed_functions.html">Removed Functions</a>. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> these on the behalf of the user for the entire deprecation window and at least one version following the removal.</p>
 <section id="integer-division-with-operator" class="level2">
 <h2 class="anchored" data-anchor-id="integer-division-with-operator">Integer division with <code>operator/</code></h2>
 <p><em>Deprecated</em>: Using <code>/</code> with two integer arguments is interpreted as integer floor division, such that</p>
@@ -729,25 +732,6 @@ <h4 class="anchored" data-anchor-id="sizes-and-parallel-arrays-1">Sizes and para
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -768,6 +752,150 @@ <h4 class="anchored" data-anchor-id="sizes-and-parallel-arrays-1">Sizes and para
     }
   }
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+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/distributions_over_unbounded_vectors.html b/docs/2_34/functions-reference/distributions_over_unbounded_vectors.html
index f991c8da3..61c77adbf 100644
--- a/docs/2_34/functions-reference/distributions_over_unbounded_vectors.html
+++ b/docs/2_34/functions-reference/distributions_over_unbounded_vectors.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -996,25 +999,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-7">Stan functions</h3>
 Durbin, J., and S. J. Koopman. 2001. <em>Time Series Analysis by State Space Methods</em>. New York: Oxford University Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1035,6 +1019,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-7">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/functions_index.html b/docs/2_34/functions-reference/functions_index.html
index ae9014a08..024648881 100644
--- a/docs/2_34/functions-reference/functions_index.html
+++ b/docs/2_34/functions-reference/functions_index.html
@@ -37,10 +37,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -3773,25 +3776,6 @@ <h2 class="anchored" data-anchor-id="z">Z</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -3812,6 +3796,150 @@ <h2 class="anchored" data-anchor-id="z">Z</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
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diff --git a/docs/2_34/functions-reference/hidden_markov_models.html b/docs/2_34/functions-reference/hidden_markov_models.html
index 3f0184954..684d9abf7 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -550,25 +553,6 @@ <h2 class="anchored" data-anchor-id="hmm-stan-functions">Stan functions</h2>
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diff --git a/docs/2_34/functions-reference/higher-order_functions.html b/docs/2_34/functions-reference/higher-order_functions.html
index 0a4b8f31e..65a29180c 100644
--- a/docs/2_34/functions-reference/higher-order_functions.html
+++ b/docs/2_34/functions-reference/higher-order_functions.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1036,25 +1039,6 @@ <h3 class="anchored" data-anchor-id="rectangular-map">Rectangular map</h3>
 Tanaka, Ken’ichiro, Masaaki Sugihara, Kazuo Murota, and Masatake Mori. 2009. <span>“Function Classes for Double Exponential Integration Formulas.”</span> <em>Numerische Mathematik</em> 111 (4): 631–55. <a href="https://doi.org/10.1007/s00211-008-0195-1">https://doi.org/10.1007/s00211-008-0195-1</a>.
 </div>
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@@ -1075,6 +1059,150 @@ <h3 class="anchored" data-anchor-id="rectangular-map">Rectangular map</h3>
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+  const disableStylesheet = (stylesheets) => {
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diff --git a/docs/2_34/functions-reference/index.html b/docs/2_34/functions-reference/index.html
index 5d1162140..6a1cee9e0 100644
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+++ b/docs/2_34/functions-reference/index.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -491,7 +494,7 @@ <h1 class="title">Stan Functions Reference</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the reference for the functions defined in the Stan math library and available in the Stan programming language.</p>
+<p>This is the reference for the functions defined in the <a href="https://github.com/stan-dev/math">Stan math library</a> and available in the <a href="https://mc-stan.org/">Stan</a> programming language.</p>
 <p>For more information the Stan language and inference engines and how to use Stan for Bayesian inference, see</p>
 <ul>
 <li><p>the <a href="https://mc-stan.org/docs/stan-users-guide/index.html"><strong>Stan User’s Guide</strong></a>. The Stan user’s guide provides example models and programming techniques for coding statistical models in Stan. It also serves as an example-driven introduction to Bayesian modeling and inference:</p></li>
@@ -517,25 +520,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -556,6 +540,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/integer-valued_basic_functions.html b/docs/2_34/functions-reference/integer-valued_basic_functions.html
index b78ab5cbf..72326a0ee 100644
--- a/docs/2_34/functions-reference/integer-valued_basic_functions.html
+++ b/docs/2_34/functions-reference/integer-valued_basic_functions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -633,7 +636,7 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
 <p></p>
 <p><code>int</code> <strong><code>to_int</code></strong><code>(data real x)</code><br></p>
 <p>Return the value <code>x</code> truncated to an integer. This will throw an error if the value of <code>x</code> is too big to represent as a 32-bit signed integer.</p>
-<p>This is similar to <code>trunc</code> (see [Rounding functions]) but the return type is of type <code>int</code>. For example, <code>to_int(3.9)</code> is <code>3</code>, and <code>to_int(-3.9)</code> is <code>-3</code>.</p>
+<p>This is similar to <code>trunc</code> (see <a href="../functions-reference/real-valued_basic_functions.html#rounding-functions">Rounding functions</a>) but the return type is of type <code>int</code>. For example, <code>to_int(3.9)</code> is <code>3</code>, and <code>to_int(-3.9)</code> is <code>-3</code>.</p>
 <small><i>Available since 2.31</i></small>
 <a id="index-entry-04b9411b8a7b904eccd2c168519bf4ccca638602" class="index-entry"></a> <!-- int; to_int; (data real x); -->
 <p></p>
@@ -646,25 +649,6 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -685,6 +669,150 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/mathematical_functions.html b/docs/2_34/functions-reference/mathematical_functions.html
index 439154ee6..0342c4e4e 100644
--- a/docs/2_34/functions-reference/mathematical_functions.html
+++ b/docs/2_34/functions-reference/mathematical_functions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -550,25 +553,6 @@ <h2 class="anchored" data-anchor-id="digamma-appendix">Digamma</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -589,6 +573,150 @@ <h2 class="anchored" data-anchor-id="digamma-appendix">Digamma</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/matrix_operations.html b/docs/2_34/functions-reference/matrix_operations.html
index 332a60138..ea9d1b0ec 100644
--- a/docs/2_34/functions-reference/matrix_operations.html
+++ b/docs/2_34/functions-reference/matrix_operations.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -2014,25 +2017,6 @@ <h2 class="anchored" data-anchor-id="reverse-functions">Reverse functions</h2>
 = 1\)</span> and <span class="math inline">\(\theta_m = 0\)</span> for <span class="math inline">\(m \neq n\)</span>, thus providing a “soft” version of the maximum function.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -2053,6 +2037,150 @@ <h2 class="anchored" data-anchor-id="reverse-functions">Reverse functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/mixed_operations.html b/docs/2_34/functions-reference/mixed_operations.html
index e62055dd9..e15f7e695 100644
--- a/docs/2_34/functions-reference/mixed_operations.html
+++ b/docs/2_34/functions-reference/mixed_operations.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -750,25 +753,6 @@ <h1>Mixed Operations</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -789,6 +773,150 @@ <h1>Mixed Operations</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/multivariate_discrete_distributions.html b/docs/2_34/functions-reference/multivariate_discrete_distributions.html
index 8fcdc5034..1556b6f38 100644
--- a/docs/2_34/functions-reference/multivariate_discrete_distributions.html
+++ b/docs/2_34/functions-reference/multivariate_discrete_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -645,25 +648,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -684,6 +668,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/positive_continuous_distributions.html b/docs/2_34/functions-reference/positive_continuous_distributions.html
index 2cf5d4db9..792bbcf22 100644
--- a/docs/2_34/functions-reference/positive_continuous_distributions.html
+++ b/docs/2_34/functions-reference/positive_continuous_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1070,25 +1073,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1109,6 +1093,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/positive_lower-bounded_distributions.html b/docs/2_34/functions-reference/positive_lower-bounded_distributions.html
index baa17dce5..8439cd3e1 100644
--- a/docs/2_34/functions-reference/positive_lower-bounded_distributions.html
+++ b/docs/2_34/functions-reference/positive_lower-bounded_distributions.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -698,25 +701,6 @@ <h3 class="anchored" data-anchor-id="boundaries">Boundaries</h3>
 Vandekerckhove, Joachim, and Dominik Wabersich. 2014. <span>“The <span>R</span><span>W</span>iener Package: An <span>R</span> Package Providing Distribution Functions for the <span>W</span>iener Diffusion Model.”</span> <em>The R Journal</em> 6/1. <a href="http://journal.r-project.org/archive/2014-1/vandekerckhove-wabersich.pdf">http://journal.r-project.org/archive/2014-1/vandekerckhove-wabersich.pdf</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -737,6 +721,150 @@ <h3 class="anchored" data-anchor-id="boundaries">Boundaries</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
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diff --git a/docs/2_34/functions-reference/real-valued_basic_functions.html b/docs/2_34/functions-reference/real-valued_basic_functions.html
index 1aaebcdbf..8722e902b 100644
--- a/docs/2_34/functions-reference/real-valued_basic_functions.html
+++ b/docs/2_34/functions-reference/real-valued_basic_functions.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1579,25 +1582,6 @@ <h2 class="anchored" data-anchor-id="special-functions">Special functions</h2>
 Wichura, Michael J. 1988. <span>“Algorithm <span>AS</span> 241: The Percentage Points of the Normal Distribution.”</span> <em>Journal of the Royal Statistical Society. Series C (Applied Statistics)</em> 37 (3): 477–84. <a href="http://www.jstor.org/stable/2347330">http://www.jstor.org/stable/2347330</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1618,6 +1602,150 @@ <h2 class="anchored" data-anchor-id="special-functions">Special functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/references.html b/docs/2_34/functions-reference/references.html
index 989877854..b2208f997 100644
--- a/docs/2_34/functions-reference/references.html
+++ b/docs/2_34/functions-reference/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/removed_functions.html b/docs/2_34/functions-reference/removed_functions.html
index d00785035..2da919b82 100644
--- a/docs/2_34/functions-reference/removed_functions.html
+++ b/docs/2_34/functions-reference/removed_functions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -581,25 +584,6 @@ <h2 class="anchored" data-anchor-id="real-arguments-to-logical-operators-operato
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -620,6 +604,150 @@ <h2 class="anchored" data-anchor-id="real-arguments-to-logical-operators-operato
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/simplex_distributions.html b/docs/2_34/functions-reference/simplex_distributions.html
index bc3dfe75e..8b27dbc57 100644
--- a/docs/2_34/functions-reference/simplex_distributions.html
+++ b/docs/2_34/functions-reference/simplex_distributions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -628,25 +631,6 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -667,6 +651,150 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/functions-reference/sparse_matrix_operations.html b/docs/2_34/functions-reference/sparse_matrix_operations.html
index 04372d93e..d40c24f11 100644
--- a/docs/2_34/functions-reference/sparse_matrix_operations.html
+++ b/docs/2_34/functions-reference/sparse_matrix_operations.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -589,25 +592,6 @@ <h3 class="anchored" data-anchor-id="sparse-matrix-multiplication">Sparse matrix
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -628,6 +612,150 @@ <h3 class="anchored" data-anchor-id="sparse-matrix-multiplication">Sparse matrix
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
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diff --git a/docs/2_34/functions-reference/unbounded_continuous_distributions.html b/docs/2_34/functions-reference/unbounded_continuous_distributions.html
index 59cc3496b..35ec48ad4 100644
--- a/docs/2_34/functions-reference/unbounded_continuous_distributions.html
+++ b/docs/2_34/functions-reference/unbounded_continuous_distributions.html
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 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1176,25 +1179,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 Ding, Peng, and Joseph K. Blitzstein. 2018. <span>“On the Gaussian Mixture Representation of the Laplace Distribution.”</span> <em>The American Statistician</em> 72 (2): 172–74. <a href="https://doi.org/10.1080/00031305.2017.1291448">https://doi.org/10.1080/00031305.2017.1291448</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1215,6 +1199,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
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+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+      return false;
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+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+  };
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/unbounded_discrete_distributions.html b/docs/2_34/functions-reference/unbounded_discrete_distributions.html
index 1c20b1ea7..eddc5546b 100644
--- a/docs/2_34/functions-reference/unbounded_discrete_distributions.html
+++ b/docs/2_34/functions-reference/unbounded_discrete_distributions.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -934,25 +937,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-6">Stan functions</h3>
 Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. <em>Bayesian Data Analysis</em>. Third Edition. London: Chapman &amp; Hall / CRC Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
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-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -973,6 +957,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-6">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
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+      }
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
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+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/functions-reference/void_functions.html b/docs/2_34/functions-reference/void_functions.html
index b21010d4f..871d6a0f1 100644
--- a/docs/2_34/functions-reference/void_functions.html
+++ b/docs/2_34/functions-reference/void_functions.html
@@ -38,10 +38,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
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@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
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 </section>
 
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -550,6 +534,150 @@ <h2 class="anchored" data-anchor-id="reject-statement">Reject statement</h2>
     }
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   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
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diff --git a/docs/2_34/index.html b/docs/2_34/index.html
index 9e277b867..fead176e3 100644
--- a/docs/2_34/index.html
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -230,7 +233,7 @@ <h1 class="title">Stan Documentation</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official documentation for Stan.</p>
+<p>This is the official documentation for <a href="https://mc-stan.org/">Stan</a>.</p>
 <ul>
 <li><p>The <a href="stan-users-guide/index.html"><strong>Stan User’s Guide</strong></a> (<a href="https://mc-stan.org/docs/2_34/stan-users-guide-2_34.pdf">pdf</a>) provides example models and programming techniques for coding statistical models in Stan.</p></li>
 <li><p>The <a href="reference-manual/index.html"><strong>Stan Reference Manual</strong></a> (<a href="https://mc-stan.org/docs/2_34/reference-manual-2_34.pdf">pdf</a>) specifies the Stan programming language and inference algorithms.</p></li>
@@ -256,25 +259,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
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@@ -295,6 +279,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
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diff --git a/docs/2_34/reference-manual/analysis.html b/docs/2_34/reference-manual/analysis.html
index 18c53e280..6c7dc7af4 100644
--- a/docs/2_34/reference-manual/analysis.html
+++ b/docs/2_34/reference-manual/analysis.html
@@ -58,10 +58,12 @@
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
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@@ -119,7 +121,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -713,25 +716,6 @@ <h3 class="unnumbered anchored" data-anchor-id="thinning-samples">Thinning sampl
 <li id="fn3"><p>The <code>lp__</code> value also represents the potential energy in the Hamiltonian system and is rate bounded by the randomly supplied kinetic energy each iteration, which follows a Chi-square distribution in the number of parameters.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -752,6 +736,150 @@ <h3 class="unnumbered anchored" data-anchor-id="thinning-samples">Thinning sampl
     }
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+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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+    const a = window.document.createElement('a');
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+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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diff --git a/docs/2_34/reference-manual/blocks.html b/docs/2_34/reference-manual/blocks.html
index 83b5370a7..d7c0d2ae2 100644
--- a/docs/2_34/reference-manual/blocks.html
+++ b/docs/2_34/reference-manual/blocks.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -648,7 +651,7 @@ <h3 class="unnumbered anchored" data-anchor-id="variable-read-write-and-definiti
 <td style="text-align: left;">write / sample</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>\slshape (initialization)</code></td>
+<td style="text-align: left;"><em><code>(initialization)</code></em></td>
 <td style="text-align: center;">n/a</td>
 <td style="text-align: left;">read, transform / chain</td>
 </tr>
@@ -802,7 +805,7 @@ <h2 class="anchored" data-anchor-id="statistical-variable-taxonomy">Statistical
 <span id="cb2-29"><a href="#cb2-29" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> variance_y;       <span class="co">// derived quantity (transform)</span></span>
 <span id="cb2-30"><a href="#cb2-30" aria-hidden="true" tabindex="-1"></a>  variance_y = sigma_y * sigma_y;</span>
 <span id="cb2-31"><a href="#cb2-31" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>In this example, <code>y[N]</code> is a modeled data vector. Although it is specified in the <code>data</code> block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.</p>
+<p>In this example, <code>y</code> is an array of modeled data. Although it is specified in the <code>data</code> block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.</p>
 <p>The variable <code>N</code> is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.</p>
 <p>The other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the <code>data</code> block may be specified on a per-model execution basis as part of the data read. In the above model, <code>mu_mu</code> and <code>sigma_mu</code> are configurable unmodeled parameters.</p>
 <p>Unmodeled parameters that are hard coded in the model must be declared in the <code>transformed data</code> block. For example, the unmodeled parameters <code>alpha</code> and <code>beta</code> are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the <code>data</code> block, like the variables <code>mu_mu</code> and <code>sigma_mu</code>.</p>
@@ -934,25 +937,6 @@ <h2 class="anchored" data-anchor-id="program-block-generated-quantities">Program
 <li id="fn3"><p>With multiple threads, or even running chains sequentially in a single thread, data could be read only once per set of chains. Stan was designed to be thread safe and future versions will provide a multithreading option for Markov chains.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -973,6 +957,150 @@ <h2 class="anchored" data-anchor-id="program-block-generated-quantities">Program
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
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+    return window.location.protocol === 'file:';
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+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/reference-manual/comments.html b/docs/2_34/reference-manual/comments.html
index 6fd2ba64e..753ceacf2 100644
--- a/docs/2_34/reference-manual/comments.html
+++ b/docs/2_34/reference-manual/comments.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
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@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -469,25 +472,6 @@ <h2 class="anchored" data-anchor-id="bracketed-comments">Bracketed comments</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -508,6 +492,150 @@ <h2 class="anchored" data-anchor-id="bracketed-comments">Bracketed comments</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/deprecations.html b/docs/2_34/reference-manual/deprecations.html
index 6f5108bfe..b9172cc10 100644
--- a/docs/2_34/reference-manual/deprecations.html
+++ b/docs/2_34/reference-manual/deprecations.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -452,7 +455,7 @@ <h2 id="toc-title">On this page</h2>
 <section id="deprecated-features" class="level1">
 <h1>Deprecated Features</h1>
 <p>This appendix lists currently deprecated functionality along with how to replace it.</p>
-<p>Starting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Features]. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> many of these on the behalf of the user for at least one version after they are removed.</p>
+<p>Starting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in <a href="../reference-manual/removals.html">Removed Features</a>. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> many of these on the behalf of the user for at least one version after they are removed.</p>
 <p>Any feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).</p>
 <section id="lkj_cov-distribution" class="level2">
 <h2 class="anchored" data-anchor-id="lkj_cov-distribution"><code>lkj_cov</code> distribution</h2>
@@ -482,25 +485,6 @@ <h2 class="anchored" data-anchor-id="deprecated-functions">Deprecated Functions<
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -521,6 +505,150 @@ <h2 class="anchored" data-anchor-id="deprecated-functions">Deprecated Functions<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/diagnostics.html b/docs/2_34/reference-manual/diagnostics.html
index 5159736ae..aece8cae8 100644
--- a/docs/2_34/reference-manual/diagnostics.html
+++ b/docs/2_34/reference-manual/diagnostics.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -494,7 +497,7 @@ <h3 class="unnumbered anchored" data-anchor-id="unconstrained-scale">Unconstrain
 </section>
 <section id="includes-jacobian" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="includes-jacobian">Includes Jacobian</h3>
-<p>The log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.</p>
+<p>The log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.</p>
 </section>
 </section>
 <section id="configuration-options" class="level2">
@@ -510,25 +513,6 @@ <h2 class="anchored" data-anchor-id="speed-warning-and-data-trimming">Speed warn
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -549,6 +533,150 @@ <h2 class="anchored" data-anchor-id="speed-warning-and-data-trimming">Speed warn
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
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+      const toggle = toggles[i];
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+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
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+        manageTransitions("body", true);
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+    }
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+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
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+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  // Switch to dark mode if need be
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+    toggleColorMode(true);
+  } else {
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diff --git a/docs/2_34/reference-manual/encoding.html b/docs/2_34/reference-manual/encoding.html
index 21dea1990..f4d855dae 100644
--- a/docs/2_34/reference-manual/encoding.html
+++ b/docs/2_34/reference-manual/encoding.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -70,7 +72,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -444,25 +447,6 @@ <h2 class="anchored" data-anchor-id="string-literals">String literals</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -483,6 +467,150 @@ <h2 class="anchored" data-anchor-id="string-literals">String literals</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
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+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
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+        manageTransitions("body", true);
+      }, 40);  
+    }
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+  const hasAlternateSentinel = () => {  
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+  const setStyleSentinel = (alternate) => {
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+  const darkModeDefault = false;
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+  // Dark / light mode switch
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+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/reference-manual/execution.html b/docs/2_34/reference-manual/execution.html
index 157e155e9..b553daa79 100644
--- a/docs/2_34/reference-manual/execution.html
+++ b/docs/2_34/reference-manual/execution.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -543,7 +546,7 @@ <h3 class="unnumbered anchored" data-anchor-id="define-transformed-data">Define
 </section>
 <section id="initialization" class="level2">
 <h2 class="anchored" data-anchor-id="initialization">Initialization</h2>
-<p>Initialization is the same for sampling, optimization, and diagnosis</p>
+<p>Initialization is the same for all of Stan’s algorithms.</p>
 <section id="user-supplied-initial-values" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="user-supplied-initial-values">User-supplied initial values</h3>
 <p>If there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.</p>
@@ -641,25 +644,6 @@ <h3 class="unnumbered anchored" data-anchor-id="write">Write</h3>
 <li id="fn2"><p>In the R version of Stan, the values may either be written to a CSV file or directly back to R’s memory.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -680,6 +664,150 @@ <h3 class="unnumbered anchored" data-anchor-id="write">Write</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
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+          theme: newTheme
+        }
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+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/expressions.html b/docs/2_34/reference-manual/expressions.html
index 65a57a8e7..a20b159ab 100644
--- a/docs/2_34/reference-manual/expressions.html
+++ b/docs/2_34/reference-manual/expressions.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -622,8 +625,7 @@ <h4 class="unnumbered anchored" data-anchor-id="reserved-words-from-stan-languag
 cholesky_factor_corr, cholesky_factor_cov,
 corr_matrix, cov_matrix, array</code></pre>
 <p>The following built in functions are also reserved and cannot be used as variable names:</p>
-<pre><code>print, reject, profile, get_lp, increment_log_prob,
-target</code></pre>
+<pre><code>print, reject, profile, target</code></pre>
 <p>The following block identifiers are reserved and cannot be used as variable names:</p>
 <pre><code>functions, model, data, parameters, quantities,
 transformed, generated</code></pre>
@@ -1384,6 +1386,10 @@ <h3 class="unnumbered anchored" data-anchor-id="promotion">Promotion</h3>
 <li><code>array[...] U</code> may be promoted to <code>array[...] T</code> if <code>U</code> can be promoted to <code>T</code>.</li>
 </ol>
 <p>For example, this means <code>array[,] int</code> may be used where <code>array [,] real</code> or <code>array [,] complex</code> is required; as another example, <code>array[] real</code> may be used anywhere <code>array[] complex</code> is required.</p>
+<p>Tuples have the natural extension of the above rules, applied to all sub-types at once</p>
+<ol start="8" type="1">
+<li>A <code>tuple(U1, ..., UN)</code> may be promoted to a <code>tuple(T1, ..., TN)</code> if every <code>Un</code> can be promoted to <code>Tn</code> for <code>n</code> in <code>1:N</code></li>
+</ol>
 <section id="literals" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="literals">Literals</h4>
 <p>An integer literal expression such as <code>42</code> is of type <code>int</code>. Real literals such as <code>42.0</code> are of type <code>real</code>. Imaginary literals such as <code>-17i</code> are of type <code>complex</code>. the expression <code>7 - 2i</code> acts like a complex literal, but technically it combines a real literal <code>7</code> and an imaginary literal <code>2i</code> through subtraction.</p>
@@ -1619,25 +1625,6 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-problems-with-derivat
 <li id="fn2"><p>Internally, they are implemented as their own expression types because Stan doesn’t have object-level functional types (yet).<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1658,6 +1645,150 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-problems-with-derivat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/includes.html b/docs/2_34/reference-manual/includes.html
index 3dd17252c..fdaa12503 100644
--- a/docs/2_34/reference-manual/includes.html
+++ b/docs/2_34/reference-manual/includes.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -501,11 +504,21 @@ <h3 class="unnumbered anchored" data-anchor-id="comments-after-includes">Comment
 </section>
 <section id="recursive-includes" class="level2">
 <h2 class="anchored" data-anchor-id="recursive-includes">Recursive includes</h2>
-<p>Recursive includes will be ignored. For example, suppose <code>a.stan</code> contains</p>
+<p>Recursive includes will lead to a compiler error. For example, suppose <code>a.stan</code> contains</p>
 <div class="sourceCode" id="cb6"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co">#include b.stan</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>and <code>b.stan</code> contains</p>
 <div class="sourceCode" id="cb7"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co">#include a.stan</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>The result of processing this file will be empty, because <code>a.stan</code> will include <code>b.stan</code>, from which the include of <code>a.stan</code> is ignored and a warning printed.</p>
+<p>This will result in an error explaining the circular dependency:</p>
+<pre><code>Syntax error in './b.stan', line 1, column 0, included from
+'./a.stan', line 1, column 0, included from
+'./b.stan', line 1, column 0, included from
+'a.stan', line 1, column 0, include error:
+   -------------------------------------------------
+     1:  #include a.stan
+         ^
+   -------------------------------------------------
+
+File a.stan recursively included itself.</code></pre>
 </section>
 <section id="include-paths" class="level2">
 <h2 class="anchored" data-anchor-id="include-paths">Include paths</h2>
@@ -520,25 +533,6 @@ <h3 class="unnumbered anchored" data-anchor-id="slashes-in-include-paths">Slashe
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -559,6 +553,150 @@ <h3 class="unnumbered anchored" data-anchor-id="slashes-in-include-paths">Slashe
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/index.html b/docs/2_34/reference-manual/index.html
index dc1eb0da7..146d9e4d4 100644
--- a/docs/2_34/reference-manual/index.html
+++ b/docs/2_34/reference-manual/index.html
@@ -37,10 +37,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -421,7 +424,7 @@ <h1 class="title">Stan Reference Manual</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official reference manual for Stan’s <em>programming language</em> for coding probability models, <em>inference algorithms</em> for fitting models and making predictions, and <em>posterior analysis</em> tools for evaluating the results. This manual applies to all Stan interfaces.</p>
+<p>This is the official reference manual for <a href="https://mc-stan.org/">Stan</a>’s <em>programming language</em> for coding probability models, <em>inference algorithms</em> for fitting models and making predictions, and <em>posterior analysis</em> tools for evaluating the results. This manual applies to all Stan interfaces.</p>
 <p>The first part of the reference manual provides a full specification of the Stan programming language. The language is responsible for defining a log density function conditioned on data. Typically, this is a Bayesian posterior, but it may also be a penalized likelihood function. The second part of the manual specifies the inference algorithms and posterior inference tools. The third part provides auxiliary information about the use of Stan.</p>
 <p><a href="https://mc-stan.org/docs/2_34/reference-manual-2_34.pdf">Download the pdf version of this manual</a>.</p>
 <section id="copyright-and-trademark" class="level4 unnumbered">
@@ -443,25 +446,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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@@ -482,6 +466,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
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diff --git a/docs/2_34/reference-manual/laplace.html b/docs/2_34/reference-manual/laplace.html
index 6758dc2a1..7d5f03a6b 100644
--- a/docs/2_34/reference-manual/laplace.html
+++ b/docs/2_34/reference-manual/laplace.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -99,7 +101,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -214,6 +216,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
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@@ -456,25 +459,6 @@ <h1>Laplace Approximation</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -495,6 +479,150 @@ <h1>Laplace Approximation</h1>
     }
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   toggleBodyColorPrimary();  
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diff --git a/docs/2_34/reference-manual/licenses.html b/docs/2_34/reference-manual/licenses.html
index 4d05cdac4..c5c5b7bc7 100644
--- a/docs/2_34/reference-manual/licenses.html
+++ b/docs/2_34/reference-manual/licenses.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -69,7 +71,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -406,6 +409,7 @@ <h2 id="toc-title">On this page</h2>
   <li><a href="#boost-license" id="toc-boost-license" class="nav-link" data-scroll-target="#boost-license">Boost license</a></li>
   <li><a href="#eigen-license" id="toc-eigen-license" class="nav-link" data-scroll-target="#eigen-license">Eigen license</a></li>
   <li><a href="#sundials-license" id="toc-sundials-license" class="nav-link" data-scroll-target="#sundials-license">SUNDIALS license</a></li>
+  <li><a href="#threaded-building-blocks-tbb-license" id="toc-threaded-building-blocks-tbb-license" class="nav-link" data-scroll-target="#threaded-building-blocks-tbb-license">Threaded Building Blocks (TBB) License</a></li>
   <li><a href="#google-test-license" id="toc-google-test-license" class="nav-link" data-scroll-target="#google-test-license">Google test license</a></li>
   </ul></li>
   </ul>
@@ -454,6 +458,14 @@ <h2 class="anchored" data-anchor-id="sundials-license">SUNDIALS license</h2>
 </ul>
 <p>The copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.</p>
 </section>
+<section id="threaded-building-blocks-tbb-license" class="level2">
+<h2 class="anchored" data-anchor-id="threaded-building-blocks-tbb-license">Threaded Building Blocks (TBB) License</h2>
+<p>Stan uses the Threaded Building Blocks (TBB) library for parallel computations. TBB is distributed under the</p>
+<ul>
+<li><a href="https://opensource.org/license/apache-2-0">Apache License, version 2</a></li>
+</ul>
+<p>The copyright of TBB is owned by Intel Corporation.</p>
+</section>
 <section id="google-test-license" class="level2">
 <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h2>
 <p>Stan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the</p>
@@ -473,25 +485,6 @@ <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h
 <li id="fn1"><p>Universities or companies often own the copyright of computer programs developed by their employees.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -512,6 +505,150 @@ <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h
     }
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+  const disableStylesheet = (stylesheets) => {
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+      stylesheet.rel = 'stylesheet';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/mcmc.html b/docs/2_34/reference-manual/mcmc.html
index c929f089b..665fcbf27 100644
--- a/docs/2_34/reference-manual/mcmc.html
+++ b/docs/2_34/reference-manual/mcmc.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -929,25 +932,6 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-and-eliminating-diver
 <li id="fn5"><p>The current default threshold is a factor of <span class="math inline">\(10^3\)</span>, whereas when the leapfrog integrator is working properly, the divergences will be around <span class="math inline">\(10^{-7}\)</span> and do not compound due to the symplectic nature of the leapfrog integrator.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -968,6 +952,150 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-and-eliminating-diver
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/optimization.html b/docs/2_34/reference-manual/optimization.html
index 0678acdd2..06c273ef2 100644
--- a/docs/2_34/reference-manual/optimization.html
+++ b/docs/2_34/reference-manual/optimization.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -591,25 +594,6 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-vs.-unconstrained-pa
 Nocedal, Jorge, and Stephen J. Wright. 2006. <em>Numerical Optimization</em>. Second. Berlin: Springer-<span>V</span>erlag.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -630,6 +614,150 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-vs.-unconstrained-pa
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/pathfinder.html b/docs/2_34/reference-manual/pathfinder.html
index f54c256bc..2e23c273d 100644
--- a/docs/2_34/reference-manual/pathfinder.html
+++ b/docs/2_34/reference-manual/pathfinder.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -90,7 +92,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -205,6 +207,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -445,25 +448,6 @@ <h1>Pathfinder</h1>
 Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -484,6 +468,150 @@ <h1>Pathfinder</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/references.html b/docs/2_34/reference-manual/references.html
index 14c964857..220aec734 100644
--- a/docs/2_34/reference-manual/references.html
+++ b/docs/2_34/reference-manual/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/removals.html b/docs/2_34/reference-manual/removals.html
index 1944fb1cb..ec06d517b 100644
--- a/docs/2_34/reference-manual/removals.html
+++ b/docs/2_34/reference-manual/removals.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -581,25 +584,6 @@ <h2 class="anchored" data-anchor-id="real-values-in-conditionals">Real values in
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -620,6 +604,150 @@ <h2 class="anchored" data-anchor-id="real-values-in-conditionals">Real values in
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
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+    i.classList.add('bi');
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diff --git a/docs/2_34/reference-manual/reproducibility.html b/docs/2_34/reference-manual/reproducibility.html
index 43ff31826..6121adea7 100644
--- a/docs/2_34/reference-manual/reproducibility.html
+++ b/docs/2_34/reference-manual/reproducibility.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "copy-button": false,
@@ -90,7 +92,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -205,6 +207,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -460,25 +463,6 @@ <h1>Reproducibility</h1>
 Corden, Martyn J., and David Kreitzer. 2014. <span>“Consistency of Floating-Point Results Using the <span>I</span>ntel Compiler or <span>W</span>hy Doesn’t My Application Always Give the Same Answer?”</span> Intel Corporation. <a href="https://software.intel.com/en-us/articles/consistency-of-floating-point-results-using-the-intel-compiler">https://software.intel.com/en-us/articles/consistency-of-floating-point-results-using-the-intel-compiler</a>.
 </div>
 </div></section></div></main> <!-- /main -->
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@@ -499,6 +483,150 @@ <h1>Reproducibility</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+      stylesheet.rel = 'stylesheet';
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+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+      toggleBodyColorPrimary();
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+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
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+  // Dark / light mode switch
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+    // Read the current dark / light value 
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+  };
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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+    i.classList.add('bi');
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diff --git a/docs/2_34/reference-manual/statements.html b/docs/2_34/reference-manual/statements.html
index 143478c89..4d4bb44c4 100644
--- a/docs/2_34/reference-manual/statements.html
+++ b/docs/2_34/reference-manual/statements.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -562,10 +565,11 @@ <h3 class="anchored" data-anchor-id="promotion">Promotion</h3>
 </section>
 <section id="lvalue-summary" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="lvalue-summary">Lvalue summary</h3>
-<p>The expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,</p>
+<p>The expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,</p>
 <ul>
 <li>a variable, or</li>
-<li>a variable with one or more indices.</li>
+<li>a variable with one or more indices, or</li>
+<li>a comma separated list of lvalues surrounded by <code>(</code> and <code>)</code></li>
 </ul>
 <p>To be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.</p>
 </section>
@@ -1061,21 +1065,20 @@ <h3 class="unnumbered anchored" data-anchor-id="local-variable-declarations">Loc
 </section>
 <section id="no-constraints-on-local-variables" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="no-constraints-on-local-variables">No constraints on local variables</h3>
-<p>Local variables may not have constraints on their declaration. The only types that may be used are</p>
-<pre><code>int, real, vector[K], row_vector[K], matrix[M, N].</code></pre>
+<p>Local variables may not have constraints on their declaration. The only types that may be used are listed in the <a href="../reference-manual/types.html#id:constrained-types.figure">types table</a> under “local”.</p>
 </section>
 <section id="blocks-within-blocks" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="blocks-within-blocks">Blocks within blocks</h3>
 <p>A block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following</p>
-<div class="sourceCode" id="cb55"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (m <span class="cf">in</span> <span class="dv">1</span>:M) {</span>
-<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>  {</span>
-<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>     <span class="dt">int</span> n = <span class="dv">2</span> * m;</span>
-<span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>     sum += n;</span>
-<span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb55-7"><a href="#cb55-7" aria-hidden="true" tabindex="-1"></a>    sum += x[m, n];</span>
-<span id="cb55-8"><a href="#cb55-8" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb55-9"><a href="#cb55-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb54"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (m <span class="cf">in</span> <span class="dv">1</span>:M) {</span>
+<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a>  {</span>
+<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a>     <span class="dt">int</span> n = <span class="dv">2</span> * m;</span>
+<span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a>     sum += n;</span>
+<span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb54-6"><a href="#cb54-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb54-7"><a href="#cb54-7" aria-hidden="true" tabindex="-1"></a>    sum += x[m, n];</span>
+<span id="cb54-8"><a href="#cb54-8" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb54-9"><a href="#cb54-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The variable declaration <code>int n;</code> is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of <code>n</code> are unrelated.</p>
 </section>
 </section>
@@ -1086,56 +1089,56 @@ <h2 class="anchored" data-anchor-id="break-continue-statements">Break and contin
 <section id="break-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="break-statements">Break statements</h3>
 <p>When a <code>break</code> statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:</p>
-<div class="sourceCode" id="cb56"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (<span class="dv">1</span>) {</span>
-<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (n &lt; <span class="dv">0</span>) {</span>
-<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">break</span>;</span>
-<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
-<span id="cb56-6"><a href="#cb56-6" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
-<span id="cb56-7"><a href="#cb56-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb55"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (<span class="dv">1</span>) {</span>
+<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (n &lt; <span class="dv">0</span>) {</span>
+<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">break</span>;</span>
+<span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
+<span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
+<span id="cb55-7"><a href="#cb55-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The <code>while~(1)</code> loop is a “forever” loop, because <code>1</code> is the true value, so the test always succeeds. Within the loop, if the value of <code>n</code> is less than 0, the loop terminates, otherwise it executes <code>foo(n)</code> and then decrements <code>n</code>. The statement above does exactly the same thing as</p>
-<div class="sourceCode" id="cb57"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (n &gt;= <span class="dv">0</span>) {</span>
-<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
-<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
-<span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb56"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (n &gt;= <span class="dv">0</span>) {</span>
+<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
+<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
+<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This case is simply illustrative of the behavior; it is not a case where a <code>break</code> simplifies the loop.</p>
 </section>
 <section id="continue-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="continue-statements">Continue statements</h3>
 <p>The <code>continue</code> statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array <code>x</code>,</p>
+<div class="sourceCode" id="cb57"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
+<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
+<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
+<span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &lt;= <span class="dv">0</span>) {</span>
+<span id="cb57-5"><a href="#cb57-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">continue</span>;</span>
+<span id="cb57-6"><a href="#cb57-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb57-7"><a href="#cb57-7" aria-hidden="true" tabindex="-1"></a>  sum += x[n];</span>
+<span id="cb57-8"><a href="#cb57-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>When the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an <code>x[n]</code> less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,</p>
 <div class="sourceCode" id="cb58"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
 <span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
 <span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
-<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &lt;= <span class="dv">0</span>) {</span>
-<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">continue</span>;</span>
+<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &gt; <span class="dv">0</span>) {</span>
+<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a>    sum += x[n];</span>
 <span id="cb58-6"><a href="#cb58-6" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a>  sum += x[n];</span>
-<span id="cb58-8"><a href="#cb58-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>When the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an <code>x[n]</code> less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,</p>
-<div class="sourceCode" id="cb59"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
-<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
-<span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
-<span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &gt; <span class="dv">0</span>) {</span>
-<span id="cb59-5"><a href="#cb59-5" aria-hidden="true" tabindex="-1"></a>    sum += x[n];</span>
-<span id="cb59-6"><a href="#cb59-6" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb59-7"><a href="#cb59-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>While the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.</p>
 </section>
 <section id="breaking-and-continuing-nested-loops" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="breaking-and-continuing-nested-loops">Breaking and continuing nested loops</h3>
 <p>If there is a loop nested within a loop, a <code>break</code> or <code>continue</code> statement only breaks out of the inner loop. So</p>
-<div class="sourceCode" id="cb60"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (cond1) {</span>
-<span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb60-3"><a href="#cb60-3" aria-hidden="true" tabindex="-1"></a>  <span class="cf">while</span> (cond2) {</span>
-<span id="cb60-4"><a href="#cb60-4" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
-<span id="cb60-5"><a href="#cb60-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">if</span> (cond3) {</span>
-<span id="cb60-6"><a href="#cb60-6" aria-hidden="true" tabindex="-1"></a>      <span class="cf">break</span>;</span>
-<span id="cb60-7"><a href="#cb60-7" aria-hidden="true" tabindex="-1"></a>    }</span>
-<span id="cb60-8"><a href="#cb60-8" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
-<span id="cb60-9"><a href="#cb60-9" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb60-10"><a href="#cb60-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// execution continues here after break</span></span>
-<span id="cb60-11"><a href="#cb60-11" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb60-12"><a href="#cb60-12" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb59"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (cond1) {</span>
+<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a>  <span class="cf">while</span> (cond2) {</span>
+<span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
+<span id="cb59-5"><a href="#cb59-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">if</span> (cond3) {</span>
+<span id="cb59-6"><a href="#cb59-6" aria-hidden="true" tabindex="-1"></a>      <span class="cf">break</span>;</span>
+<span id="cb59-7"><a href="#cb59-7" aria-hidden="true" tabindex="-1"></a>    }</span>
+<span id="cb59-8"><a href="#cb59-8" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
+<span id="cb59-9"><a href="#cb59-9" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb59-10"><a href="#cb59-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// execution continues here after break</span></span>
+<span id="cb59-11"><a href="#cb59-11" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb59-12"><a href="#cb59-12" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>If the break is triggered by <code>cond3</code> being true, execution will continue after the nested loop.</p>
 <p>As with break statements, continue statements go back to the top of the most deeply nested loop in which the <code>continue</code> appears.</p>
 <p>Although break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.</p>
@@ -1144,32 +1147,32 @@ <h3 class="unnumbered anchored" data-anchor-id="breaking-and-continuing-nested-l
 <section id="print-statements.section" class="level2">
 <h2 class="anchored" data-anchor-id="print-statements.section">Print statements</h2>
 <p>Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.</p>
-<div class="sourceCode" id="cb61"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) { <span class="kw">print</span>(<span class="st">"loop iteration: "</span>, n); ... }</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb60"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) { <span class="kw">print</span>(<span class="st">"loop iteration: "</span>, n); ... }</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression <code>n</code>, followed by a new line.</p>
 <section id="print-content" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="print-content">Print content</h3>
 <p>The text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.</p>
 <p>Printing a simple <code>real</code> or <code>int</code> typed variable always prints the variable’s value.<a href="#fn5" class="footnote-ref" id="fnref5" role="doc-noteref"><sup>5</sup></a></p>
 <p>For array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as</p>
-<div class="sourceCode" id="cb62"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb61"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>and a <span class="math inline">\(2 \times 3\)</span>-matrix as</p>
-<div class="sourceCode" id="cb63"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>], [<span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>]]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb62"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>], [<span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>]]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>Complex numbers print as pairs. For example, the pair of statements</p>
-<div class="sourceCode" id="cb64"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="dt">complex</span> z = to_complex(<span class="fl">1.2</span>, -<span class="fl">3.5</span>);</span>
-<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(z)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb63"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="dt">complex</span> z = to_complex(<span class="fl">1.2</span>, -<span class="fl">3.5</span>);</span>
+<span id="cb63-2"><a href="#cb63-2" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(z)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>will print as <code>(1.2,-3.5)</code>, with no space after the comma or within the parentheses.</p>
 <p>Printing a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.</p>
-<div class="sourceCode" id="cb65"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="kw">transformed data</span> {</span>
-<span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">matrix</span>[<span class="dv">2</span>, <span class="dv">2</span>] u;</span>
-<span id="cb65-3"><a href="#cb65-3" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">1</span>, <span class="dv">1</span>] = <span class="fl">1.0</span>;   u[<span class="dv">1</span>, <span class="dv">2</span>] = <span class="fl">4.0</span>;</span>
-<span id="cb65-4"><a href="#cb65-4" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">2</span>, <span class="dv">1</span>] = <span class="fl">9.0</span>;   u[<span class="dv">2</span>, <span class="dv">2</span>] = <span class="fl">16.0</span>;</span>
-<span id="cb65-5"><a href="#cb65-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:<span class="dv">2</span>) {</span>
-<span id="cb65-6"><a href="#cb65-6" aria-hidden="true" tabindex="-1"></a>    <span class="kw">print</span>(<span class="st">"u["</span>, n, <span class="st">"] = "</span>, u[n]);</span>
-<span id="cb65-7"><a href="#cb65-7" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb65-8"><a href="#cb65-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb64"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="kw">transformed data</span> {</span>
+<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">matrix</span>[<span class="dv">2</span>, <span class="dv">2</span>] u;</span>
+<span id="cb64-3"><a href="#cb64-3" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">1</span>, <span class="dv">1</span>] = <span class="fl">1.0</span>;   u[<span class="dv">1</span>, <span class="dv">2</span>] = <span class="fl">4.0</span>;</span>
+<span id="cb64-4"><a href="#cb64-4" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">2</span>, <span class="dv">1</span>] = <span class="fl">9.0</span>;   u[<span class="dv">2</span>, <span class="dv">2</span>] = <span class="fl">16.0</span>;</span>
+<span id="cb64-5"><a href="#cb64-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:<span class="dv">2</span>) {</span>
+<span id="cb64-6"><a href="#cb64-6" aria-hidden="true" tabindex="-1"></a>    <span class="kw">print</span>(<span class="st">"u["</span>, n, <span class="st">"] = "</span>, u[n]);</span>
+<span id="cb64-7"><a href="#cb64-7" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb64-8"><a href="#cb64-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This print statement executes twice, printing the following two lines of output.</p>
-<div class="sourceCode" id="cb66"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">1</span>] = [<span class="dv">1</span>, <span class="dv">4</span>]</span>
-<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">2</span>] = [<span class="dv">9</span>, <span class="dv">16</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb65"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">1</span>] = [<span class="dv">1</span>, <span class="dv">4</span>]</span>
+<span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">2</span>] = [<span class="dv">9</span>, <span class="dv">16</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </section>
 <section id="non-void-input" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="non-void-input">Non-void input</h3>
@@ -1188,12 +1191,13 @@ <h3 class="unnumbered anchored" data-anchor-id="string-literals">String literals
 <h3 class="unnumbered anchored" data-anchor-id="debug-by-print">Debug by <code>print</code></h3>
 <p>Because Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.</p>
 <p>It is particularly useful to print the value of the target log density accumulator (through the <code>target()</code> function), as in the following example.</p>
-<div class="sourceCode" id="cb67"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="dt">vector</span>[<span class="dv">2</span>] y;</span>
-<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>y[<span class="dv">1</span>] = <span class="dv">1</span>;</span>
-<span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density before ="</span>, target());</span>
-<span id="cb67-4"><a href="#cb67-4" aria-hidden="true" tabindex="-1"></a>y ~ normal(<span class="dv">0</span>,<span class="dv">1</span>);  <span class="co">// bug!  y[2] not defined</span></span>
-<span id="cb67-5"><a href="#cb67-5" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density after ="</span>, target());</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb66"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="dt">vector</span>[<span class="dv">2</span>] y;</span>
+<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>y[<span class="dv">1</span>] = <span class="dv">1</span>;</span>
+<span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density before ="</span>, target());</span>
+<span id="cb66-4"><a href="#cb66-4" aria-hidden="true" tabindex="-1"></a>y ~ normal(<span class="dv">0</span>,<span class="dv">1</span>);  <span class="co">// bug!  y[2] not defined</span></span>
+<span id="cb66-5"><a href="#cb66-5" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density after ="</span>, target());</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The example has a bug in that <code>y[2]</code> is not defined before the vector <code>y</code> is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).</p>
+<p>Note that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.</p>
 </section>
 </section>
 <section id="reject-statements.section" class="level2">
@@ -1201,9 +1205,9 @@ <h2 class="anchored" data-anchor-id="reject-statements.section">Reject statement
 <p>The Stan <code>reject</code> statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.</p>
 <p>Like the <code>print</code> statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.</p>
 <p>Reject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable <code>x</code>’s value is negative.</p>
-<div class="sourceCode" id="cb68"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> (x &lt; <span class="dv">0</span>) {</span>
-<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">reject</span>(<span class="st">"x must not be negative; found x="</span>, x);</span>
-<span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb67"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> (x &lt; <span class="dv">0</span>) {</span>
+<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">reject</span>(<span class="st">"x must not be negative; found x="</span>, x);</span>
+<span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <section id="behavior-of-reject-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="behavior-of-reject-statements">Behavior of reject statements</h3>
 <p>Reject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the <code>normal_lpdf</code> function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.</p>
@@ -1226,26 +1230,26 @@ <h4 class="unnumbered anchored" data-anchor-id="recoverable-rejection-contexts">
 <section id="rejection-is-not-for-constraints" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="rejection-is-not-for-constraints">Rejection is not for constraints</h3>
 <p>Rejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.</p>
-<div class="sourceCode" id="cb69"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
-<span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> a;</span>
-<span id="cb69-3"><a href="#cb69-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a&gt; b;</span>
-<span id="cb69-4"><a href="#cb69-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a, <span class="kw">upper</span>=b&gt; theta;</span>
-<span id="cb69-5"><a href="#cb69-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb69-6"><a href="#cb69-6" aria-hidden="true" tabindex="-1"></a>}</span>
-<span id="cb69-7"><a href="#cb69-7" aria-hidden="true" tabindex="-1"></a><span class="kw">model</span> {</span>
-<span id="cb69-8"><a href="#cb69-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// **wrong** needs explicit truncation</span></span>
-<span id="cb69-9"><a href="#cb69-9" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span>
-<span id="cb69-10"><a href="#cb69-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb69-11"><a href="#cb69-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb68"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
+<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> a;</span>
+<span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a&gt; b;</span>
+<span id="cb68-4"><a href="#cb68-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a, <span class="kw">upper</span>=b&gt; theta;</span>
+<span id="cb68-5"><a href="#cb68-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb68-6"><a href="#cb68-6" aria-hidden="true" tabindex="-1"></a>}</span>
+<span id="cb68-7"><a href="#cb68-7" aria-hidden="true" tabindex="-1"></a><span class="kw">model</span> {</span>
+<span id="cb68-8"><a href="#cb68-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// **wrong** needs explicit truncation</span></span>
+<span id="cb68-9"><a href="#cb68-9" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span>
+<span id="cb68-10"><a href="#cb68-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb68-11"><a href="#cb68-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This program is wrong because its truncation bounds on <code>theta</code> depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.</p>
-<div class="sourceCode" id="cb70"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>) <span class="kw">T</span>[a, b];</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb69"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>) <span class="kw">T</span>[a, b];</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over <span class="math inline">\([a ,b]\)</span>; in these simple univariate cases, we can overcome that with the <code>T[ , ]</code> notation, which essentially divides by whatever the prior integrates to over <span class="math inline">\([a, b]\)</span>.</p>
 <p>This problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.</p>
-<div class="sourceCode" id="cb71"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (theta &lt; a || theta &gt; b) {</span>
-<span id="cb71-2"><a href="#cb71-2" aria-hidden="true" tabindex="-1"></a>    <span class="kw">reject</span>(<span class="st">"theta not in (a, b)"</span>);</span>
-<span id="cb71-3"><a href="#cb71-3" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb71-4"><a href="#cb71-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// still **wrong**, needs T[a,b]</span></span>
-<span id="cb71-5"><a href="#cb71-5" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb70"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (theta &lt; a || theta &gt; b) {</span>
+<span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a>    <span class="kw">reject</span>(<span class="st">"theta not in (a, b)"</span>);</span>
+<span id="cb70-3"><a href="#cb70-3" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb70-4"><a href="#cb70-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// still **wrong**, needs T[a,b]</span></span>
+<span id="cb70-5"><a href="#cb70-5" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>In this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.</p>
 <p>Even if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.</p>
 
@@ -1265,25 +1269,6 @@ <h3 class="unnumbered anchored" data-anchor-id="rejection-is-not-for-constraints
 <li id="fn5"><p>The adjoint component is always zero during execution for the algorithmic differentiation variables used to implement parameters, transformed parameters, and local variables in the model.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -1304,6 +1289,150 @@ <h3 class="unnumbered anchored" data-anchor-id="rejection-is-not-for-constraints
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diff --git a/docs/2_34/reference-manual/syntax.html b/docs/2_34/reference-manual/syntax.html
index f48ba3e91..75861137a 100644
--- a/docs/2_34/reference-manual/syntax.html
+++ b/docs/2_34/reference-manual/syntax.html
@@ -38,10 +38,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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@@ -99,7 +101,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -842,7 +845,7 @@ <h3 class="unnumbered anchored" data-anchor-id="probability-function-naming">Pro
 <section id="indexes" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
 <p>Standalone expressions used as indexes must denote either an integer (<code>int</code>) or an integer array (<code>array[] int</code>). Expressions participating in range indexes (e.g., <code>a</code> and <code>b</code> in <code>a : b</code>) must denote integers (<code>int</code>).</p>
-<p>A second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type <code>matrix[ , , ]</code>, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.</p>
+<p>A second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type <code>array[ , , ] matrix</code>, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.</p>
 
 
 </section>
@@ -850,25 +853,6 @@ <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
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@@ -889,6 +873,150 @@ <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
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diff --git a/docs/2_34/reference-manual/transforms.html b/docs/2_34/reference-manual/transforms.html
index e1a01d613..f2dc3e762 100644
--- a/docs/2_34/reference-manual/transforms.html
+++ b/docs/2_34/reference-manual/transforms.html
@@ -58,10 +58,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1405,25 +1408,6 @@ <h3 class="unnumbered anchored" data-anchor-id="absolute-jacobian-determinant-of
 <li id="fn3"><p>For convenience, a summation with no terms, such as <span class="math inline">\(\sum_{j' &lt; 1} x_{i,j'}\)</span>, is defined to be 0. This implies <span class="math inline">\(x_{1,1} = 1\)</span> and that <span class="math inline">\(x_{i,1} = z_{i,1}\)</span> for <span class="math inline">\(i &gt; 1\)</span>.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-   kbds.forEach(function(kbd) {
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-
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-   if (!el.innerText.startsWith("#|")) {
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1444,6 +1428,150 @@ <h3 class="unnumbered anchored" data-anchor-id="absolute-jacobian-determinant-of
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
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+    const els = window.document.querySelectorAll(selector);
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
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+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
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+    } else {
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+    }
+  }
+  const setStyleSentinel = (alternate) => {
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+    } else {
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+  }
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+  // Dark / light mode switch
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+    // Read the current dark / light value 
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+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  } else {
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diff --git a/docs/2_34/reference-manual/types.html b/docs/2_34/reference-manual/types.html
index e048bd438..c0910ef4a 100644
--- a/docs/2_34/reference-manual/types.html
+++ b/docs/2_34/reference-manual/types.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
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@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -600,7 +603,7 @@ <h3 class="unnumbered anchored" data-anchor-id="array-types">Array types</h3>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="dt">array</span>[<span class="dv">6</span>, <span class="dv">7</span>] <span class="dt">matrix</span>[<span class="dv">3</span>, <span class="dv">3</span>] m;</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="dt">array</span>[<span class="dv">12</span>, <span class="dv">8</span>, <span class="dv">15</span>] <span class="dt">complex</span> z;</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>declares <code>x</code> to be a one-dimensional array of size 10 containing real values, declares <code>m</code> to be a two-dimensional array of size <span class="math inline">\(6 \times 7\)</span> containing values that are <span class="math inline">\(3 \times 3\)</span> matrices, and declares <code>z</code> to be a <span class="math inline">\(12 \times 8 \times 15\)</span> array of complex numbers.</p>
-<p>Prior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.</p>
+<p>Prior to 2.26 Stan models used a different syntax which has since been removed. See the <a href="../reference-manual/removals.html">Removed Features</a> chapter for more details.</p>
 </section>
 <section id="tuple-types" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="tuple-types">Tuple types</h3>
@@ -630,8 +633,8 @@ <h2 class="anchored" data-anchor-id="numerical-data-types">Primitive numerical d
 <section id="integers" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="integers">Integers</h3>
 <p>Stan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is <span class="math inline">\(2^{31}-1\)</span>; the minimum value is <span class="math inline">\(-(2^{31})\)</span>.</p>
-<p>When integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.</p>
-<p>Integer arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see <a href="https://mc-stan.org/docs/functions-reference/integer-valued_basic_functions.html#int-arithmetic">the Stan Functions Reference integer-valued arithmetic operators section</a> for more information).</p>
+<p>When integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.</p>
+<p>Integer arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see <a href="https://mc-stan.org/docs/functions-reference/integer-valued_basic_functions.html#int-arithmetic">the Stan Functions Reference integer-valued arithmetic operators section</a> for more information).</p>
 </section>
 <section id="reals" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="reals">Reals</h3>
@@ -721,7 +724,7 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-real">Constrained re
 <div class="sourceCode" id="cb15"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span>&lt;<span class="kw">lower</span>=-<span class="dv">1</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; rho;</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <section id="infinite-constraints" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="infinite-constraints">Infinite constraints</h4>
-<p>Lower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants <code>positive_infinity()</code> and <code>negative_infinity()</code> which may be used for this purpose, or they may be read as data in the dump format.</p>
+<p>Lower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants <code>positive_infinity()</code> and <code>negative_infinity()</code> which may be used for this purpose, or they may be supplied as data.</p>
 </section>
 </section>
 <section id="affinely-transformed-real" class="level3 unnumbered">
@@ -1621,25 +1624,6 @@ <h3 class="unnumbered anchored" data-anchor-id="combining-with-other-features">C
 <li id="fn1"><p>Stan compiles integers to <code>int</code> and reals to <code>double</code> types in C++. Precise details of rounding will depend on the compiler and hardware architecture on which the code is run.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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@@ -1660,6 +1644,150 @@ <h3 class="unnumbered anchored" data-anchor-id="combining-with-other-features">C
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+  const disableStylesheet = (stylesheets) => {
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+      stylesheet.rel = 'stylesheet';
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+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/user-functions.html b/docs/2_34/reference-manual/user-functions.html
index cf0aca155..98a25d035 100644
--- a/docs/2_34/reference-manual/user-functions.html
+++ b/docs/2_34/reference-manual/user-functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -657,6 +660,7 @@ <h3 class="unnumbered anchored" data-anchor-id="return-guarantee-required">Retur
 <li>a for loop or while loop qualifies if its body qualifies, and</li>
 <li>a conditional statement qualifies if it has a default else clause and all of its body statements qualify.</li>
 </ul>
+<p>An exception is made for “obviously infinite” loops like <code>while (1)</code>, which contain a <code>return</code> statement and no <code>break</code> statements. The only way to exit such a loop is to return, so they are considered as returning statements.</p>
 <p>These rules disqualify</p>
 <div class="sourceCode" id="cb14"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> foo(<span class="dt">real</span> x) {</span>
 <span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x &gt; <span class="dv">2</span>) {</span>
@@ -727,25 +731,6 @@ <h2 class="anchored" data-anchor-id="forward-declarations.section">Declarations<
 <li id="fn1"><p>Despite being declared constant and appearing to have a pass-by-value syntax in Stan, the implementation of the language passes function arguments by constant reference in C++.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -766,6 +751,150 @@ <h2 class="anchored" data-anchor-id="forward-declarations.section">Declarations<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/reference-manual/variational.html b/docs/2_34/reference-manual/variational.html
index 6e01b5f83..8b2430d6a 100644
--- a/docs/2_34/reference-manual/variational.html
+++ b/docs/2_34/reference-manual/variational.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -505,25 +508,6 @@ <h3 class="unnumbered anchored" data-anchor-id="assessing-convergence">Assessing
 Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -544,6 +528,150 @@ <h3 class="unnumbered anchored" data-anchor-id="assessing-convergence">Assessing
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
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index 8e164b6c9..f993470dc 100644
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -437,25 +440,6 @@ <h2 class="anchored" data-anchor-id="whitespace-location">Whitespace location</h
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diff --git a/docs/2_34/search.json b/docs/2_34/search.json
index 758c4cc9e..5fcd4eb43 100644
--- a/docs/2_34/search.json
+++ b/docs/2_34/search.json
@@ -4,7 +4,7 @@
     "href": "stan-users-guide/using-stanc.html",
     "title": "Using the Stan Compiler",
     "section": "",
-    "text": "Stan is used in most of our interfaces through the Stan compiler stanc. Since version 2.22, the Stan compiler has been implemented in OCaml and is referred to as stanc3. The binary name is still simply stanc, so this document uses both stanc and stanc3 interchangeably.\n\n\nThe stanc3 compiler has the following command-line syntax:\n&gt; stanc (options) &lt;model_file&gt;\nwhere &lt;model_file&gt; is a path to either a Stan model file ending in suffix .stan or a Stan functions file which ends in .stanfunctions.\nThe stanc3 options are:\n\n--help - Displays the complete list of stanc3 options, then exits.\n--version - Display stanc version number\n--info - Print information about the model, such as the type information for variables and the list of used distributions.\n--name=&lt;model_name&gt; - Specify the name of the class used for the implementation of the Stan model in the generated C++ code.\n--o=&lt;file_name&gt; - Specify a path to an output file for generated C++ code (default = .hpp) or auto-formatting output (default: no file/print to stdout)\n--allow-undefined - Do not throw a parser error if there is a function in the Stan program that is declared but not defined in the functions block.\n--include_paths=&lt;dir1,...dirN&gt; - Takes a comma-separated list of directories that may contain a file in an #include directive.\n--use-opencl - If set, will use additional Stan OpenCL features enabled in the Stan-to-C++ compiler.\n--auto-format - Pretty prints the program to the console. See more on auto formatting.\n--canonicalize - Make changes to the program before pretty-printing by specifying options in a comma separated list. Options are ‘deprecations’, ‘parentheses’, ‘braces’, ‘includes’.\n--max-line-length=&lt;number&gt; - Set the column number at which formatting with --auto-format attempts to split lines. The default value is 78, which results in most lines being shorter than 80 characters.\n--print-canonical - Synonymous with --auto-format --canonicalize=[all   options].\n--print-cpp - If set, output the generated C++ Stan model class to stdout.\n--standalone-functions - If set, only generate the code for the functions defined in the file. This is the default behavior for .stanfunctions files.\n--O0 (Default) Do not apply optimizations to the Stan code.\n--O1 Apply level 1 compiler optimizations (only basic optimizations).\n--Oexperimental WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Allow the compiler to apply all optimizations to the Stan code.\n--O WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Same as --Oexperimental. Allow the compiler to apply all optimizations to the Stan code.\n--warn-uninitialized - Emit warnings about uninitialized variables to stderr. Currently an experimental feature.\n--warn-pedantic - Emit warnings in Pedantic mode which warns of potential issues in the meaning of your program.\n\nThe compiler also provides a number of debug options which are primarily of interest to stanc3 developers; use the --help option to see the full set.\n\n\n\nDuring model compilation, stanc can produce a variety of errors (issues that prevent the model from being compiled) and warnings (non-fatal issues that should still be considered).\n\n\nEven without the optional --warn-pedantic and --warn-uninitialized command line flags, both of which enable additional warnings, stanc can still produce warnings about your program. In particular, warnings will be produced in two situations\n\nA completely blank Stan program will produce the following warning message\nWarning: Empty file 'empty.stan' detected;\n    this is a valid stan model but likely unintended!\nThe use of any deprecated features will lead to warnings which will look as follows\nWarning in 'deprecated.stan', line 2, column 0: Comments beginning with # are\n deprecated and this syntax will be removed in Stan 2.32.0. Use // to\n begin line comments; this can be done automatically using stanc\n --auto-format\nA single Stan program can produce many warnings during compilation.\n\n\n\n\nErrors differ from warnings in their severity and format. In particular, errors are fatal and stop compilation, so at most one error is displayed per run of stanc.\nThere are five kinds of errors emitted by stanc3\n\nFile errors occur when the file passed to stanc is either missing or cannot be opened (i.e. has permissions issues). They look like\nError: file 'notfound.stan' not found or cannot be opened\nSyntactic errors occur whenever a program violates the Stan language’s syntax requirements. There are three kinds of errors within syntax errors; “lexing” errors mean that the input was unable to be read properly on the character level, “include” errors which occur when the #include directive fails, and “parsing” errors which result when the structure of the program is incorrect.\n\nThe lexing errors occur due to the use of invalid characters in a program. For example, a lexing error due to the use of $ in a variable name will look like the following.\nSyntax error in 'char.stan', line 2, column 6, lexing error:\n-------------------------------------------------\n  1:  data {\n  2:     int $ome_variable;\n             ^\n  3:  }\n-------------------------------------------------\nInvalid character found.\nWhen an include directive is used, it can lead to errors if the included file is not found, or if a file includes itself (including a recursive loop of includes, such as A -&gt; B -&gt; A).\nSyntax error in './incl.stan', line 1, column 0, included from\n'./incl.stan', line 1, column 0, included from\n'incl.stan', line 1, column 0, include error:\n-------------------------------------------------\n  1:  #include &lt;incl.stan&gt;\n      ^\n-------------------------------------------------\nFile incl.stan recursively included itself.\nIt is much more common to see parsing errors, which tend to have more in-depth explanations of the error found. For example, if a user forgets to put a size on a type like vector, as in the following, this raises a parsing (structural) error in the compiler.\nSyntax error in vec.stan', line 3, column 10 to column 11, parsing error:\n-------------------------------------------------\n  1:  data {\n  2:     int&lt;lower=0&gt; N;\n  3:     vector x;\n                ^\n  4:  }\n-------------------------------------------------\n\"[\" expression \"]\" expected for vector size.\n\nSemantic errors (also known as type errors) occur when a program is structured correctly but features an error in the type rules imposed by the language. An example of this is assigning a real value to a variable defined as an integer.\nSemantic error in 'type.stan', line 2, column 3 to column 15:\n-------------------------------------------------\n  1:  transformed data {\n  2:     int x = 1.5;\n         ^\n  3:  }\n-------------------------------------------------\nIll-typed arguments supplied to assignment operator =: lhs has\ntype int and rhs has type real\nThe compiler will raise an error for use of any removed features for at least one version following their removal. The deprecation warnings mentioned above eventually turn into this kind of error to prompt the user to update their model. After the version of removal, these errors will be converted to one of the other types listed here, depending on the feature.\nFinally, the compiler can raise an internal error. These are caused by bugs in the compiler, not your model, and we would appreciate it if you report them on the stanc3 repo with the error message provided. These errors usually say something like “This should never happen,” and we apologize if they do.\n\n\n\n\n\nPedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.\n\n\n\n\n\nIn addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.\n\n\n\n\n\nThe stanc3 compiler can optimize the code of Stan model during compilation. The optimized model code behaves the same as unoptimized code, but it may be faster, more memory efficient, or more numerically stable.\nThis section introduces the available optimization options and describes their effect.\nTo print out a representation of the optimized Stan program, use the stanc3 command-line flag --debug-optimized-mir-pretty. To print an analogous representation of the Stan program prior to optimization, use the flag --debug-transformed-mir-pretty.\n\n\nTo turn optimizations on, the user specifies the desired optimization level. The level specifies the set of optimizations to use. The chosen optimizations are used in a specific order, with some of them applied repeatedly.\nOptimization levels are specified by the numbers 0 and 1 and the ‘experimental’ tag:\n\nO0 No optimizations are applied.\nO1 Optimizations that are simple, do not dramatically change the program, and are unlikely to noticeably slow down compile times are applied.\nOexperimental All optimizations are applied. Some of these are not thorougly tested and may not always improve a programs performance.\n\nO0 is the default setting.\nThe levels include these optimizations:\n\nO0 includes no optimizations.\nO1 includes:\n\nDead code elimination\nCopy propagation\nConstant propagation\n\nOexperimental includes optimizations specified by O1 and also:\n\nAutomatic-differentiation level optimization\nOne step loop unrolling\nExpression propagation\nPartial evaluation\nLazy code motion\nFunction inlining\nStatic loop unrolling\n\n\nIn addition, Oexperimental will apply more repetitions of the optimizations, which may increase compile times.\n\n\n\n\n\nDead code is code that does not affect the behavior of the program. Code is not dead if it affects target, the value of any outside-observable variable like transformed parameters or generated quantities, or side effects such as print statements. Removing dead code can speed up a program by avoiding unnecessary computations.\nExample Stan program:\nmodel {\n  int i;\n  i = 5;\n  for (j in 1:10);\n  if (0) {\n    print(\"Dead code\");\n  } else {\n    print(\"Hi!\");\n  }\n}\nCompiler representation of program before dead code elimination (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  int i = 5;\n  for(j in 1:10) {\n    ;\n  }\n  if(0) {\n    FnPrint__(\"Dead code\");\n  } else {\n    FnPrint__(\"Hi!\");\n  }\n}\nCompiler representation of program after dead code elimination (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i;\n  FnPrint__(\"Hi!\");\n}\n\n\n\nConstant propagation replaces uses of a variable which is known to have a constant value C with that constant C. This removes the overhead of looking up the variable, and also makes many other optimizations possible (such as static loop unrolling and partial evaluation).\nExample Stan program:\ntransformed data {\n  int n = 100;\n  int a[n];\n  for (i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before constant propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, n] a;\n  for(i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after constant propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, 100] a;\n  for(i in 1:100) {\n    a[i] = i;\n  }\n}\n\n\n\nCopy propagation is similar to expression propagation, but only propagates variables rather than arbitrary expressions. This can reduce the complexity of the code for other optimizations such as expression propagation.\nExample Stan program:\nmodel {\n  int i = 1;\n  int j = i;\n  int k = i + j;\n}\nCompiler representation of program before copy propagation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n    int i = 1;\n    int j = i;\n    int k = (i + j);\n}\nCompiler representation of program after copy propagation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i = 1;\n  int j = i;\n  int k = (i + i);\n}\n\n\n\nPartial evaluation searches for expressions that we can replace with a faster, simpler, more memory efficient, or more numerically stable expression with the same meaning.\nExample Stan program:\nmodel {\n  real a = 1 + 1;\n  real b = log(1 - a);\n  real c = a + b * 5;\n}\nCompiler representation of program before partial evaluation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real a = (1 + 1);\n  real b = log((1 - a));\n  real c = (a + (b * 5));\n}\nCompiler representation of program after partial evaluation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  real a = 2;\n  real b = log1m(a);\n  real c = fma(b, 5, a);\n}\n\n\n\n\n\n\nStan variables can have two auto-differentiation (AD) levels: AD or non-AD. AD variables carry gradient information with them, which allows Stan to calculate the log-density gradient, but they also have more overhead than non-AD variables. It is therefore inefficient for a variable to be AD unnecessarily. AD-level optimization sets every variable to be a floating point type unless its gradient is necessary.\nExample Stan program:\ndata {\n  real y;\n}\nmodel {\n  real x = y + 1;\n}\nCompiler representation of program before AD-level optimization (simplified from the output of --debug-transformed-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  real x = (y + 1);\n}\nCompiler representation of program after AD-level optimization (simplified from the output of --debug-optimized-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  data real x = (y + 1);\n}\n\n\n\nOne step loop unrolling is similar to static loop unrolling. However, this optimization only ‘unrolls’ the first loop iteration, and can therefore work even when the total number of iterations is not predictable. This can speed up a program by providing more opportunities for further optimizations such as partial evaluation and lazy code motion.\nExample Stan program:\ndata {\n  int n;\n}\ntransformed data {\n  int x = 0;\n  for (i in 1:n) {\n    x += i;\n  }\n}\nCompiler representation of program before one step static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  data int x = 0;\n  for(i in 1:n) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after one step static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  int x = 0;\n  if((n &gt;= 1)) {\n    x = (x + 1);\n    for(i in (1 + 1):n) {\n      x = (x + i);\n    }\n  }\n}\n\n\n\nConstant propagation replaces the uses of a variable which is known to have a constant value E with that constant E. This often results in recalculating the expression, but provides more opportunities for further optimizations such as partial evaluation. Expression propagation is always followed by lazy code motion to avoid unnecessarily recomputing expressions.\nExample Stan program:\ndata {\n  int m;\n}\ntransformed data {\n  int n = m+1;\n  int a[n];\n  for (i in 1:n-1) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before expression propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, n] a;\n  for(i in 1:(n - 1)) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after expression propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, (m + 1)] a;\n  for(i in 1:((m + 1) - 1)) {\n    a[i] = i;\n  }\n}\n\n\n\nLazy code motion rearranges the statements and expressions in a program with the goals of:\n\nAvoiding computing expressions more than once, and\nComputing expressions as late as possible (to minimize the strain on the working memory set).\n\nTo accomplish these goals, lazy code motion will perform optimizations such as:\n\nMoving a repeatedly calculated expression to its own variable (also referred to as common-subexpression elimination)\nMoving an expression outside of a loop if it does not need to be in the loop (also referred to as loop-invariant code motion)\n\nLazy code motion can make some programs significantly more efficient by avoiding redundant or early computations.\nExample Stan program:\nmodel {\n  real x;\n  real y;\n  real z;\n\n  for (i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program before lazy code motion (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real x;\n  real y;\n  real z;\n  for(i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program after lazy code motion (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  data real lcm_sym4__;\n  data real lcm_sym3__;\n  real x;\n  real y;\n  lcm_sym4__ = sqrt(10);\n  real z;\n  for(i in 1:10) {\n    x = lcm_sym4__;\n    y = sqrt(i);\n  }\n  z = lcm_sym4__;\n}\n\n\n\nFunction inlining replaces each function call to each user-defined function f with the body of f. It does this by copying the function body to the call site and doing appropriately renaming the argument variables. This optimization can speed up a program by avoiding the overhead of a function call and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return x + y;\n  }\n}\ntransformed data {\n  int a = 2;\n  int b = incr(a);\n}\nCompiler representation of program before function inlining (simplified from the output of --debug-transformed-mir-pretty):\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return (x + y);\n  }\n}\n\nprepare_data {\n  data int a = 2;\n  data int b = incr(a);\n}\nCompiler representation of program after function inlining (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int a;\n  a = 2;\n  data int b;\n  data int inline_sym1__;\n  data int inline_sym3__;\n  inline_sym3__ = 0;\n  for(inline_sym4__ in 1:1) {\n    int inline_sym2__;\n    inline_sym2__ = 1;\n    inline_sym3__ = 1;\n    inline_sym1__ = (a + inline_sym2__);\n    break;\n  }\n  b = inline_sym1__;\n}\nIn this code, the for loop and break is used to simulate the behavior of a return statement. The value to be returned is held in inline_sym1__. The flag variable inline_sym3__ indicates whether a return has occurred and is necessary to handle return statements nested inside loops within the function body.\n\n\n\nStatic loop unrolling takes a loop with a predictable number of iterations X and replaces it by writing out the loop body X times. The loop index in each repeat is replaced with the appropriate constant. This optimization can speed up a program by avoiding the overhead of a loop and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\ntransformed data {\n  int x = 0;\n  for (i in 1:4) {\n    x += i;\n  }\n}\nCompiler representation of program before static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int x = 0;\n  for(i in 1:4) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int x;\n  x = 0;\n  x = (x + 1);\n  x = (x + 2);\n  x = (x + 3);\n  x = (x + 4);\n}",
+    "text": "Stan is used in most of our interfaces through the Stan compiler stanc. Since version 2.22, the Stan compiler has been implemented in OCaml and is referred to as stanc3. The binary name is still simply stanc, so this document uses both stanc and stanc3 interchangeably.\n\n\nThe stanc3 compiler has the following command-line syntax:\n&gt; stanc (options) &lt;model_file&gt;\nwhere &lt;model_file&gt; is a path to either a Stan model file ending in suffix .stan or a Stan functions file which ends in .stanfunctions.\nThe stanc3 options are:\n\n--help - Displays the complete list of stanc3 options, then exits.\n--version - Display stanc version number\n--info - Print information about the model, such as the type information for variables and the list of used distributions.\n--name=&lt;model_name&gt; - Specify the name of the class used for the implementation of the Stan model in the generated C++ code.\n--o=&lt;file_name&gt; - Specify a path to an output file for generated C++ code (default = .hpp) or auto-formatting output (default: no file/print to stdout)\n--allow-undefined - Do not throw a parser error if there is a function in the Stan program that is declared but not defined in the functions block.\n--include_paths=&lt;dir1,...dirN&gt; - Takes a comma-separated list of directories that may contain a file in an #include directive.\n--use-opencl - If set, will use additional Stan OpenCL features enabled in the Stan-to-C++ compiler.\n--auto-format - Pretty prints the program to the console. See more on auto formatting.\n--canonicalize - Make changes to the program before pretty-printing by specifying options in a comma separated list. Options are ‘deprecations’, ‘parentheses’, ‘braces’, ‘includes’.\n--max-line-length=&lt;number&gt; - Set the column number at which formatting with --auto-format attempts to split lines. The default value is 78, which results in most lines being shorter than 80 characters.\n--print-canonical - Synonymous with --auto-format --canonicalize=[all   options].\n--print-cpp - If set, output the generated C++ Stan model class to stdout.\n--standalone-functions - If set, only generate the code for the functions defined in the file. This is the default behavior for .stanfunctions files.\n--O0 (Default) Do not apply optimizations to the Stan code.\n--O1 Apply level 1 compiler optimizations (only basic optimizations).\n--Oexperimental WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Allow the compiler to apply all optimizations to the Stan code.\n--O WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Same as --Oexperimental. Allow the compiler to apply all optimizations to the Stan code.\n--warn-uninitialized - Emit warnings about uninitialized variables to stderr. Currently an experimental feature.\n--warn-pedantic - Emit warnings in Pedantic mode which warns of potential issues in the meaning of your program.\n\nThe compiler also provides a number of debug options which are primarily of interest to stanc3 developers; use the --help option to see the full set.\n\n\n\nDuring model compilation, stanc can produce a variety of errors (issues that prevent the model from being compiled) and warnings (non-fatal issues that should still be considered).\n\n\nEven without the optional --warn-pedantic and --warn-uninitialized command line flags, both of which enable additional warnings, stanc can still produce warnings about your program. In particular, warnings will be produced in two situations\n\nA completely blank Stan program will produce the following warning message\nWarning: Empty file 'empty.stan' detected;\n    this is a valid stan model but likely unintended!\nThe use of any deprecated features will lead to warnings which will look as follows\nWarning in 'deprecated.stan', line 2, column 0: Comments beginning with # are\n deprecated and this syntax will be removed in Stan 2.32.0. Use // to\n begin line comments; this can be done automatically using stanc\n --auto-format\nA single Stan program can produce many warnings during compilation.\n\n\n\n\nErrors differ from warnings in their severity and format. In particular, errors are fatal and stop compilation, so at most one error is displayed per run of stanc.\nThere are five kinds of errors emitted by stanc3\n\nFile errors occur when the file passed to stanc is either missing or cannot be opened (i.e. has permissions issues). They look like\nError: file 'notfound.stan' not found or cannot be opened\nSyntactic errors occur whenever a program violates the Stan language’s syntax requirements. There are three kinds of errors within syntax errors; “lexing” errors mean that the input was unable to be read properly on the character level, “include” errors which occur when the #include directive fails, and “parsing” errors which result when the structure of the program is incorrect.\n\nThe lexing errors occur due to the use of invalid characters in a program. For example, a lexing error due to the use of $ in a variable name will look like the following.\nSyntax error in 'char.stan', line 2, column 6, lexing error:\n-------------------------------------------------\n  1:  data {\n  2:     int $ome_variable;\n             ^\n  3:  }\n-------------------------------------------------\nInvalid character found.\nWhen an include directive is used, it can lead to errors if the included file is not found, or if a file includes itself (including a recursive loop of includes, such as A -&gt; B -&gt; A).\nSyntax error in './incl.stan', line 1, column 0, included from\n'./incl.stan', line 1, column 0, included from\n'incl.stan', line 1, column 0, include error:\n-------------------------------------------------\n  1:  #include &lt;incl.stan&gt;\n      ^\n-------------------------------------------------\nFile incl.stan recursively included itself.\nIt is much more common to see parsing errors, which tend to have more in-depth explanations of the error found. For example, if a user forgets to put a size on a type like vector, as in the following, this raises a parsing (structural) error in the compiler.\nSyntax error in vec.stan', line 3, column 10 to column 11, parsing error:\n-------------------------------------------------\n  1:  data {\n  2:     int&lt;lower=0&gt; N;\n  3:     vector x;\n                ^\n  4:  }\n-------------------------------------------------\n\"[\" expression \"]\" expected for vector size.\n\nSemantic errors (also known as type errors) occur when a program is structured correctly but features an error in the type rules imposed by the language. An example of this is assigning a real value to a variable defined as an integer.\nSemantic error in 'type.stan', line 2, column 3 to column 15:\n-------------------------------------------------\n  1:  transformed data {\n  2:     int x = 1.5;\n         ^\n  3:  }\n-------------------------------------------------\nIll-typed arguments supplied to assignment operator =: lhs has\ntype int and rhs has type real\nThe compiler will raise an error for use of any removed features for at least one version following their removal. The deprecation warnings mentioned above eventually turn into this kind of error to prompt the user to update their model. After the version of removal, these errors will be converted to one of the other types listed here, depending on the feature.\nFinally, the compiler can raise an internal error. These are caused by bugs in the compiler, not your model, and we would appreciate it if you report them on the stanc3 repo with the error message provided. These errors usually say something like “This should never happen,” and we apologize if they do.\n\n\n\n\n\nPedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the Changes of variables section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.\n\n\n\n\n\nIn addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the Stan Program Style Guide. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.\n\n\n\n\n\nThe stanc3 compiler can optimize the code of Stan model during compilation. The optimized model code behaves the same as unoptimized code, but it may be faster, more memory efficient, or more numerically stable.\nThis section introduces the available optimization options and describes their effect.\nTo print out a representation of the optimized Stan program, use the stanc3 command-line flag --debug-optimized-mir-pretty. To print an analogous representation of the Stan program prior to optimization, use the flag --debug-transformed-mir-pretty.\n\n\nTo turn optimizations on, the user specifies the desired optimization level. The level specifies the set of optimizations to use. The chosen optimizations are used in a specific order, with some of them applied repeatedly.\nOptimization levels are specified by the numbers 0 and 1 and the ‘experimental’ tag:\n\nO0 No optimizations are applied.\nO1 Optimizations that are simple, do not dramatically change the program, and are unlikely to noticeably slow down compile times are applied.\nOexperimental All optimizations are applied. Some of these are not thorougly tested and may not always improve a programs performance.\n\nO0 is the default setting.\nThe levels include these optimizations:\n\nO0 includes no optimizations.\nO1 includes:\n\nDead code elimination\nCopy propagation\nConstant propagation\n\nOexperimental includes optimizations specified by O1 and also:\n\nAutomatic-differentiation level optimization\nOne step loop unrolling\nExpression propagation\nPartial evaluation\nLazy code motion\nFunction inlining\nStatic loop unrolling\n\n\nIn addition, Oexperimental will apply more repetitions of the optimizations, which may increase compile times.\n\n\n\n\n\nDead code is code that does not affect the behavior of the program. Code is not dead if it affects target, the value of any outside-observable variable like transformed parameters or generated quantities, or side effects such as print statements. Removing dead code can speed up a program by avoiding unnecessary computations.\nExample Stan program:\nmodel {\n  int i;\n  i = 5;\n  for (j in 1:10);\n  if (0) {\n    print(\"Dead code\");\n  } else {\n    print(\"Hi!\");\n  }\n}\nCompiler representation of program before dead code elimination (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  int i = 5;\n  for(j in 1:10) {\n    ;\n  }\n  if(0) {\n    FnPrint__(\"Dead code\");\n  } else {\n    FnPrint__(\"Hi!\");\n  }\n}\nCompiler representation of program after dead code elimination (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i;\n  FnPrint__(\"Hi!\");\n}\n\n\n\nConstant propagation replaces uses of a variable which is known to have a constant value C with that constant C. This removes the overhead of looking up the variable, and also makes many other optimizations possible (such as static loop unrolling and partial evaluation).\nExample Stan program:\ntransformed data {\n  int n = 100;\n  int a[n];\n  for (i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before constant propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, n] a;\n  for(i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after constant propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, 100] a;\n  for(i in 1:100) {\n    a[i] = i;\n  }\n}\n\n\n\nCopy propagation is similar to expression propagation, but only propagates variables rather than arbitrary expressions. This can reduce the complexity of the code for other optimizations such as expression propagation.\nExample Stan program:\nmodel {\n  int i = 1;\n  int j = i;\n  int k = i + j;\n}\nCompiler representation of program before copy propagation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n    int i = 1;\n    int j = i;\n    int k = (i + j);\n}\nCompiler representation of program after copy propagation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i = 1;\n  int j = i;\n  int k = (i + i);\n}\n\n\n\nPartial evaluation searches for expressions that we can replace with a faster, simpler, more memory efficient, or more numerically stable expression with the same meaning.\nExample Stan program:\nmodel {\n  real a = 1 + 1;\n  real b = log(1 - a);\n  real c = a + b * 5;\n}\nCompiler representation of program before partial evaluation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real a = (1 + 1);\n  real b = log((1 - a));\n  real c = (a + (b * 5));\n}\nCompiler representation of program after partial evaluation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  real a = 2;\n  real b = log1m(a);\n  real c = fma(b, 5, a);\n}\n\n\n\n\n\n\nStan variables can have two auto-differentiation (AD) levels: AD or non-AD. AD variables carry gradient information with them, which allows Stan to calculate the log-density gradient, but they also have more overhead than non-AD variables. It is therefore inefficient for a variable to be AD unnecessarily. AD-level optimization sets every variable to be a floating point type unless its gradient is necessary.\nExample Stan program:\ndata {\n  real y;\n}\nmodel {\n  real x = y + 1;\n}\nCompiler representation of program before AD-level optimization (simplified from the output of --debug-transformed-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  real x = (y + 1);\n}\nCompiler representation of program after AD-level optimization (simplified from the output of --debug-optimized-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  data real x = (y + 1);\n}\n\n\n\nOne step loop unrolling is similar to static loop unrolling. However, this optimization only ‘unrolls’ the first loop iteration, and can therefore work even when the total number of iterations is not predictable. This can speed up a program by providing more opportunities for further optimizations such as partial evaluation and lazy code motion.\nExample Stan program:\ndata {\n  int n;\n}\ntransformed data {\n  int x = 0;\n  for (i in 1:n) {\n    x += i;\n  }\n}\nCompiler representation of program before one step static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  data int x = 0;\n  for(i in 1:n) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after one step static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  int x = 0;\n  if((n &gt;= 1)) {\n    x = (x + 1);\n    for(i in (1 + 1):n) {\n      x = (x + i);\n    }\n  }\n}\n\n\n\nConstant propagation replaces the uses of a variable which is known to have a constant value E with that constant E. This often results in recalculating the expression, but provides more opportunities for further optimizations such as partial evaluation. Expression propagation is always followed by lazy code motion to avoid unnecessarily recomputing expressions.\nExample Stan program:\ndata {\n  int m;\n}\ntransformed data {\n  int n = m+1;\n  int a[n];\n  for (i in 1:n-1) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before expression propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, n] a;\n  for(i in 1:(n - 1)) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after expression propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, (m + 1)] a;\n  for(i in 1:((m + 1) - 1)) {\n    a[i] = i;\n  }\n}\n\n\n\nLazy code motion rearranges the statements and expressions in a program with the goals of:\n\nAvoiding computing expressions more than once, and\nComputing expressions as late as possible (to minimize the strain on the working memory set).\n\nTo accomplish these goals, lazy code motion will perform optimizations such as:\n\nMoving a repeatedly calculated expression to its own variable (also referred to as common-subexpression elimination)\nMoving an expression outside of a loop if it does not need to be in the loop (also referred to as loop-invariant code motion)\n\nLazy code motion can make some programs significantly more efficient by avoiding redundant or early computations.\nExample Stan program:\nmodel {\n  real x;\n  real y;\n  real z;\n\n  for (i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program before lazy code motion (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real x;\n  real y;\n  real z;\n  for(i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program after lazy code motion (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  data real lcm_sym4__;\n  data real lcm_sym3__;\n  real x;\n  real y;\n  lcm_sym4__ = sqrt(10);\n  real z;\n  for(i in 1:10) {\n    x = lcm_sym4__;\n    y = sqrt(i);\n  }\n  z = lcm_sym4__;\n}\n\n\n\nFunction inlining replaces each function call to each user-defined function f with the body of f. It does this by copying the function body to the call site and doing appropriately renaming the argument variables. This optimization can speed up a program by avoiding the overhead of a function call and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return x + y;\n  }\n}\ntransformed data {\n  int a = 2;\n  int b = incr(a);\n}\nCompiler representation of program before function inlining (simplified from the output of --debug-transformed-mir-pretty):\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return (x + y);\n  }\n}\n\nprepare_data {\n  data int a = 2;\n  data int b = incr(a);\n}\nCompiler representation of program after function inlining (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int a;\n  a = 2;\n  data int b;\n  data int inline_sym1__;\n  data int inline_sym3__;\n  inline_sym3__ = 0;\n  for(inline_sym4__ in 1:1) {\n    int inline_sym2__;\n    inline_sym2__ = 1;\n    inline_sym3__ = 1;\n    inline_sym1__ = (a + inline_sym2__);\n    break;\n  }\n  b = inline_sym1__;\n}\nIn this code, the for loop and break is used to simulate the behavior of a return statement. The value to be returned is held in inline_sym1__. The flag variable inline_sym3__ indicates whether a return has occurred and is necessary to handle return statements nested inside loops within the function body.\n\n\n\nStatic loop unrolling takes a loop with a predictable number of iterations X and replaces it by writing out the loop body X times. The loop index in each repeat is replaced with the appropriate constant. This optimization can speed up a program by avoiding the overhead of a loop and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\ntransformed data {\n  int x = 0;\n  for (i in 1:4) {\n    x += i;\n  }\n}\nCompiler representation of program before static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int x = 0;\n  for(i in 1:4) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int x;\n  x = 0;\n  x = (x + 1);\n  x = (x + 2);\n  x = (x + 3);\n  x = (x + 4);\n}",
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-    "text": "Pedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.",
+    "text": "Pedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the Changes of variables section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.",
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@@ -52,7 +52,7 @@
     "href": "stan-users-guide/using-stanc.html#stanc-pretty-printing",
     "title": "Using the Stan Compiler",
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-    "text": "In addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.",
+    "text": "In addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the Stan Program Style Guide. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.",
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@@ -419,7 +419,7 @@
     "href": "stan-users-guide/posterior-predictive-checks.html",
     "title": "Posterior and Prior Predictive Checks",
     "section": "",
-    "text": "Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.\nPrior predictive checks evaluate the prior the same way. Specifically, they evaluate what data sets would be consistent with the prior. They will not be calibrated with actual data, but extreme values help diagnose priors that are either too strong, too weak, poorly shaped, or poorly located.\nPrior and posterior predictive checks are two cases of the general concept of predictive checks, just conditioning on different things (no data and the observed data, respectively). For hierarchical models, there are intermediate versions, as discussed in the section on hierarchical models and mixed replication.\n\n\nThe posterior predictive distribution is the distribution over new observations given previous observations. It’s predictive in the sense that it’s predicting behavior on new data that is not part of the training set. It’s posterior in that everything is conditioned on observed data \\(y\\).\nThe posterior predictive distribution for replications \\(y^{\\textrm{rep}}\\) of the original data set \\(y\\) given model parameters \\(\\theta\\) is defined by \\[\np(y^{\\textrm{rep}} \\mid y)\n= \\int p(y^{\\textrm{rep}} \\mid \\theta)\n       \\cdot p(\\theta \\mid y) \\, \\textrm{d}\\theta.\n\\]\nAs with other posterior predictive quantities, generating a replicated data set \\(y^{\\textrm{rep}}\\) from the posterior predictive distribution is straightforward using the generated quantities block. Consider a simple regression model with parameters \\(\\theta = (\\alpha, \\beta, \\sigma).\\)\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n  vector[N] y;\n}\nparameters {\n  real alpha;\n  real beta;\n  real&lt;lower=0&gt; sigma;\n}\nmodel {\n  alpha ~ normal(0, 2);\n  beta ~ normal(0, 1);\n  sigma ~ normal(0, 1);\n  y ~ normal(alpha + beta * x, sigma);\n}\nTo generate a replicated data set y_rep for this simple model, the following generated quantities block suffices.\ngenerated quantities {\n  array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n}\nThe vectorized form of the normal random number generator is used with the original predictors x and the model parameters alpha, beta, and sigma. The replicated data variable y_rep is declared to be the same size as the original data y, but instead of a vector type, it is declared to be an array of reals to match the return type of the function normal_rng. Because the vector and real array types have the same dimensions and layout, they can be plotted against one another and otherwise compared during downstream processing.\nThe posterior predictive sampling for posterior predictive checks is different from usual posterior predictive sampling discussed in the chapter on posterior predictions in that the original predictors \\(x\\) are used. That is, the posterior predictions are for the original data.\n\n\n\nA standard posterior predictive check would plot a histogram of each replicated data set along with the original data set and compare them by eye. For this purpose, only a few replications are needed. These should be taken by thinning a larger set of replications down to the size needed to ensure rough independence of the replications.\nHere’s a complete example where the model is a simple Poisson with a weakly informative exponential prior with a mean of 10 and standard deviation of 10.\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0&gt; y;\n}\ntransformed data {\n  real&lt;lower=0&gt; mean_y = mean(to_vector(y));\n  real&lt;lower=0&gt; sd_y = sd(to_vector(y));\n}\nparameters {\n  real&lt;lower=0&gt; lambda;\n}\nmodel {\n  y ~ poisson(lambda);\n  lambda ~ exponential(0.2);\n}\ngenerated quantities {\n  array[N] int&lt;lower=0&gt; y_rep = poisson_rng(rep_array(lambda, N));\n  real&lt;lower=0&gt; mean_y_rep = mean(to_vector(y_rep));\n  real&lt;lower=0&gt; sd_y_rep = sd(to_vector(y_rep));\n  int&lt;lower=0, upper=1&gt; mean_gte = (mean_y_rep &gt;= mean_y);\n  int&lt;lower=0, upper=1&gt; sd_gte = (sd_y_rep &gt;= sd_y);\n}\nThe generated quantities block creates a variable y_rep for the replicated data, variables mean_y_rep and sd_y_rep for the statistics of the replicated data, and indicator variables mean_gte and sd_gte for whether the replicated statistic is greater than or equal to the statistic applied to the original data.\nNow consider generating data \\(y \\sim \\textrm{Poisson}(5)\\). The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\n\n\nWith a Poisson data-generating process and Poisson model, the posterior replications look similar to the original data. If it were easy to pick the original data out of the lineup, there would be a problem.\nNow consider generating over-dispersed data \\(y \\sim \\textrm{negative-binomial2}(5, 1).\\) This has the same mean as \\(\\textrm{Poisson}(5)\\), namely \\(5\\), but a standard deviation of \\(\\sqrt{5 + 5^2 /1} \\approx 5.5.\\) There is no way to fit this data with the Poisson model, because a variable distributed as \\(\\textrm{Poisson}(\\lambda)\\) has mean \\(\\lambda\\) and standard deviation \\(\\sqrt{\\lambda},\\) which is \\(\\sqrt{5}\\) for \\(\\textrm{Poisson}(5).\\) Here’s the resulting small multiples plot, again with original data in the upper left.\n\n\n\nPosterior predictive checks for negative binomial data generating process and Poisson model.\n\n\nThis time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.\n\n\n\nIf a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.\n\n\n\n\nPrior predictive checks generate data according to the prior in order to asses whether a prior is appropriate (Gabry et al. 2019). A posterior predictive check generates replicated data according to the posterior predictive distribution. In contrast, the prior predictive check generates data according to the prior predictive distribution, \\[\ny^{\\textrm{sim}} \\sim p(y).\n\\] The prior predictive distribution is just like the posterior predictive distribution with no observed data, so that a prior predictive check is nothing more than the limiting case of a posterior predictive check with no data.\nThis is easy to carry out mechanically by simulating parameters \\[\n\\theta^{\\textrm{sim}} \\sim p(\\theta)\n\\] according to the priors, then simulating data \\[\ny^{\\textrm{sim}} \\sim p(y \\mid \\theta^{\\textrm{sim}})\n\\] according to the sampling distribution given the simulated parameters. The result is a simulation from the joint distribution, \\[\n(y^{\\textrm{sim}}, \\theta^{\\textrm{sim}}) \\sim p(y, \\theta)\n\\] and thus \\[\ny^{\\textrm{sim}} \\sim p(y)\n\\] is a simulation from the prior predictive distribution.\n\n\nA prior predictive check is coded just like a posterior predictive check. If a posterior predictive check has already been coded and it’s possible to set the data to be empty, then no additional coding is necessary. The disadvantage to coding prior predictive checks as posterior predictive checks with no data is that Markov chain Monte Carlo will be used to sample the parameters, which is less efficient than taking independent draws using random number generation.\nPrior predictive checks can be coded entirely within the generated quantities block using random number generation. The resulting draws will be independent. Predictors must be read in from the actual data set—they do not have a generative model from which to be simulated. For a Poisson regression, prior predictive sampling can be encoded as the following complete Stan program.\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n}\ngenerated quantities {\n  real alpha = normal_rng(0, 1);\n  real beta = normal_rng(0, 1);\n  array[N] real y_sim = poisson_log_rng(alpha + beta * x);\n}\nRunning this program using Stan’s fixed-parameter sampler yields draws from the prior. These may be plotted to consider their appropriateness.\n\n\n\n\nSuppose we have a model for a football (aka soccer) league where there are \\(J\\) teams. Each team has a scoring rate \\(\\lambda_j\\) and in each game will be assumed to score \\(\\textrm{poisson}(\\lambda_j)\\) points. Yes, this model completely ignores defense. Suppose the modeler does not want to “put their thumb on the scale” and would rather “let the data speak for themselves” and so uses a prior with very wide tails, because it seems uninformative, such as the widely deployed \\[\n\\lambda_j \\sim \\textrm{gamma}(\\epsilon_1, \\epsilon_2).\n\\] This is not just a manufactured example; The BUGS Book recommends setting \\(\\epsilon = (0.5, 0.00001)\\), which corresponds to a Jeffreys prior for a Poisson rate parameter prior (Lunn et al. 2012, 85).\nSuppose the league plays a round-robin tournament wherein every team plays every other team. The following Stan model generates random team abilities and the results of such a round-robin tournament, which may be used to perform prior predictive checks.\ndata {\n  int&lt;lower=0&gt; J;\n  array[2] real&lt;lower=0&gt; epsilon;\n}\ngenerated quantities {\n  array[J] real&lt;lower=0&gt; lambda;\n  array[J, J] int y;\n  for (j in 1:J) lambda[j] = gamma_rng(epsilon[1], epsilon[2]);\n  for (i in 1:J) {\n    for (j in 1:J) {\n      y[i, j] = poisson_rng(lambda[i]) - poisson_rng(lambda[j]);\n    }\n  }\n}\nIn this simulation, teams play each other twice and play themselves once. This could be made more realistic by controlling the combinatorics to only generate a single result for each pair of teams, of which there are \\(\\binom{J}{2} = \\frac{J \\cdot (J - 1)}{2}.\\)\nUsing the \\(\\textrm{gamma}(0.5, 0.00001)\\) reference prior on team abilities, the following are the first 20 simulated point differences for the match between the first two teams, \\(y^{(1:20)}_{1, 2}\\).\n2597 -26000   5725  22496   1270   1072   4502  -2809   -302   4987\n7513   7527  -3268 -12374   3828   -158 -29889   2986  -1392     66\nThat’s some pretty highly scoring football games being simulated; all but one has a score differential greater than 100! In other words, this \\(\\textrm{gamma}(0.5, 0.00001)\\) prior is putting around 95% of its weight on score differentials above 100. Given that two teams combined rarely score 10 points, this prior is way out of line with prior knowledge about football matches; it is not only consistent with outcomes that have never occurred in the history of the sport, it puts most of the prior probability mass there.\nThe posterior predictive distribution can be strongly affected by the prior when there is not much observed data and substantial prior mass is concentrated around infeasible values (A. Gelman 2006).\nJust as with posterior predictive distributions, any statistics of the generated data may be evaluated. Here, the focus was on score difference between a single pair of teams, but it could’ve been on maximums, minimums, averages, variances, etc.\nIn this textbook example, the prior is univariate and directly related to the expected number of points scored, and could thus be directly inspected for consistency with prior knowledge about scoring rates in football. There will not be the same kind of direct connection when the prior and sampling distributions are multivariate. In these more challenging situations, prior predictive checks are an easy way to get a handle on the implications of a prior in terms of what it says the data is going to look like; for a more complex application involving spatially heterogeneous air pollution concentration, see (Gabry et al. 2019).\nPrior predictive checks can also be compared with the data, but one should not expect them to be calibrated in the same way as posterior predictive checks. That would require guessing the posterior and encoding it in the prior. The goal is make sure the prior is not so wide that it will pull probability mass away from feasible values.\n\n\n\nAndrew Gelman, Meng, and Stern (1996) discuss the case of mixed replication for hierarchical models in which the hyperparameters remain fixed, but varying effects are replicated. This is neither a purely prior nor purely posterior predictive check, but falls somewhere in between.\nFor example, consider a simple varying intercept logistic regression, with intercepts \\(\\alpha_k\\) for \\(k \\in 1:K\\). Each data item \\(y_n \\in \\{ 0, 1 \\}\\) is assumed to correspond to group \\(kk_n \\in 1:K.\\) The sampling distribution is thus \\[\ny_n \\sim \\textrm{bernoulli}(\\textrm{logit}^{-1}(\\alpha_{kk[n]})).\n\\] The varying intercepts have a hierarchical normal prior, \\[\n\\alpha_k \\sim \\textrm{normal}(\\mu, \\sigma).\n\\] The hyperparameters are themselves given weakly informative priors, \\[\\begin{eqnarray*}\n\\mu & \\sim & \\textrm{normal}(0, 2)\n\\\\[4pt]\n\\sigma & \\sim & \\textrm{lognormal}(0, 1).\n\\end{eqnarray*}\\]\nLike in a posterior predictive check, the hyperparameters \\(\\mu\\) and \\(\\sigma\\) are drawn from the posterior, \\[\n\\mu^{(m)}, \\sigma^{(m)} \\sim p(\\mu, \\sigma \\mid y)\n\\] Like in a prior predictive check, replicated values of \\(\\alpha\\) are drawn from the hyperparameters, \\[\n\\alpha^{\\textrm{rep}(m)}_k \\sim \\textrm{normal}(\\alpha_k \\mid\n\\mu^{(m)}, \\sigma^{(m)}).\n\\] The data items are then each replicated using the replicated intercepts, \\[\ny^{\\textrm{rep}(m)}_n \\sim\n\\textrm{bernoulli}\n  (\\textrm{logit}^{-1}(\\alpha^{\\textrm{rep}(m)}_{kk[n]})).\n\\] Thus the \\(y^{\\textrm{rep}(m)}\\) can be seen as a kind of posterior predictive replication of observations from new groups that were not among the original \\(K\\) groups.\nIn Stan, mixed predictive replications \\(y^{\\textrm{rep}(m)}\\) can be programmed directly.\ndata {\n  int&lt;lower=0&gt; K;\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=1, upper=K&gt; kk;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real mu;\n  real&lt;lower=0&gt; sigma;\n  vector&lt;offset=mu, multiplier=sigma&gt;[K] alpha;\n}\nmodel {\n  mu ~ normal(0, 2);               // hyperprior\n  sigma ~ lognormal(0, 1);\n  alpha ~ normal(mu, sigma);       // hierarchical prior\n  y ~ bernoulli_logit(alpha[kk]);  // sampling distribution\n}\ngenerated quantities {\n  // alpha replicated;  mu and sigma not replicated\n  array[K] real alpha_rep\n    = normal_rng(rep_vector(mu, K), sigma);\n  array[N] int&lt;lower=0, upper=1&gt; y_rep\n    = bernoulli_logit_rng(alpha_rep[kk]);\n}\n\n\n\nFollowing Andrew Gelman, Meng, and Stern (1996), prior, posterior, and mixed replications may all be defined as posteriors from joint models over parameters and observed and replicated data.\n\n\nFor example, posterior predictive replication may be formulated using sampling notation as follows. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] The heavily overloaded sampling notation is meant to indicate that both \\(y\\) and \\(y^{\\textrm{rep}}\\) are drawn from the same distribution, or more formally using capital letters to distinguish random variables, that the conditional densities \\(p_{Y^{\\textrm{rep}} \\mid\n\\Theta}\\) and \\(p_{Y \\mid \\Theta}\\) are the same.\nThe joint density is \\[\np(\\theta, y, y^{\\textrm{rep}})\n= p(\\theta) \\cdot p(y \\mid \\theta) \\cdot p(y^{\\textrm{rep}} \\mid \\theta).\n\\] This again is assuming that the two distributions for \\(y\\) and \\(y^{\\textrm{rep}}\\) are identical.\nThe variable \\(y\\) is observed, with the predictive simulation \\(y^{\\textrm{rep}}\\) and parameter vector \\(\\theta\\) not observed. The posterior is \\(p(y^{\\textrm{rep}}, \\theta \\mid y)\\). Given draws from the posterior, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are retained.\n\n\n\nThe prior predictive model simply drops the data component of the posterior predictive model. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] This corresponds to the joint density \\[\np(\\theta, y^{\\textrm{rep}}) = p(\\theta) \\cdot p(y^{\\textrm{rep}} \\mid\n\\theta).\n\\]\nIt is typically straightforward to draw \\(\\theta\\) from the prior and \\(y^{\\textrm{rep}}\\) from the sampling distribution given \\(\\theta\\) efficiently. In cases where it is not, the model may be coded and executed just as the posterior predictive model, only with no data.\n\n\n\nThe mixed replication corresponds to the model \\[\\begin{eqnarray*}\n\\phi & \\sim & p(\\phi)\n\\\\[2pt]\n\\alpha & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\alpha)\n\\\\[2pt]\n\\alpha^{\\textrm{rep}} & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\phi)\n\\end{eqnarray*}\\] The notation here is meant to indicate that \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\) have identical distributions, as do \\(y\\) and \\(y^{\\textrm{rep}}\\).\nThis corresponds to a joint model \\[\np(\\phi, \\alpha, \\alpha^{\\textrm{rep}}, y, y^{\\textrm{rep}})\n=\np(\\phi)\n\\cdot p(\\alpha \\mid \\phi)\n\\cdot p(y \\mid \\alpha)\n\\cdot p(\\alpha^{\\textrm{rep}} \\mid \\phi)\n\\cdot p(y^{\\textrm{rep}} \\mid \\alpha^{\\textrm{rep}}),\n\\] where \\(y\\) is the only observed variable, \\(\\alpha\\) contains the lower-level parameters and \\(\\phi\\) the hyperparameters. Note that \\(\\phi\\) is not replicated and instead appears in the distribution for both \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\).\nThe posterior is \\(p(\\phi, \\alpha, \\alpha^{\\textrm{rep}},\ny^{\\textrm{rep}} \\mid y)\\). From posterior draws, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are kept.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\nPosterior predictive checks for negative binomial data generating process and Poisson model.\nHistogram of means of replicated data sets; vertical red line at mean of original data.\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.",
+    "text": "Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.\nPrior predictive checks evaluate the prior the same way. Specifically, they evaluate what data sets would be consistent with the prior. They will not be calibrated with actual data, but extreme values help diagnose priors that are either too strong, too weak, poorly shaped, or poorly located.\nPrior and posterior predictive checks are two cases of the general concept of predictive checks, just conditioning on different things (no data and the observed data, respectively). For hierarchical models, there are intermediate versions, as discussed in the section on hierarchical models and mixed replication.\n\n\nThe posterior predictive distribution is the distribution over new observations given previous observations. It’s predictive in the sense that it’s predicting behavior on new data that is not part of the training set. It’s posterior in that everything is conditioned on observed data \\(y\\).\nThe posterior predictive distribution for replications \\(y^{\\textrm{rep}}\\) of the original data set \\(y\\) given model parameters \\(\\theta\\) is defined by \\[\np(y^{\\textrm{rep}} \\mid y)\n= \\int p(y^{\\textrm{rep}} \\mid \\theta)\n       \\cdot p(\\theta \\mid y) \\, \\textrm{d}\\theta.\n\\]\nAs with other posterior predictive quantities, generating a replicated data set \\(y^{\\textrm{rep}}\\) from the posterior predictive distribution is straightforward using the generated quantities block. Consider a simple regression model with parameters \\(\\theta = (\\alpha, \\beta, \\sigma).\\)\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n  vector[N] y;\n}\nparameters {\n  real alpha;\n  real beta;\n  real&lt;lower=0&gt; sigma;\n}\nmodel {\n  alpha ~ normal(0, 2);\n  beta ~ normal(0, 1);\n  sigma ~ normal(0, 1);\n  y ~ normal(alpha + beta * x, sigma);\n}\nTo generate a replicated data set y_rep for this simple model, the following generated quantities block suffices.\ngenerated quantities {\n  array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n}\nThe vectorized form of the normal random number generator is used with the original predictors x and the model parameters alpha, beta, and sigma. The replicated data variable y_rep is declared to be the same size as the original data y, but instead of a vector type, it is declared to be an array of reals to match the return type of the function normal_rng. Because the vector and real array types have the same dimensions and layout, they can be plotted against one another and otherwise compared during downstream processing.\nThe posterior predictive sampling for posterior predictive checks is different from usual posterior predictive sampling discussed in the chapter on posterior predictions in that the original predictors \\(x\\) are used. That is, the posterior predictions are for the original data.\n\n\n\nA standard posterior predictive check would plot a histogram of each replicated data set along with the original data set and compare them by eye. For this purpose, only a few replications are needed. These should be taken by thinning a larger set of replications down to the size needed to ensure rough independence of the replications.\nHere’s a complete example where the model is a simple Poisson with a weakly informative exponential prior with a mean of 10 and standard deviation of 10.\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0&gt; y;\n}\ntransformed data {\n  real&lt;lower=0&gt; mean_y = mean(to_vector(y));\n  real&lt;lower=0&gt; sd_y = sd(to_vector(y));\n}\nparameters {\n  real&lt;lower=0&gt; lambda;\n}\nmodel {\n  y ~ poisson(lambda);\n  lambda ~ exponential(0.2);\n}\ngenerated quantities {\n  array[N] int&lt;lower=0&gt; y_rep = poisson_rng(rep_array(lambda, N));\n  real&lt;lower=0&gt; mean_y_rep = mean(to_vector(y_rep));\n  real&lt;lower=0&gt; sd_y_rep = sd(to_vector(y_rep));\n  int&lt;lower=0, upper=1&gt; mean_gte = (mean_y_rep &gt;= mean_y);\n  int&lt;lower=0, upper=1&gt; sd_gte = (sd_y_rep &gt;= sd_y);\n}\nThe generated quantities block creates a variable y_rep for the replicated data, variables mean_y_rep and sd_y_rep for the statistics of the replicated data, and indicator variables mean_gte and sd_gte for whether the replicated statistic is greater than or equal to the statistic applied to the original data.\nNow consider generating data \\(y \\sim \\textrm{Poisson}(5)\\). The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\n\n\nWith a Poisson data-generating process and Poisson model, the posterior replications look similar to the original data. If it were easy to pick the original data out of the lineup, there would be a problem.\nNow consider generating over-dispersed data \\(y \\sim \\textrm{negative-binomial2}(5, 1).\\) This has the same mean as \\(\\textrm{Poisson}(5)\\), namely \\(5\\), but a standard deviation of \\(\\sqrt{5 + 5^2 /1} \\approx 5.5.\\) There is no way to fit this data with the Poisson model, because a variable distributed as \\(\\textrm{Poisson}(\\lambda)\\) has mean \\(\\lambda\\) and standard deviation \\(\\sqrt{\\lambda},\\) which is \\(\\sqrt{5}\\) for \\(\\textrm{Poisson}(5).\\) Here’s the resulting small multiples plot, again with original data in the upper left.\n\n\n\nPosterior predictive checks for negative binomial data generating process and Poisson model.\n\n\nThis time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.\n\n\n\nIf a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.\n\n\n\n\nPrior predictive checks generate data according to the prior in order to asses whether a prior is appropriate (Gabry et al. 2019). A posterior predictive check generates replicated data according to the posterior predictive distribution. In contrast, the prior predictive check generates data according to the prior predictive distribution, \\[\ny^{\\textrm{sim}} \\sim p(y).\n\\] The prior predictive distribution is just like the posterior predictive distribution with no observed data, so that a prior predictive check is nothing more than the limiting case of a posterior predictive check with no data.\nThis is easy to carry out mechanically by simulating parameters \\[\n\\theta^{\\textrm{sim}} \\sim p(\\theta)\n\\] according to the priors, then simulating data \\[\ny^{\\textrm{sim}} \\sim p(y \\mid \\theta^{\\textrm{sim}})\n\\] according to the sampling distribution given the simulated parameters. The result is a simulation from the joint distribution, \\[\n(y^{\\textrm{sim}}, \\theta^{\\textrm{sim}}) \\sim p(y, \\theta)\n\\] and thus \\[\ny^{\\textrm{sim}} \\sim p(y)\n\\] is a simulation from the prior predictive distribution.\n\n\nA prior predictive check is coded just like a posterior predictive check. If a posterior predictive check has already been coded and it’s possible to set the data to be empty, then no additional coding is necessary. The disadvantage to coding prior predictive checks as posterior predictive checks with no data is that Markov chain Monte Carlo will be used to sample the parameters, which is less efficient than taking independent draws using random number generation.\nPrior predictive checks can be coded entirely within the generated quantities block using random number generation. The resulting draws will be independent. Predictors must be read in from the actual data set—they do not have a generative model from which to be simulated. For a Poisson regression, prior predictive sampling can be encoded as the following complete Stan program.\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n}\ngenerated quantities {\n  real alpha = normal_rng(0, 1);\n  real beta = normal_rng(0, 1);\n  array[N] real y_sim = poisson_log_rng(alpha + beta * x);\n}\nRunning this program using Stan’s fixed-parameter sampler yields draws from the prior. These may be plotted to consider their appropriateness.\n\n\n\n\nSuppose we have a model for a football (aka soccer) league where there are \\(J\\) teams. Each team has a scoring rate \\(\\lambda_j\\) and in each game will be assumed to score \\(\\textrm{poisson}(\\lambda_j)\\) points. Yes, this model completely ignores defense. Suppose the modeler does not want to “put their thumb on the scale” and would rather “let the data speak for themselves” and so uses a prior with very wide tails, because it seems uninformative, such as the widely deployed \\[\n\\lambda_j \\sim \\textrm{gamma}(\\epsilon_1, \\epsilon_2).\n\\] This is not just a manufactured example; The BUGS Book recommends setting \\(\\epsilon = (0.5, 0.00001)\\), which corresponds to a Jeffreys prior for a Poisson rate parameter prior (Lunn et al. 2012, 85).\nSuppose the league plays a round-robin tournament wherein every team plays every other team. The following Stan model generates random team abilities and the results of such a round-robin tournament, which may be used to perform prior predictive checks.\ndata {\n  int&lt;lower=0&gt; J;\n  array[2] real&lt;lower=0&gt; epsilon;\n}\ngenerated quantities {\n  array[J] real&lt;lower=0&gt; lambda;\n  array[J, J] int y;\n  for (j in 1:J) lambda[j] = gamma_rng(epsilon[1], epsilon[2]);\n  for (i in 1:J) {\n    for (j in 1:J) {\n      y[i, j] = poisson_rng(lambda[i]) - poisson_rng(lambda[j]);\n    }\n  }\n}\nIn this simulation, teams play each other twice and play themselves once. This could be made more realistic by controlling the combinatorics to only generate a single result for each pair of teams, of which there are \\(\\binom{J}{2} = \\frac{J \\cdot (J - 1)}{2}.\\)\nUsing the \\(\\textrm{gamma}(0.5, 0.00001)\\) reference prior on team abilities, the following are the first 20 simulated point differences for the match between the first two teams, \\(y^{(1:20)}_{1, 2}\\).\n2597 -26000   5725  22496   1270   1072   4502  -2809   -302   4987\n7513   7527  -3268 -12374   3828   -158 -29889   2986  -1392     66\nThat’s some pretty highly scoring football games being simulated; all but one has a score differential greater than 100! In other words, this \\(\\textrm{gamma}(0.5, 0.00001)\\) prior is putting around 95% of its weight on score differentials above 100. Given that two teams combined rarely score 10 points, this prior is way out of line with prior knowledge about football matches; it is not only consistent with outcomes that have never occurred in the history of the sport, it puts most of the prior probability mass there.\nThe posterior predictive distribution can be strongly affected by the prior when there is not much observed data and substantial prior mass is concentrated around infeasible values (A. Gelman 2006).\nJust as with posterior predictive distributions, any statistics of the generated data may be evaluated. Here, the focus was on score difference between a single pair of teams, but it could’ve been on maximums, minimums, averages, variances, etc.\nIn this textbook example, the prior is univariate and directly related to the expected number of points scored, and could thus be directly inspected for consistency with prior knowledge about scoring rates in football. There will not be the same kind of direct connection when the prior and sampling distributions are multivariate. In these more challenging situations, prior predictive checks are an easy way to get a handle on the implications of a prior in terms of what it says the data is going to look like; for a more complex application involving spatially heterogeneous air pollution concentration, see (Gabry et al. 2019).\nPrior predictive checks can also be compared with the data, but one should not expect them to be calibrated in the same way as posterior predictive checks. That would require guessing the posterior and encoding it in the prior. The goal is make sure the prior is not so wide that it will pull probability mass away from feasible values.\n\n\n\nAndrew Gelman, Meng, and Stern (1996) discuss the case of mixed replication for hierarchical models in which the hyperparameters remain fixed, but varying effects are replicated. This is neither a purely prior nor purely posterior predictive check, but falls somewhere in between.\nFor example, consider a simple varying intercept logistic regression, with intercepts \\(\\alpha_k\\) for \\(k \\in 1:K\\). Each data item \\(y_n \\in \\{ 0, 1 \\}\\) is assumed to correspond to group \\(kk_n \\in 1:K.\\) The sampling distribution is thus \\[\ny_n \\sim \\textrm{bernoulli}(\\textrm{logit}^{-1}(\\alpha_{kk[n]})).\n\\] The varying intercepts have a hierarchical normal prior, \\[\n\\alpha_k \\sim \\textrm{normal}(\\mu, \\sigma).\n\\] The hyperparameters are themselves given weakly informative priors, \\[\\begin{eqnarray*}\n\\mu & \\sim & \\textrm{normal}(0, 2)\n\\\\[4pt]\n\\sigma & \\sim & \\textrm{lognormal}(0, 1).\n\\end{eqnarray*}\\]\nLike in a posterior predictive check, the hyperparameters \\(\\mu\\) and \\(\\sigma\\) are drawn from the posterior, \\[\n\\mu^{(m)}, \\sigma^{(m)} \\sim p(\\mu, \\sigma \\mid y)\n\\] Like in a prior predictive check, replicated values of \\(\\alpha\\) are drawn from the hyperparameters, \\[\n\\alpha^{\\textrm{rep}(m)}_k \\sim \\textrm{normal}(\\alpha_k \\mid\n\\mu^{(m)}, \\sigma^{(m)}).\n\\] The data items are then each replicated using the replicated intercepts, \\[\ny^{\\textrm{rep}(m)}_n \\sim\n\\textrm{bernoulli}\n  (\\textrm{logit}^{-1}(\\alpha^{\\textrm{rep}(m)}_{kk[n]})).\n\\] Thus the \\(y^{\\textrm{rep}(m)}\\) can be seen as a kind of posterior predictive replication of observations from new groups that were not among the original \\(K\\) groups.\nIn Stan, mixed predictive replications \\(y^{\\textrm{rep}(m)}\\) can be programmed directly.\ndata {\n  int&lt;lower=0&gt; K;\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=1, upper=K&gt; kk;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real mu;\n  real&lt;lower=0&gt; sigma;\n  vector&lt;offset=mu, multiplier=sigma&gt;[K] alpha;\n}\nmodel {\n  mu ~ normal(0, 2);               // hyperprior\n  sigma ~ lognormal(0, 1);\n  alpha ~ normal(mu, sigma);       // hierarchical prior\n  y ~ bernoulli_logit(alpha[kk]);  // sampling distribution\n}\ngenerated quantities {\n  // alpha replicated;  mu and sigma not replicated\n  array[K] real alpha_rep\n    = normal_rng(rep_vector(mu, K), sigma);\n  array[N] int&lt;lower=0, upper=1&gt; y_rep\n    = bernoulli_logit_rng(alpha_rep[kk]);\n}\n\n\n\nFollowing Andrew Gelman, Meng, and Stern (1996), prior, posterior, and mixed replications may all be defined as posteriors from joint models over parameters and observed and replicated data.\n\n\nFor example, posterior predictive replication may be formulated using sampling notation as follows. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] The heavily overloaded sampling notation is meant to indicate that both \\(y\\) and \\(y^{\\textrm{rep}}\\) are drawn from the same distribution, or more formally using capital letters to distinguish random variables, that the conditional densities \\(p_{Y^{\\textrm{rep}} \\mid\n\\Theta}\\) and \\(p_{Y \\mid \\Theta}\\) are the same.\nThe joint density is \\[\np(\\theta, y, y^{\\textrm{rep}})\n= p(\\theta) \\cdot p(y \\mid \\theta) \\cdot p(y^{\\textrm{rep}} \\mid \\theta).\n\\] This again is assuming that the two distributions for \\(y\\) and \\(y^{\\textrm{rep}}\\) are identical.\nThe variable \\(y\\) is observed, with the predictive simulation \\(y^{\\textrm{rep}}\\) and parameter vector \\(\\theta\\) not observed. The posterior is \\(p(y^{\\textrm{rep}}, \\theta \\mid y)\\). Given draws from the posterior, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are retained.\n\n\n\nThe prior predictive model simply drops the data component of the posterior predictive model. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] This corresponds to the joint density \\[\np(\\theta, y^{\\textrm{rep}}) = p(\\theta) \\cdot p(y^{\\textrm{rep}} \\mid\n\\theta).\n\\]\nIt is typically straightforward to draw \\(\\theta\\) from the prior and \\(y^{\\textrm{rep}}\\) from the sampling distribution given \\(\\theta\\) efficiently. In cases where it is not, the model may be coded and executed just as the posterior predictive model, only with no data.\n\n\n\nThe mixed replication corresponds to the model \\[\\begin{eqnarray*}\n\\phi & \\sim & p(\\phi)\n\\\\[2pt]\n\\alpha & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\alpha)\n\\\\[2pt]\n\\alpha^{\\textrm{rep}} & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\phi)\n\\end{eqnarray*}\\] The notation here is meant to indicate that \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\) have identical distributions, as do \\(y\\) and \\(y^{\\textrm{rep}}\\).\nThis corresponds to a joint model \\[\np(\\phi, \\alpha, \\alpha^{\\textrm{rep}}, y, y^{\\textrm{rep}})\n=\np(\\phi)\n\\cdot p(\\alpha \\mid \\phi)\n\\cdot p(y \\mid \\alpha)\n\\cdot p(\\alpha^{\\textrm{rep}} \\mid \\phi)\n\\cdot p(y^{\\textrm{rep}} \\mid \\alpha^{\\textrm{rep}}),\n\\] where \\(y\\) is the only observed variable, \\(\\alpha\\) contains the lower-level parameters and \\(\\phi\\) the hyperparameters. Note that \\(\\phi\\) is not replicated and instead appears in the distribution for both \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\).\nThe posterior is \\(p(\\phi, \\alpha, \\alpha^{\\textrm{rep}},\ny^{\\textrm{rep}} \\mid y)\\). From posterior draws, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are kept.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\nPosterior predictive checks for negative binomial data generating process and Poisson model.\nHistogram of means of replicated data sets; vertical red line at mean of original data.\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.",
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     "href": "stan-users-guide/posterior-predictive-checks.html#posterior-p-values",
     "title": "Posterior and Prior Predictive Checks",
     "section": "",
-    "text": "If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.",
+    "text": "If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.",
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     "href": "reference-manual/types.html",
     "title": "Data Types and Declarations",
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-    "text": "This chapter covers the data types for expressions in Stan. Every variable used in a Stan program must have a declared data type. Only values of that type will be assignable to the variable (except for temporary states of transformed data and transformed parameter values). This follows the convention of programming languages like C++, not the conventions of scripting languages like Python or statistical languages such as R or BUGS.\nThe motivation for strong, static typing is threefold.\n\nStrong typing forces the programmer’s intent to be declared with the variable, making programs easier to comprehend and hence easier to debug and maintain.\nStrong typing allows programming errors relative to the declared intent to be caught sooner (at compile time) rather than later (at run time). The Stan compiler (called through an interface such as CmdStan, RStan, or PyStan) will flag any type errors and indicate the offending expressions quickly when the program is compiled.\nConstrained types will catch runtime data, initialization, and intermediate value errors as soon as they occur rather than allowing them to propagate and potentially pollute final results.\n\nStrong typing disallows assigning the same variable to objects of different types at different points in the program or in different invocations of the program.\n\n\nArguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.\n\n\n\n\nUnfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.\n\n\n\n\nThe complex data type is a scalar, but unlike real and int types, it contains two components, a real and imaginary component, both of which are of type real. That is, the real and imaginary components of a complex number are 64-bit, IEEE 754-complaint floating point numbers.\n\n\nImaginary literals are written in mathematical notation using a numeral followed by the suffix i. For example, the following example constructs a complex number \\(2 - 1.3i\\) and assigns it to the variable z.\ncomplex z = 2 - 1.3i;\nreal re = get_real(z);  // re has value 2.0\nreal im = get_imag(z);  // im has value -1.3\nThe getter functions then extract the real and imaginary components of z and assign them to re and im respectively.\nThe function to_complex constructs a complex number from its real and imaginary components. The functional form needs to be used whenever the components are not literal numerals, as in the following example.\nvector[K] re;\nvector[K] im;\n// ...\nfor (k in 1:K) {\n  complex z = to_complex(re[k], im[k]);\n  // ...\n}\n\n\n\nExpressions of type real may be assigned to variables of type complex. For example, the following is a valid sequence of Stan statements.\nreal x = 5.0;\ncomplex z = x;  // get_real(z) == 5.0, get_imag(z) == 0\nThe real number assigned to a complex number determine’s the complex number’s real component, with the imaginary component set to zero.\nAssignability is transitive, so that expressions of type int may also be assigned to variables of type complex, as in the following example.\nint n = 2;\ncomplex z = n;\nFunction arguments also support promotion of integer or real typed expressions to type complex.\n\n\n\n\nAll variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be read as data in the dump format.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.\n\n\n\n\nStan provides three types of container objects: arrays, vectors, and matrices. Vectors and matrices are more limited kinds of data structures than arrays. Vectors are intrinsically one-dimensional collections of real or complex values, whereas matrices are intrinsically two dimensional. Vectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical. A \\(3 \\times 4\\) matrix is a different kind of object in Stan than a \\(3\n\\times 4\\) array.\nThe intention of using matrix types is to call out their usage in the code. There are three situations in Stan where only vectors and matrices may be used,\n\nmatrix arithmetic operations (e.g., matrix multiplication)\nlinear algebra functions (e.g., eigenvalues and determinants), and\nmultivariate function parameters and outcomes (e.g., multivariate normal distribution arguments).\n\nVectors and matrices cannot be typed to return integer values. They are restricted to real and complex values.\nFor constructing vectors and matrices in Stan, see Vector, Matrix, and Array Expressions.\n\n\nVectors and matrices, as well as arrays, are indexed starting from one (1) in Stan. This follows the convention in statistics and linear algebra as well as their implementations in the statistical software packages R, MATLAB, BUGS, and JAGS. General computer programming languages, on the other hand, such as C++ and Python, index arrays starting from zero.\n\n\n\nVectors in Stan are column vectors; see below for information on row vectors. Vectors are declared with a size (i.e., a dimensionality). For example, a 3-dimensional real vector is declared with the keyword vector, as follows.\nvector[3] u;\nVectors may also be declared with constraints, as in the following declaration of a 3-vector of non-negative values.\nvector&lt;lower=0&gt;[3] u;\nSimilarly, they may be declared with a offset and/or multiplier, as in the following example\nvector&lt;offset=42, multiplier=3&gt;[3] u;\n\n\n\nLike real vectors, complex vectors are column vectors and are declared with a size. For example, a 3-dimensional complex vector is declared with the keyword complex_vector, as follows.\ncomplex_vector[3] v;\nComplex vector declarations do not support any constraints.\n\n\n\nA unit simplex is a vector with non-negative values whose entries sum to 1. For instance, \\([0.2,0.3,0.4,0.1]^{\\top}\\) is a unit 4-simplex. Unit simplexes are most often used as parameters in categorical or multinomial distributions, and they are also the sampled variate in a Dirichlet distribution. Simplexes are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-simplex by\nsimplex[5] theta;\nUnit simplexes are implemented as vectors and may be assigned to other vectors and vice-versa. Simplex variables, like other constrained variables, are validated to ensure they contain simplex values; for simplexes, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\nIn high dimensional problems, simplexes may require smaller step sizes in the inference algorithms in order to remain stable; this can be achieved through higher target acceptance rates for samplers and longer warmup periods, tighter tolerances for optimization with more iterations, and in either case, with less dispersed parameter initialization or custom initialization if there are informative priors for some parameters.\n\n\n\nA unit vector is a vector with a norm of one. For instance, \\([0.5,\n0.5, 0.5, 0.5]^{\\top}\\) is a unit 4-vector. Unit vectors are sometimes used in directional statistics. Unit vectors are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-vector by\nunit_vector[5] theta;\nUnit vectors are implemented as vectors and may be assigned to other vectors and vice-versa. Unit vector variables, like other constrained variables, are validated to ensure that they are indeed unit length; for unit vectors, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\n\n\n\nAn ordered vector type in Stan represents a vector whose entries are sorted in ascending order. For instance, \\((-1.3,2.7,2.71)^{\\top}\\) is an ordered 3-vector. Ordered vectors are most often employed as cut points in ordered logistic regression models (see section).\nThe variable c is declared as an ordered 5-vector by\nordered[5] c;\nAfter their declaration, ordered vectors, like unit simplexes, may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nThere is also a positive, ordered vector type which operates similarly to ordered vectors, but all entries are constrained to be positive. For instance, \\((2,3.7,4,12.9)\\) is a positive, ordered 4-vector.\nThe variable d is declared as a positive, ordered 5-vector by\npositive_ordered[5] d;\nLike ordered vectors, after their declaration, positive ordered vectors may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nRow vectors are declared with the keyword row_vector. Like (column) vectors, they are declared with a size. For example, a 1093-dimensional row vector u would be declared as\nrow_vector[1093] u;\nConstraints are declared as for vectors, as in the following example of a 10-vector with values between -1 and 1.\nrow_vector&lt;lower=-1, upper=1&gt;[10] u;\nOffset and multiplier are also similar as for the following 3-row-vector with offset -42 and multiplier 3.\nrow_vector&lt;offset=-42, multiplier=3&gt;[3] u;\nRow vectors may not be assigned to column vectors, nor may column vectors be assigned to row vectors. If assignments are required, they may be accommodated through the transposition operator.\n\n\n\nComplex row vectors are declared with the keyword complex_row_vector and given a size in basic declarations. For example, a 12-dimensional complex row vector v would be declared as\ncomplex_row_vector[12] v;\nComplex row vectors do not allow constraints.\n\n\n\nMatrices are declared with the keyword matrix along with a number of rows and number of columns. For example,\nmatrix[3, 3] A;\nmatrix[M, N] B;\ndeclares A to be a \\(3 \\times 3\\) matrix and B to be a \\(M\n\\times N\\) matrix. For the second declaration to be well formed, the variables M and N must be declared as integers in either the data or transformed data block and before the matrix declaration.\nMatrices may also be declared with constraints, as in this (\\(3 \\times 4\\)) matrix of non-positive values.\nmatrix&lt;upper=0&gt;[3, 4] B;\nSimilarly, matrices can be declared to have a set offset and/or multiplier, as in this matrix with multiplier 5.\nmatrix&lt;multiplier=5&gt;[3, 4] B;\n\n\nRows of a matrix can be assigned by indexing the left-hand side of an assignment statement. For example, this is possible.\nmatrix[M, N] a;\nrow_vector[N] b;\n// ...\na[1] = b;\nThis copies the values from row vector b to a[1], which is the first row of the matrix a. If the number of columns in a is not the same as the size of b, a run-time error is raised; the number of columns of a is N, which is also the number of columns of b.\nAssignment works by copying values in Stan. That means any subsequent assignment to a[1] does not affect b, nor does an assignment to b affect a.\n\n\n\n\nComplex matrices are declared with the keyword complex_matrix and a number of rows and columns. For example,\ncomplex_matrix[3, 3] C;\nComplex matrices do not allow constraints.\n\n\n\nMatrix variables may be constrained to represent covariance matrices. A matrix is a covariance matrix if it is symmetric and positive definite. Like correlation matrices, covariance matrices only need a single dimension in their declaration. For instance,\ncov_matrix[K] Omega;\ndeclares Omega to be a \\(K \\times K\\) covariance matrix, where \\(K\\) is the value of the data variable K.\n\n\n\nMatrix variables may be constrained to represent correlation matrices. A matrix is a correlation matrix if it is symmetric and positive definite, has entries between \\(-1\\) and \\(1\\), and has a unit diagonal. Because correlation matrices are square, only one dimension needs to be declared. For example,\ncorr_matrix[3] Sigma;\ndeclares Sigma to be a \\(3 \\times 3\\) correlation matrix.\nCorrelation matrices may be assigned to other matrices, including unconstrained matrices, if their dimensions match, and vice-versa.\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a covariance matrix. This is often more convenient or more efficient than representing covariance matrices directly.\nA Cholesky factor \\(L\\) is an \\(M \\times N\\) lower-triangular matrix (if \\(m &lt; n\\) then \\(L[m, n] =0\\)) with a strictly positive diagonal (\\(L[k, k]\n&gt; 0\\)) and \\(M \\geq N\\). If \\(L\\) is a Cholesky factor, then \\(\\Sigma = L\n\\, L^{\\top}\\) is a covariance matrix (i.e., it is positive definite). The mapping between positive definite matrices and their Cholesky factors is bijective—every covariance matrix has a unique Cholesky factorization.\nThe typical case of a square Cholesky factor may be declared with a single dimension,\ncholesky_factor_cov[4] L;\n\n\nIn general, two dimensions may be declared, with the above being equal to cholesky_factor_cov[4, 4]. The type cholesky_factor_cov[M, N] may be used for the general \\(M \\times N\\) case to produce positive semi-definite matrices of rank \\(M\\).\n\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a correlation matrix.\nA Cholesky factor for a correlation matrix \\(L\\) is a \\(K \\times K\\) lower-triangular matrix with positive diagonal entries and rows that are of length 1 (i.e., \\(\\sum_{n=1}^K L_{m,n}^2 = 1\\)). If \\(L\\) is a Cholesky factor for a correlation matrix, then \\(L\\,L^{\\top}\\) is a correlation matrix (i.e., symmetric positive definite with a unit diagonal).\nTo declare the variable L to be a K by K Cholesky factor of a correlation matrix, the following code may be used.\ncholesky_factor_corr[K] L;\n\n\n\nConstrained variables of all types may be assigned to other variables of the same unconstrained type and vice-versa. Matching is interpreted strictly as having the same basic type and number of array dimensions. Constraints are not considered, but basic data types are. For instance, a variable declared to be real&lt;lower=0, upper=1&gt; could be assigned to a variable declared as real and vice-versa. Similarly, a variable declared as matrix[3, 3] may be assigned to a variable declared as cov_matrix[3] or cholesky_factor_cov[3], and vice-versa.\nChecks are carried out at the end of each relevant block of statements to ensure constraints are enforced. This includes run-time size checks. The Stan compiler isn’t able to catch the fact that an attempt may be made to assign a matrix of one dimensionality to a matrix of mismatching dimensionality.\n\n\n\nReal-valued vectors, row vectors and matrices may be assigned to complex-valued vectors, row vectors and matrices, respectively. For example, the following is legal.\nvector[N] v = ...;\ncomplex_vector[N] u = 2 * v;\nRow vectors and matrices work the same way.\n\n\n\nVariables may be declared with sizes given by expressions. Such expressions are constrained to only contain data or transformed data variables. This ensures that all sizes are determined once the data is read in and transformed data variables defined by their statements. For example, the following is legal.\ndata {\n  int&lt;lower=0&gt; N_observed, N_missing;\n  // ...\ntransformed parameters {\n  vector[N_observed + N_missing] y;\n  // ...\n\n\n\nIf v is a column vector or row vector, then v[2] is the second element in the vector. If m is a matrix, then m[2, 3] is the value in the second row and third column.\nProviding a matrix with a single index returns the specified row. For instance, if m is a matrix, then m[2] is the second row. This allows Stan blocks such as\nmatrix[M, N] m;\nrow_vector[N] v;\nreal x;\n// ...\nv = m[2];\nx = v[3];   // x == m[2][3] == m[2, 3]\nThe type of m[2] is row_vector because it is the second row of m. Thus it is possible to write m[2][3] instead of m[2, 3] to access the third element in the second row. When given a choice, the form m[2, 3] is preferred.\nComplex versions work the same way,\ncomplex_matrix[M, N] m = ...;\ncomplex_row_vector[N] u = m[3];\ncomplex_vector[M] v = m[ , 2];\n\n\nThe form m[2, 3] is more efficient because it does not require the creation and use of an intermediate expression template for m[2]. In later versions, explicit calls to m[2][3] may be optimized to be as efficient as m[2, 3] by the Stan compiler.\n\n\n\n\nAn integer expression is used to pick out the sizes of vectors, matrices, and arrays. For instance, we can declare a vector of size M + N using\nvector[M + N] y;\nAny integer-denoting expression may be used for the size declaration, providing all variables involved are either data, transformed data, or local variables. That is, expressions used for size declarations may not include parameters or transformed parameters or generated quantities.\n\n\n\n\nStan supports arrays of arbitrary dimension. The values in an array can be any type, so that arrays may contain values that are simple reals or integers, vectors, matrices, or other arrays. Arrays are the only way to store sequences of integers, and some functions in Stan, such as discrete distributions, require integer arguments.\nA two-dimensional array is just an array of arrays, both conceptually and in terms of current implementation. When an index is supplied to an array, it returns the value at that index. When more than one index is supplied, this indexing operation is chained. For example, if a is a two-dimensional array, then a[m, n] is just a convenient shorthand for a[m][n].\nVectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical.\nFor constructing arrays in Stan, see Vector, Matrix, and Array Expressions.\n\n\nArrays are declared with the keyword array followed by the dimensions enclosed in square brackets, the element type, and the name of the variable.\nThe variable n is declared as an array of five integers as follows.\narray[5] int n;\nA two-dimensional array of complex values with three rows and four columns is declared as follows.\narray[3, 4] complex a;\nA three-dimensional array z of positive reals with five rows, four columns, and two shelves can be declared as follows.\narray[5, 4, 2] real&lt;lower=0&gt; z;\nArrays may also be declared to contain vectors. For example,\narray[3] vector[7] mu;\ndeclares mu to be an array of size 3 containing vectors with 7 elements. Arrays may also contain matrices. The example\narray[15, 12] complex_matrix[7, 2] mu;\ndeclares a 15 by 12 array of \\(7 \\times 2\\) complex matrices. Any of the constrained types may also be used in arrays, as in the declaration\narray[2, 3, 4] cholesky_factor_cov[5, 6] mu;\nof a \\(2 \\times 3 \\times 4\\) array of \\(5 \\times 6\\) Cholesky factors of covariance matrices.\n\n\n\nIf x is a 1-dimensional array of length 5, then x[1] is the first element in the array and x[5] is the last. For a \\(3\n\\times 4\\) array y of two dimensions, y[1, 1] is the first element and y[3, 4] the last element. For a three-dimensional array z, the first element is z[1, 1, 1], and so on.\nSubarrays of arrays may be accessed by providing fewer than the full number of indexes. For example, suppose y is a two-dimensional array with three rows and four columns. Then y[3] is one-dimensional array of length four. This means that y[3][1] may be used instead of y[3, 1] to access the value of the first column of the third row of y. The form y[3, 1] is the preferred form (see note in this chapter).\n\n\n\nSubarrays may be manipulated and assigned just like any other variables. Similar to the behavior of matrices, Stan allows blocks such as\narray[9, 10, 11] real w;\narray[10, 11] real x;\narray[11] real y;\nreal z;\n// ...\nx = w[5];\ny = x[4];  // y == w[5][4] == w[5, 4]\nz = y[3];  // z == w[5][4][3] == w[5, 4, 3]\nComplex-valued arrays work the same way.\n\n\n\nArrays of vectors and matrices are accessed in the same way as arrays of doubles. Consider the following vector and scalar declarations.\narray[3, 4] vector[5] a;\narray[4] vector[5] b;\nvector[5] c;\nreal x;\nWith these declarations, the following assignments are legal.\nb = a[1];       // result is array of vectors\nc = a[1, 3];    // result is vector\nc = b[3];       //   same result as above\nx = a[1, 3, 5]; // result is scalar\nx = b[3, 5];    //   same result as above\nx = c[5];       //   same result as above\nRow vectors and other derived vector types (simplex and ordered) behave the same way in terms of indexing.\nConsider the following matrix, vector and scalar declarations.\narray[3, 4] matrix[6, 5] d;\narray[4] matrix[6, 5] e;\nmatrix[6, 5] f;\nrow_vector[5] g;\nreal x;\nWith these declarations, the following definitions are legal.\ne = d[1];           // result is array of matrices\nf = d[1, 3];        // result is matrix\nf = e[3];           //   same result as above\ng = d[1, 3, 2];     // result is row vector\ng = e[3, 2];        //   same result as above\ng = f[2];           //   same result as above\nx = d[1, 3, 5, 2];  // result is scalar\nx = e[3, 5, 2];     //   same result as above\nx = f[5, 2];        //   same result as above\nx = g[2];           //   same result as above\nAs shown, the result f[2] of supplying a single index to a matrix is the indexed row, here row 2 of matrix f.\n\n\n\nSubarrays of arrays may be assigned by indexing on the left-hand side of an assignment statement. For example, the following is legal.\narray[I, J, K] real x;\narray[J, K] real y;\narray[K] real z;\n// ...\nx[1] = y;\nx[1, 1] = z;\nThe sizes must match. Here, x[1] is a J by K array, as is y.\nPartial array assignment also works for arrays of matrices, vectors, and row vectors.\n\n\n\nArrays, row vectors, column vectors and matrices are not interchangeable in Stan. Thus a variable of any one of these fundamental types is not assignable to any of the others, nor may it be used as an argument where the other is required (use as arguments follows the assignment rules).\n\n\nFor example, vectors cannot be assigned to arrays or vice-versa.\narray[4] real a;\nvector[4] b;\nrow_vector[4] c;\n// ...\na = b; // illegal assignment of vector to array\nb = a; // illegal assignment of array to vector\na = c; // illegal assignment of row vector to array\nc = a; // illegal assignment of array to row vector\n\n\n\nIt is not even legal to assign row vectors to column vectors or vice versa.\nvector[4] b;\nrow_vector[4] c;\n// ...\nb = c; // illegal assignment of row vector to column vector\nc = b; // illegal assignment of column vector to row vector\n\n\n\nThe same holds for matrices, where 2-dimensional arrays may not be assigned to matrices or vice-versa.\narray[3, 4] real a;\nmatrix[3, 4] b;\n// ...\na = b;  // illegal assignment of matrix to array\nb = a;  // illegal assignment of array to matrix\n\n\n\nA \\(1 \\times N\\) matrix cannot be assigned a row vector or vice versa.\nmatrix[1, 4] a;\nrow_vector[4] b;\n// ...\na = b;  // illegal assignment of row vector to matrix\nb = a;  // illegal assignment of matrix to row vector\nSimilarly, an \\(M \\times 1\\) matrix may not be assigned to a column vector.\nmatrix[4, 1] a;\nvector[4] b;\n// ...\na = b;  // illegal assignment of column vector to matrix\nb = a;  // illegal assignment of matrix to column vector\n\n\n\n\nAn integer expression is used to pick out the sizes of arrays. The same restrictions as for vector and matrix sizes apply, namely that the size is declared with an integer-denoting expression that does not contain any parameters, transformed parameters, or generated quantities.\n\n\n\nIf any of an array’s dimensions is size zero, the entire array will be of size zero. That is, if we declare\narray[3, 0] real a;\nthen the resulting size of a is zero and querying any of its dimensions at run time will result in the value zero. Declared as above, a[1] will be a size-zero one-dimensional array. For comparison, declaring\narray[0, 3] real b;\nalso produces an array with an overall size of zero, but in this case, there is no way to index legally into b, because b[0] is undefined. The array will behave at run time as if it’s a \\(0 \\times\n0\\) array. For example, the result of to_matrix(b) will be a \\(0 \\times 0\\) matrix, not a \\(0 \\times 3\\) matrix.\n\n\n\n\nStan supports tuples of arbitrary size. The values in a tuple can be of arbitrary type, but the component types must be declared along with the declaration of the tuple. Tuples can be manipulated as a whole, or their elements may be accessed and set individually.\n\n\nTuples are declared with the keyword tuple followed by a parenthesized sequence of types, which determine the types of the respective tuple entries. For example, a tuple with three elements may be declared as\ntuple(int, vector[3], complex) abc;\nTuples must have at least two entries, so the following declarations are illegal.\ntuple() nil;  // ILLEGAL\ntuple(int) n;  // ILLEGAL\nTuples can be assigned as a whole if their elements can be assigned individually. For example, a can be assigned to b in the following example because int can be promoted to complex.\ntuple(int, real) a;\n...\ntuple(complex, real) b = a;\nTuple types may have elements which are declared as tuples, such as the following example.\ntuple(int, tuple(real, complex)) x;\nIn this case, it would probably be simpler to use a 3-tuple type, tuple(int, real, complex).\nTuples can be declared with constraints anywhere that ordinary variables can (i.e., as top-level block variables). That means any context in which it is legal to have a declaration\nreal&lt;lower=0&gt; sigma;\nreal&lt;lower=0, upper=1&gt; theta;\nit is legal to have a tuple with constraints such as\ntuple(real&lt;lower=0&gt;, real&lt;lower=0, upper=1&gt;) sigma_theta;\n\n\n\nTuple elements may be accessed directly. For example, with our declaration of abc from the last section, Stan uses abc.1 for the first element, abc.2 for the second, and abc.3 for the third. These numbers must be integer literals (i.e., they cannot be variables), and must be within the size of the number of elements of tuples. The types of elements are as declared, so that abc.1 is of type int, abc.2 of type vector[3] and abc.3 of type complex.\n\n\n\nTuple elements can be assigned individually, allowing, e.g.,\ntuple(int, real) ab;\nab.1 = 123;\nab.2 = 12.9;\nAs with other assignments, promotions will happen if necessary (of int to real and of real to complex, along with the corresponding container type promotions).\n\n\n\nFor convenience of using values stored in tuples, Stan supports “unpacking” (or “destructuring”) of tuples in an assignment statement.\nGiven a tuple t of type tuple(T1, ..., Tn) and a sequence of assignable expressions of types v1, …, vn, where each vi has a type which is assignable from type Ti, individual elements of the tuple may be assigned to the corresponding variables in the sequence by the statement\n(v1, /*...*/, vn) = t;\nNote that the above parenthesis are required, unlike in some other languages with similar features (e.g., Python).\nThese unpacking assignments can be nested if the tuple on the right hand side contains nested tuples.\nFor example, if T is a tuple of type tuple(int, (real, real), complex), then the program\nint i;\nreal x, y;\ncomplex z;\n\n(i, (x, y), z) = T;\nAssigns the result of T.1 to i, the result of T.2.1 to x, the result of T.2.2 to y, and the result of T.3 to z.\nThe left hand side must match in size the tuple on the right. Additionally, the same variable may not appear more than once in the left hand side of an unpacking assignment.\n\n\n\n\nThe type information associated with a variable only contains the underlying type and dimensionality of the variable.\n\n\nThe size associated with a given variable is not part of its data type. For example, declaring a variable using\narray[3] real a;\ndeclares the variable a to be an array. The fact that it was declared to have size 3 is part of its declaration, but not part of its underlying type.\n\n\nSizes are determined dynamically (at run time) and thus cannot be type-checked statically when the program is compiled. As a result, any conformance error on size will raise a run-time error. For example, trying to assign an array of size 5 to an array of size 6 will cause a run-time error. Similarly, multiplying an \\(N\n\\times M\\) by a \\(J \\times K\\) matrix will raise a run-time error if \\(M\n\\neq J\\).\n\n\n\n\nLike sizes, constraints are not treated as part of a variable’s type in Stan when it comes to the compile-time check of operations it may participate in. Anywhere Stan accepts a matrix as an argument, it will syntactically accept a correlation matrix or covariance matrix or Cholesky factor. Thus a covariance matrix may be assigned to a matrix and vice-versa.\nSimilarly, a bounded real may be assigned to an unconstrained real and vice-versa.\n\n\nFor arguments to functions, constraints are sometimes, but not always checked when the function is called. Exclusions include C++ standard library functions. All probability functions and cumulative distribution functions check that their arguments are appropriate at run time as the function is called.\n\n\n\nFor data variables, constraints are checked after the variable is read from a data file or other source. For transformed data variables, the check is done after the statements in the transformed data block have executed. Thus it is legal for intermediate values of variables to not satisfy declared constraints.\nFor parameters, constraints are enforced by the transform applied and do not need to be checked. For transformed parameters, the check is done after the statements in the transformed parameter block have executed.\nFor all blocks defining variables (transformed data, transformed parameters, generated quantities), real values are initialized to NaN and integer values are initialized to the smallest legal integer (i.e., a large absolute value negative number).\nFor generated quantities, constraints are enforced after the statements in the generated quantities block have executed.\n\n\n\n\nIn order to refer to data types, it is convenient to have a way to refer to them. The type naming notation outlined in this section is not part of the Stan programming language, but rather a convention adopted in this document to enable a concise description of a type.\nBecause size information is not part of a data type, data types will be written without size information. For instance, array[] real is the type of one-dimensional array of reals and matrix is the type of matrices. The three-dimensional integer array type is written as array[,,] int, indicating the number slots available for indexing. Similarly, array[,] vector is the type of a two-dimensional array of vectors.\n\n\n\n\nVariables in Stan are declared by giving a type and a name. For example\nint N;\nvector[N] y;\narray[5] matrix[3, 4] A;\ndeclares a variable N that is an integer, a variable y that is a vector of length N (the previously declared variable), and a variable A, which is a length-5 array where each element is a 3 by 4 matrix.\nThere are several different places a variable is declared in Stan. They are block variables, like those inside data, which can have constraints and must include sizes for their types, like in the above examples. Local variables, like those defined inside loops or local blocks cannot be constrained, but still include sizes. Finally, variables declared as function parameters are not constrained types and exclude sizes.\n// valid block variables, but not locals or function parameters\nvector&lt;lower=0&gt;[N] u;\n\n// valid as a block or local variable, but not a function parameter\narray[3] int is;\n\n// function parameters exclude sizes and cannot be constrained\nvoid pretty_print_tri_lower(matrix x) { ... }\nIn the following table, the leftmost column is a list of the unconstrained and undimensioned basic types; these are used as function return types and argument types. The middle column is of unconstrained types with dimensions; these are used as local variable types. The variables M and N indicate number of columns and rows, respectively. The variable K is used for square matrices, i.e., K denotes both the number of rows and columns. The rightmost column lists the corresponding constrained types. An expression of any right-hand column type may be assigned to its corresponding left-hand column basic type. At runtime, dimensions are checked for consistency for all variables; containers of any sizes may be assigned to function arguments. The constrained matrix types cov_matrix[K], corr_matrix[K], cholesky_factor_cov[K], and cholesky_factor_corr[K] are only assignable to matrices of dimensions matrix[K, K] types.\n\n\n\n\n\n\n\n\nFunction Argument (unsized)\n   Local\n(unconstrained)\n              Block\n          (constrained)\n\n\n\n\nint\nint\nint\n\n\n\n\nint&lt;lower=L&gt;\n\n\n\n\nint&lt;upper=U&gt;\n\n\n\n\nint&lt;lower=L, upper=U&gt;\n\n\n\n\nint&lt;offset=O&gt;\n\n\n\n\nint&lt;multiplier=M&gt;\n\n\n\n\nint&lt;offset=O, multiplier=M&gt;\n\n\nreal\nreal\nreal\n\n\n\n\nreal&lt;lower=L&gt;\n\n\n\n\nreal&lt;upper=U&gt;\n\n\n\n\nreal&lt;lower=L, upper=U&gt;\n\n\n\n\nreal&lt;offset=O&gt;\n\n\n\n\nreal&lt;multiplier=M&gt;\n\n\n\n\nreal&lt;offset=O, multiplier=M&gt;\n\n\ncomplex\ncomplex\ncomplex\n\n\nvector\nvector[N]\nvector[N]\n\n\n\n\nvector[N]&lt;lower=L&gt;\n\n\n\n\nvector[N]&lt;upper=U&gt;\n\n\n\n\nvector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nvector[N]&lt;offset=O&gt;\n\n\n\n\nvector[N]&lt;multiplier=M&gt;\n\n\n\n\nvector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\nordered[N]\n\n\n\n\npositive_ordered[N]\n\n\n\n\nsimplex[N]\n\n\n\n\nunit_vector[N]\n\n\nrow_vector\nrow_vector[N]\nrow_vector[N]\n\n\n\n\nrow_vector[N]&lt;lower=L&gt;\n\n\n\n\nrow_vector[N]&lt;upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O&gt;\n\n\n\n\nrow_vector[N]&lt;multiplier=M&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O, multiplier=M&gt;\n\n\nmatrix\nmatrix[M, N]\nmatrix[M, N]\n\n\n\n\nmatrix[M, N]&lt;lower=L&gt;\n\n\n\n\nmatrix[M, N]&lt;upper=U&gt;\n\n\n\n\nmatrix[M, N]&lt;lower=L, uppers=U&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O&gt;\n\n\n\n\nmatrix[M, N]&lt;multiplier=M&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O, multiplier=M&gt;\n\n\n\nmatrix[K, K]\ncorr_matrix[K]\n\n\n\nmatrix[K, K]\ncov_matrix[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_corr[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_cov[K]\n\n\ncomplex_vector\ncomplex_vector[M]\ncomplex_vector[M]\n\n\ncomplex_row_vector\ncomplex_row_vector[N]\ncomplex_row_vector[N]\n\n\ncomplex_matrix\ncomplex_matrix[M, N]\ncomplex_matrix[M,N]\n\n\narray[] vector\narray[M] vector[N]\narray[M] vector[N]\n\n\n\n\narray[M] vector[N]&lt;lower=L&gt;\n\n\n\n\narray[M] vector[N]&lt;upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O&gt;\n\n\n\n\narray[M] vector[N]&lt;multiplier=M&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\narray[M] ordered[N]\n\n\n\n\narray[M] positive_ordered[N]\n\n\n\n\narray[M] simplex[N]\n\n\n\n\narray[M] unit_vector[N]\n\n\n\n\nAdditional array types follow the same basic template as the final example in the table and can contain any of the previous types. The unsized version of arrays with more than one dimension is specified by using commas, e.g. array[ , ] is a 2-D array.\nFor more on how function arguments and return types are declared, consult the User’s Guide chapter on functions.\n\n\n\nStan allows assignable variables to be declared and defined in a single statement. Assignable variables are\n\nlocal variables, and\nvariables declared in the transformed data, transformed parameters, or generated quantities blocks.\n\nFor example, the statement\nint N = 5;\ndeclares the variable N to be an integer scalar type and at the same time defines it to be the value of the expression 5.\n\n\nThe type of the expression on the right-hand side of the assignment must be assignable to the type of the variable being declared. For example, it is legal to have\nreal sum = 0;\neven though 0 is of type int and sum is of type real, because integer-typed scalar expressions can be assigned to real-valued scalar variables. In all other cases, the type of the expression on the right-hand side of the assignment must be identical to the type of the variable being declared.\nVariables of any type may have values assigned to them. For example,\nmatrix[3, 2] a = b;\ndeclares a \\(3 \\times 2\\) matrix variable a and assigns a copy of the value of b to the variable a. The variable b must be of type matrix for the statement to be well formed. For the code to execute successfully, b must be the same shape as a, but this cannot be validated until run time. Because a copy is assigned, subsequent changes to a do not affect b and subsequent changes to b do not affect a.\n\n\n\nThe right-hand side may be any expression which has a type which is assignable to the variable being declared. For example,\nmatrix[3, 2] a = 0.5 * (b + c);\nassigns the matrix variable a to half of the sum of b and c. The only requirement on b and c is that the expression b + c be of type matrix. For example, b could be of type matrix and c of type real, because adding a matrix to a scalar produces a matrix, and the multiplying by a scalar produces another matrix.\nSimilarly,\ncomplex z = 2 + 3i;\nassigns the the complex number \\(2 + 3i\\) to the complex scalar z. The right-hand side expression can be a call to a user defined function, allowing general algorithms to be applied that might not be otherwise expressible as simple expressions (e.g., iterative or recursive algorithms).\n\n\n\nAny variable that is in scope and any function that is available in the block in which the compound declaration and definition appears may be used in the expression on the right-hand side of the compound declaration and definition statement.\n\n\n\n\nStan will interpret multiple comma-separated variable names following a single type as declaring multiple new variables. This is available for all variable declarations in all blocks.\n\n\nThe code:\nreal x, y;\nis equivalent to\nreal x;\nreal y;\nAs a result, all declarations on the same line must be of the same type.\n\n\n\nThe ability to declare multiple variables can be combined with assignments whenever a declare-define is valid, as documented in the section introducing compound declarations and definitions :\nreal x = 3, y = 5.6;\nConstrained data types can also be declared together, so long as the constraint for each variable is the same:\nreal&lt;lower=0&gt; x, y;",
+    "text": "This chapter covers the data types for expressions in Stan. Every variable used in a Stan program must have a declared data type. Only values of that type will be assignable to the variable (except for temporary states of transformed data and transformed parameter values). This follows the convention of programming languages like C++, not the conventions of scripting languages like Python or statistical languages such as R or BUGS.\nThe motivation for strong, static typing is threefold.\n\nStrong typing forces the programmer’s intent to be declared with the variable, making programs easier to comprehend and hence easier to debug and maintain.\nStrong typing allows programming errors relative to the declared intent to be caught sooner (at compile time) rather than later (at run time). The Stan compiler (called through an interface such as CmdStan, RStan, or PyStan) will flag any type errors and indicate the offending expressions quickly when the program is compiled.\nConstrained types will catch runtime data, initialization, and intermediate value errors as soon as they occur rather than allowing them to propagate and potentially pollute final results.\n\nStrong typing disallows assigning the same variable to objects of different types at different points in the program or in different invocations of the program.\n\n\nArguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the Removed Features chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.\n\n\n\n\nUnfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.\n\n\n\n\nThe complex data type is a scalar, but unlike real and int types, it contains two components, a real and imaginary component, both of which are of type real. That is, the real and imaginary components of a complex number are 64-bit, IEEE 754-complaint floating point numbers.\n\n\nImaginary literals are written in mathematical notation using a numeral followed by the suffix i. For example, the following example constructs a complex number \\(2 - 1.3i\\) and assigns it to the variable z.\ncomplex z = 2 - 1.3i;\nreal re = get_real(z);  // re has value 2.0\nreal im = get_imag(z);  // im has value -1.3\nThe getter functions then extract the real and imaginary components of z and assign them to re and im respectively.\nThe function to_complex constructs a complex number from its real and imaginary components. The functional form needs to be used whenever the components are not literal numerals, as in the following example.\nvector[K] re;\nvector[K] im;\n// ...\nfor (k in 1:K) {\n  complex z = to_complex(re[k], im[k]);\n  // ...\n}\n\n\n\nExpressions of type real may be assigned to variables of type complex. For example, the following is a valid sequence of Stan statements.\nreal x = 5.0;\ncomplex z = x;  // get_real(z) == 5.0, get_imag(z) == 0\nThe real number assigned to a complex number determine’s the complex number’s real component, with the imaginary component set to zero.\nAssignability is transitive, so that expressions of type int may also be assigned to variables of type complex, as in the following example.\nint n = 2;\ncomplex z = n;\nFunction arguments also support promotion of integer or real typed expressions to type complex.\n\n\n\n\nAll variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be supplied as data.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.\n\n\n\n\nStan provides three types of container objects: arrays, vectors, and matrices. Vectors and matrices are more limited kinds of data structures than arrays. Vectors are intrinsically one-dimensional collections of real or complex values, whereas matrices are intrinsically two dimensional. Vectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical. A \\(3 \\times 4\\) matrix is a different kind of object in Stan than a \\(3\n\\times 4\\) array.\nThe intention of using matrix types is to call out their usage in the code. There are three situations in Stan where only vectors and matrices may be used,\n\nmatrix arithmetic operations (e.g., matrix multiplication)\nlinear algebra functions (e.g., eigenvalues and determinants), and\nmultivariate function parameters and outcomes (e.g., multivariate normal distribution arguments).\n\nVectors and matrices cannot be typed to return integer values. They are restricted to real and complex values.\nFor constructing vectors and matrices in Stan, see Vector, Matrix, and Array Expressions.\n\n\nVectors and matrices, as well as arrays, are indexed starting from one (1) in Stan. This follows the convention in statistics and linear algebra as well as their implementations in the statistical software packages R, MATLAB, BUGS, and JAGS. General computer programming languages, on the other hand, such as C++ and Python, index arrays starting from zero.\n\n\n\nVectors in Stan are column vectors; see below for information on row vectors. Vectors are declared with a size (i.e., a dimensionality). For example, a 3-dimensional real vector is declared with the keyword vector, as follows.\nvector[3] u;\nVectors may also be declared with constraints, as in the following declaration of a 3-vector of non-negative values.\nvector&lt;lower=0&gt;[3] u;\nSimilarly, they may be declared with a offset and/or multiplier, as in the following example\nvector&lt;offset=42, multiplier=3&gt;[3] u;\n\n\n\nLike real vectors, complex vectors are column vectors and are declared with a size. For example, a 3-dimensional complex vector is declared with the keyword complex_vector, as follows.\ncomplex_vector[3] v;\nComplex vector declarations do not support any constraints.\n\n\n\nA unit simplex is a vector with non-negative values whose entries sum to 1. For instance, \\([0.2,0.3,0.4,0.1]^{\\top}\\) is a unit 4-simplex. Unit simplexes are most often used as parameters in categorical or multinomial distributions, and they are also the sampled variate in a Dirichlet distribution. Simplexes are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-simplex by\nsimplex[5] theta;\nUnit simplexes are implemented as vectors and may be assigned to other vectors and vice-versa. Simplex variables, like other constrained variables, are validated to ensure they contain simplex values; for simplexes, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\nIn high dimensional problems, simplexes may require smaller step sizes in the inference algorithms in order to remain stable; this can be achieved through higher target acceptance rates for samplers and longer warmup periods, tighter tolerances for optimization with more iterations, and in either case, with less dispersed parameter initialization or custom initialization if there are informative priors for some parameters.\n\n\n\nA unit vector is a vector with a norm of one. For instance, \\([0.5,\n0.5, 0.5, 0.5]^{\\top}\\) is a unit 4-vector. Unit vectors are sometimes used in directional statistics. Unit vectors are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-vector by\nunit_vector[5] theta;\nUnit vectors are implemented as vectors and may be assigned to other vectors and vice-versa. Unit vector variables, like other constrained variables, are validated to ensure that they are indeed unit length; for unit vectors, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\n\n\n\nAn ordered vector type in Stan represents a vector whose entries are sorted in ascending order. For instance, \\((-1.3,2.7,2.71)^{\\top}\\) is an ordered 3-vector. Ordered vectors are most often employed as cut points in ordered logistic regression models (see section).\nThe variable c is declared as an ordered 5-vector by\nordered[5] c;\nAfter their declaration, ordered vectors, like unit simplexes, may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nThere is also a positive, ordered vector type which operates similarly to ordered vectors, but all entries are constrained to be positive. For instance, \\((2,3.7,4,12.9)\\) is a positive, ordered 4-vector.\nThe variable d is declared as a positive, ordered 5-vector by\npositive_ordered[5] d;\nLike ordered vectors, after their declaration, positive ordered vectors may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nRow vectors are declared with the keyword row_vector. Like (column) vectors, they are declared with a size. For example, a 1093-dimensional row vector u would be declared as\nrow_vector[1093] u;\nConstraints are declared as for vectors, as in the following example of a 10-vector with values between -1 and 1.\nrow_vector&lt;lower=-1, upper=1&gt;[10] u;\nOffset and multiplier are also similar as for the following 3-row-vector with offset -42 and multiplier 3.\nrow_vector&lt;offset=-42, multiplier=3&gt;[3] u;\nRow vectors may not be assigned to column vectors, nor may column vectors be assigned to row vectors. If assignments are required, they may be accommodated through the transposition operator.\n\n\n\nComplex row vectors are declared with the keyword complex_row_vector and given a size in basic declarations. For example, a 12-dimensional complex row vector v would be declared as\ncomplex_row_vector[12] v;\nComplex row vectors do not allow constraints.\n\n\n\nMatrices are declared with the keyword matrix along with a number of rows and number of columns. For example,\nmatrix[3, 3] A;\nmatrix[M, N] B;\ndeclares A to be a \\(3 \\times 3\\) matrix and B to be a \\(M\n\\times N\\) matrix. For the second declaration to be well formed, the variables M and N must be declared as integers in either the data or transformed data block and before the matrix declaration.\nMatrices may also be declared with constraints, as in this (\\(3 \\times 4\\)) matrix of non-positive values.\nmatrix&lt;upper=0&gt;[3, 4] B;\nSimilarly, matrices can be declared to have a set offset and/or multiplier, as in this matrix with multiplier 5.\nmatrix&lt;multiplier=5&gt;[3, 4] B;\n\n\nRows of a matrix can be assigned by indexing the left-hand side of an assignment statement. For example, this is possible.\nmatrix[M, N] a;\nrow_vector[N] b;\n// ...\na[1] = b;\nThis copies the values from row vector b to a[1], which is the first row of the matrix a. If the number of columns in a is not the same as the size of b, a run-time error is raised; the number of columns of a is N, which is also the number of columns of b.\nAssignment works by copying values in Stan. That means any subsequent assignment to a[1] does not affect b, nor does an assignment to b affect a.\n\n\n\n\nComplex matrices are declared with the keyword complex_matrix and a number of rows and columns. For example,\ncomplex_matrix[3, 3] C;\nComplex matrices do not allow constraints.\n\n\n\nMatrix variables may be constrained to represent covariance matrices. A matrix is a covariance matrix if it is symmetric and positive definite. Like correlation matrices, covariance matrices only need a single dimension in their declaration. For instance,\ncov_matrix[K] Omega;\ndeclares Omega to be a \\(K \\times K\\) covariance matrix, where \\(K\\) is the value of the data variable K.\n\n\n\nMatrix variables may be constrained to represent correlation matrices. A matrix is a correlation matrix if it is symmetric and positive definite, has entries between \\(-1\\) and \\(1\\), and has a unit diagonal. Because correlation matrices are square, only one dimension needs to be declared. For example,\ncorr_matrix[3] Sigma;\ndeclares Sigma to be a \\(3 \\times 3\\) correlation matrix.\nCorrelation matrices may be assigned to other matrices, including unconstrained matrices, if their dimensions match, and vice-versa.\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a covariance matrix. This is often more convenient or more efficient than representing covariance matrices directly.\nA Cholesky factor \\(L\\) is an \\(M \\times N\\) lower-triangular matrix (if \\(m &lt; n\\) then \\(L[m, n] =0\\)) with a strictly positive diagonal (\\(L[k, k]\n&gt; 0\\)) and \\(M \\geq N\\). If \\(L\\) is a Cholesky factor, then \\(\\Sigma = L\n\\, L^{\\top}\\) is a covariance matrix (i.e., it is positive definite). The mapping between positive definite matrices and their Cholesky factors is bijective—every covariance matrix has a unique Cholesky factorization.\nThe typical case of a square Cholesky factor may be declared with a single dimension,\ncholesky_factor_cov[4] L;\n\n\nIn general, two dimensions may be declared, with the above being equal to cholesky_factor_cov[4, 4]. The type cholesky_factor_cov[M, N] may be used for the general \\(M \\times N\\) case to produce positive semi-definite matrices of rank \\(M\\).\n\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a correlation matrix.\nA Cholesky factor for a correlation matrix \\(L\\) is a \\(K \\times K\\) lower-triangular matrix with positive diagonal entries and rows that are of length 1 (i.e., \\(\\sum_{n=1}^K L_{m,n}^2 = 1\\)). If \\(L\\) is a Cholesky factor for a correlation matrix, then \\(L\\,L^{\\top}\\) is a correlation matrix (i.e., symmetric positive definite with a unit diagonal).\nTo declare the variable L to be a K by K Cholesky factor of a correlation matrix, the following code may be used.\ncholesky_factor_corr[K] L;\n\n\n\nConstrained variables of all types may be assigned to other variables of the same unconstrained type and vice-versa. Matching is interpreted strictly as having the same basic type and number of array dimensions. Constraints are not considered, but basic data types are. For instance, a variable declared to be real&lt;lower=0, upper=1&gt; could be assigned to a variable declared as real and vice-versa. Similarly, a variable declared as matrix[3, 3] may be assigned to a variable declared as cov_matrix[3] or cholesky_factor_cov[3], and vice-versa.\nChecks are carried out at the end of each relevant block of statements to ensure constraints are enforced. This includes run-time size checks. The Stan compiler isn’t able to catch the fact that an attempt may be made to assign a matrix of one dimensionality to a matrix of mismatching dimensionality.\n\n\n\nReal-valued vectors, row vectors and matrices may be assigned to complex-valued vectors, row vectors and matrices, respectively. For example, the following is legal.\nvector[N] v = ...;\ncomplex_vector[N] u = 2 * v;\nRow vectors and matrices work the same way.\n\n\n\nVariables may be declared with sizes given by expressions. Such expressions are constrained to only contain data or transformed data variables. This ensures that all sizes are determined once the data is read in and transformed data variables defined by their statements. For example, the following is legal.\ndata {\n  int&lt;lower=0&gt; N_observed, N_missing;\n  // ...\ntransformed parameters {\n  vector[N_observed + N_missing] y;\n  // ...\n\n\n\nIf v is a column vector or row vector, then v[2] is the second element in the vector. If m is a matrix, then m[2, 3] is the value in the second row and third column.\nProviding a matrix with a single index returns the specified row. For instance, if m is a matrix, then m[2] is the second row. This allows Stan blocks such as\nmatrix[M, N] m;\nrow_vector[N] v;\nreal x;\n// ...\nv = m[2];\nx = v[3];   // x == m[2][3] == m[2, 3]\nThe type of m[2] is row_vector because it is the second row of m. Thus it is possible to write m[2][3] instead of m[2, 3] to access the third element in the second row. When given a choice, the form m[2, 3] is preferred.\nComplex versions work the same way,\ncomplex_matrix[M, N] m = ...;\ncomplex_row_vector[N] u = m[3];\ncomplex_vector[M] v = m[ , 2];\n\n\nThe form m[2, 3] is more efficient because it does not require the creation and use of an intermediate expression template for m[2]. In later versions, explicit calls to m[2][3] may be optimized to be as efficient as m[2, 3] by the Stan compiler.\n\n\n\n\nAn integer expression is used to pick out the sizes of vectors, matrices, and arrays. For instance, we can declare a vector of size M + N using\nvector[M + N] y;\nAny integer-denoting expression may be used for the size declaration, providing all variables involved are either data, transformed data, or local variables. That is, expressions used for size declarations may not include parameters or transformed parameters or generated quantities.\n\n\n\n\nStan supports arrays of arbitrary dimension. The values in an array can be any type, so that arrays may contain values that are simple reals or integers, vectors, matrices, or other arrays. Arrays are the only way to store sequences of integers, and some functions in Stan, such as discrete distributions, require integer arguments.\nA two-dimensional array is just an array of arrays, both conceptually and in terms of current implementation. When an index is supplied to an array, it returns the value at that index. When more than one index is supplied, this indexing operation is chained. For example, if a is a two-dimensional array, then a[m, n] is just a convenient shorthand for a[m][n].\nVectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical.\nFor constructing arrays in Stan, see Vector, Matrix, and Array Expressions.\n\n\nArrays are declared with the keyword array followed by the dimensions enclosed in square brackets, the element type, and the name of the variable.\nThe variable n is declared as an array of five integers as follows.\narray[5] int n;\nA two-dimensional array of complex values with three rows and four columns is declared as follows.\narray[3, 4] complex a;\nA three-dimensional array z of positive reals with five rows, four columns, and two shelves can be declared as follows.\narray[5, 4, 2] real&lt;lower=0&gt; z;\nArrays may also be declared to contain vectors. For example,\narray[3] vector[7] mu;\ndeclares mu to be an array of size 3 containing vectors with 7 elements. Arrays may also contain matrices. The example\narray[15, 12] complex_matrix[7, 2] mu;\ndeclares a 15 by 12 array of \\(7 \\times 2\\) complex matrices. Any of the constrained types may also be used in arrays, as in the declaration\narray[2, 3, 4] cholesky_factor_cov[5, 6] mu;\nof a \\(2 \\times 3 \\times 4\\) array of \\(5 \\times 6\\) Cholesky factors of covariance matrices.\n\n\n\nIf x is a 1-dimensional array of length 5, then x[1] is the first element in the array and x[5] is the last. For a \\(3\n\\times 4\\) array y of two dimensions, y[1, 1] is the first element and y[3, 4] the last element. For a three-dimensional array z, the first element is z[1, 1, 1], and so on.\nSubarrays of arrays may be accessed by providing fewer than the full number of indexes. For example, suppose y is a two-dimensional array with three rows and four columns. Then y[3] is one-dimensional array of length four. This means that y[3][1] may be used instead of y[3, 1] to access the value of the first column of the third row of y. The form y[3, 1] is the preferred form (see note in this chapter).\n\n\n\nSubarrays may be manipulated and assigned just like any other variables. Similar to the behavior of matrices, Stan allows blocks such as\narray[9, 10, 11] real w;\narray[10, 11] real x;\narray[11] real y;\nreal z;\n// ...\nx = w[5];\ny = x[4];  // y == w[5][4] == w[5, 4]\nz = y[3];  // z == w[5][4][3] == w[5, 4, 3]\nComplex-valued arrays work the same way.\n\n\n\nArrays of vectors and matrices are accessed in the same way as arrays of doubles. Consider the following vector and scalar declarations.\narray[3, 4] vector[5] a;\narray[4] vector[5] b;\nvector[5] c;\nreal x;\nWith these declarations, the following assignments are legal.\nb = a[1];       // result is array of vectors\nc = a[1, 3];    // result is vector\nc = b[3];       //   same result as above\nx = a[1, 3, 5]; // result is scalar\nx = b[3, 5];    //   same result as above\nx = c[5];       //   same result as above\nRow vectors and other derived vector types (simplex and ordered) behave the same way in terms of indexing.\nConsider the following matrix, vector and scalar declarations.\narray[3, 4] matrix[6, 5] d;\narray[4] matrix[6, 5] e;\nmatrix[6, 5] f;\nrow_vector[5] g;\nreal x;\nWith these declarations, the following definitions are legal.\ne = d[1];           // result is array of matrices\nf = d[1, 3];        // result is matrix\nf = e[3];           //   same result as above\ng = d[1, 3, 2];     // result is row vector\ng = e[3, 2];        //   same result as above\ng = f[2];           //   same result as above\nx = d[1, 3, 5, 2];  // result is scalar\nx = e[3, 5, 2];     //   same result as above\nx = f[5, 2];        //   same result as above\nx = g[2];           //   same result as above\nAs shown, the result f[2] of supplying a single index to a matrix is the indexed row, here row 2 of matrix f.\n\n\n\nSubarrays of arrays may be assigned by indexing on the left-hand side of an assignment statement. For example, the following is legal.\narray[I, J, K] real x;\narray[J, K] real y;\narray[K] real z;\n// ...\nx[1] = y;\nx[1, 1] = z;\nThe sizes must match. Here, x[1] is a J by K array, as is y.\nPartial array assignment also works for arrays of matrices, vectors, and row vectors.\n\n\n\nArrays, row vectors, column vectors and matrices are not interchangeable in Stan. Thus a variable of any one of these fundamental types is not assignable to any of the others, nor may it be used as an argument where the other is required (use as arguments follows the assignment rules).\n\n\nFor example, vectors cannot be assigned to arrays or vice-versa.\narray[4] real a;\nvector[4] b;\nrow_vector[4] c;\n// ...\na = b; // illegal assignment of vector to array\nb = a; // illegal assignment of array to vector\na = c; // illegal assignment of row vector to array\nc = a; // illegal assignment of array to row vector\n\n\n\nIt is not even legal to assign row vectors to column vectors or vice versa.\nvector[4] b;\nrow_vector[4] c;\n// ...\nb = c; // illegal assignment of row vector to column vector\nc = b; // illegal assignment of column vector to row vector\n\n\n\nThe same holds for matrices, where 2-dimensional arrays may not be assigned to matrices or vice-versa.\narray[3, 4] real a;\nmatrix[3, 4] b;\n// ...\na = b;  // illegal assignment of matrix to array\nb = a;  // illegal assignment of array to matrix\n\n\n\nA \\(1 \\times N\\) matrix cannot be assigned a row vector or vice versa.\nmatrix[1, 4] a;\nrow_vector[4] b;\n// ...\na = b;  // illegal assignment of row vector to matrix\nb = a;  // illegal assignment of matrix to row vector\nSimilarly, an \\(M \\times 1\\) matrix may not be assigned to a column vector.\nmatrix[4, 1] a;\nvector[4] b;\n// ...\na = b;  // illegal assignment of column vector to matrix\nb = a;  // illegal assignment of matrix to column vector\n\n\n\n\nAn integer expression is used to pick out the sizes of arrays. The same restrictions as for vector and matrix sizes apply, namely that the size is declared with an integer-denoting expression that does not contain any parameters, transformed parameters, or generated quantities.\n\n\n\nIf any of an array’s dimensions is size zero, the entire array will be of size zero. That is, if we declare\narray[3, 0] real a;\nthen the resulting size of a is zero and querying any of its dimensions at run time will result in the value zero. Declared as above, a[1] will be a size-zero one-dimensional array. For comparison, declaring\narray[0, 3] real b;\nalso produces an array with an overall size of zero, but in this case, there is no way to index legally into b, because b[0] is undefined. The array will behave at run time as if it’s a \\(0 \\times\n0\\) array. For example, the result of to_matrix(b) will be a \\(0 \\times 0\\) matrix, not a \\(0 \\times 3\\) matrix.\n\n\n\n\nStan supports tuples of arbitrary size. The values in a tuple can be of arbitrary type, but the component types must be declared along with the declaration of the tuple. Tuples can be manipulated as a whole, or their elements may be accessed and set individually.\n\n\nTuples are declared with the keyword tuple followed by a parenthesized sequence of types, which determine the types of the respective tuple entries. For example, a tuple with three elements may be declared as\ntuple(int, vector[3], complex) abc;\nTuples must have at least two entries, so the following declarations are illegal.\ntuple() nil;  // ILLEGAL\ntuple(int) n;  // ILLEGAL\nTuples can be assigned as a whole if their elements can be assigned individually. For example, a can be assigned to b in the following example because int can be promoted to complex.\ntuple(int, real) a;\n...\ntuple(complex, real) b = a;\nTuple types may have elements which are declared as tuples, such as the following example.\ntuple(int, tuple(real, complex)) x;\nIn this case, it would probably be simpler to use a 3-tuple type, tuple(int, real, complex).\nTuples can be declared with constraints anywhere that ordinary variables can (i.e., as top-level block variables). That means any context in which it is legal to have a declaration\nreal&lt;lower=0&gt; sigma;\nreal&lt;lower=0, upper=1&gt; theta;\nit is legal to have a tuple with constraints such as\ntuple(real&lt;lower=0&gt;, real&lt;lower=0, upper=1&gt;) sigma_theta;\n\n\n\nTuple elements may be accessed directly. For example, with our declaration of abc from the last section, Stan uses abc.1 for the first element, abc.2 for the second, and abc.3 for the third. These numbers must be integer literals (i.e., they cannot be variables), and must be within the size of the number of elements of tuples. The types of elements are as declared, so that abc.1 is of type int, abc.2 of type vector[3] and abc.3 of type complex.\n\n\n\nTuple elements can be assigned individually, allowing, e.g.,\ntuple(int, real) ab;\nab.1 = 123;\nab.2 = 12.9;\nAs with other assignments, promotions will happen if necessary (of int to real and of real to complex, along with the corresponding container type promotions).\n\n\n\nFor convenience of using values stored in tuples, Stan supports “unpacking” (or “destructuring”) of tuples in an assignment statement.\nGiven a tuple t of type tuple(T1, ..., Tn) and a sequence of assignable expressions of types v1, …, vn, where each vi has a type which is assignable from type Ti, individual elements of the tuple may be assigned to the corresponding variables in the sequence by the statement\n(v1, /*...*/, vn) = t;\nNote that the above parenthesis are required, unlike in some other languages with similar features (e.g., Python).\nThese unpacking assignments can be nested if the tuple on the right hand side contains nested tuples.\nFor example, if T is a tuple of type tuple(int, (real, real), complex), then the program\nint i;\nreal x, y;\ncomplex z;\n\n(i, (x, y), z) = T;\nAssigns the result of T.1 to i, the result of T.2.1 to x, the result of T.2.2 to y, and the result of T.3 to z.\nThe left hand side must match in size the tuple on the right. Additionally, the same variable may not appear more than once in the left hand side of an unpacking assignment.\n\n\n\n\nThe type information associated with a variable only contains the underlying type and dimensionality of the variable.\n\n\nThe size associated with a given variable is not part of its data type. For example, declaring a variable using\narray[3] real a;\ndeclares the variable a to be an array. The fact that it was declared to have size 3 is part of its declaration, but not part of its underlying type.\n\n\nSizes are determined dynamically (at run time) and thus cannot be type-checked statically when the program is compiled. As a result, any conformance error on size will raise a run-time error. For example, trying to assign an array of size 5 to an array of size 6 will cause a run-time error. Similarly, multiplying an \\(N\n\\times M\\) by a \\(J \\times K\\) matrix will raise a run-time error if \\(M\n\\neq J\\).\n\n\n\n\nLike sizes, constraints are not treated as part of a variable’s type in Stan when it comes to the compile-time check of operations it may participate in. Anywhere Stan accepts a matrix as an argument, it will syntactically accept a correlation matrix or covariance matrix or Cholesky factor. Thus a covariance matrix may be assigned to a matrix and vice-versa.\nSimilarly, a bounded real may be assigned to an unconstrained real and vice-versa.\n\n\nFor arguments to functions, constraints are sometimes, but not always checked when the function is called. Exclusions include C++ standard library functions. All probability functions and cumulative distribution functions check that their arguments are appropriate at run time as the function is called.\n\n\n\nFor data variables, constraints are checked after the variable is read from a data file or other source. For transformed data variables, the check is done after the statements in the transformed data block have executed. Thus it is legal for intermediate values of variables to not satisfy declared constraints.\nFor parameters, constraints are enforced by the transform applied and do not need to be checked. For transformed parameters, the check is done after the statements in the transformed parameter block have executed.\nFor all blocks defining variables (transformed data, transformed parameters, generated quantities), real values are initialized to NaN and integer values are initialized to the smallest legal integer (i.e., a large absolute value negative number).\nFor generated quantities, constraints are enforced after the statements in the generated quantities block have executed.\n\n\n\n\nIn order to refer to data types, it is convenient to have a way to refer to them. The type naming notation outlined in this section is not part of the Stan programming language, but rather a convention adopted in this document to enable a concise description of a type.\nBecause size information is not part of a data type, data types will be written without size information. For instance, array[] real is the type of one-dimensional array of reals and matrix is the type of matrices. The three-dimensional integer array type is written as array[,,] int, indicating the number slots available for indexing. Similarly, array[,] vector is the type of a two-dimensional array of vectors.\n\n\n\n\nVariables in Stan are declared by giving a type and a name. For example\nint N;\nvector[N] y;\narray[5] matrix[3, 4] A;\ndeclares a variable N that is an integer, a variable y that is a vector of length N (the previously declared variable), and a variable A, which is a length-5 array where each element is a 3 by 4 matrix.\nThere are several different places a variable is declared in Stan. They are block variables, like those inside data, which can have constraints and must include sizes for their types, like in the above examples. Local variables, like those defined inside loops or local blocks cannot be constrained, but still include sizes. Finally, variables declared as function parameters are not constrained types and exclude sizes.\n// valid block variables, but not locals or function parameters\nvector&lt;lower=0&gt;[N] u;\n\n// valid as a block or local variable, but not a function parameter\narray[3] int is;\n\n// function parameters exclude sizes and cannot be constrained\nvoid pretty_print_tri_lower(matrix x) { ... }\nIn the following table, the leftmost column is a list of the unconstrained and undimensioned basic types; these are used as function return types and argument types. The middle column is of unconstrained types with dimensions; these are used as local variable types. The variables M and N indicate number of columns and rows, respectively. The variable K is used for square matrices, i.e., K denotes both the number of rows and columns. The rightmost column lists the corresponding constrained types. An expression of any right-hand column type may be assigned to its corresponding left-hand column basic type. At runtime, dimensions are checked for consistency for all variables; containers of any sizes may be assigned to function arguments. The constrained matrix types cov_matrix[K], corr_matrix[K], cholesky_factor_cov[K], and cholesky_factor_corr[K] are only assignable to matrices of dimensions matrix[K, K] types.\n\n\n\n\n\n\n\n\nFunction Argument (unsized)\n   Local\n(unconstrained)\n              Block\n          (constrained)\n\n\n\n\nint\nint\nint\n\n\n\n\nint&lt;lower=L&gt;\n\n\n\n\nint&lt;upper=U&gt;\n\n\n\n\nint&lt;lower=L, upper=U&gt;\n\n\n\n\nint&lt;offset=O&gt;\n\n\n\n\nint&lt;multiplier=M&gt;\n\n\n\n\nint&lt;offset=O, multiplier=M&gt;\n\n\nreal\nreal\nreal\n\n\n\n\nreal&lt;lower=L&gt;\n\n\n\n\nreal&lt;upper=U&gt;\n\n\n\n\nreal&lt;lower=L, upper=U&gt;\n\n\n\n\nreal&lt;offset=O&gt;\n\n\n\n\nreal&lt;multiplier=M&gt;\n\n\n\n\nreal&lt;offset=O, multiplier=M&gt;\n\n\ncomplex\ncomplex\ncomplex\n\n\nvector\nvector[N]\nvector[N]\n\n\n\n\nvector[N]&lt;lower=L&gt;\n\n\n\n\nvector[N]&lt;upper=U&gt;\n\n\n\n\nvector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nvector[N]&lt;offset=O&gt;\n\n\n\n\nvector[N]&lt;multiplier=M&gt;\n\n\n\n\nvector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\nordered[N]\n\n\n\n\npositive_ordered[N]\n\n\n\n\nsimplex[N]\n\n\n\n\nunit_vector[N]\n\n\nrow_vector\nrow_vector[N]\nrow_vector[N]\n\n\n\n\nrow_vector[N]&lt;lower=L&gt;\n\n\n\n\nrow_vector[N]&lt;upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O&gt;\n\n\n\n\nrow_vector[N]&lt;multiplier=M&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O, multiplier=M&gt;\n\n\nmatrix\nmatrix[M, N]\nmatrix[M, N]\n\n\n\n\nmatrix[M, N]&lt;lower=L&gt;\n\n\n\n\nmatrix[M, N]&lt;upper=U&gt;\n\n\n\n\nmatrix[M, N]&lt;lower=L, uppers=U&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O&gt;\n\n\n\n\nmatrix[M, N]&lt;multiplier=M&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O, multiplier=M&gt;\n\n\n\nmatrix[K, K]\ncorr_matrix[K]\n\n\n\nmatrix[K, K]\ncov_matrix[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_corr[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_cov[K]\n\n\ncomplex_vector\ncomplex_vector[M]\ncomplex_vector[M]\n\n\ncomplex_row_vector\ncomplex_row_vector[N]\ncomplex_row_vector[N]\n\n\ncomplex_matrix\ncomplex_matrix[M, N]\ncomplex_matrix[M,N]\n\n\narray[] vector\narray[M] vector[N]\narray[M] vector[N]\n\n\n\n\narray[M] vector[N]&lt;lower=L&gt;\n\n\n\n\narray[M] vector[N]&lt;upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O&gt;\n\n\n\n\narray[M] vector[N]&lt;multiplier=M&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\narray[M] ordered[N]\n\n\n\n\narray[M] positive_ordered[N]\n\n\n\n\narray[M] simplex[N]\n\n\n\n\narray[M] unit_vector[N]\n\n\n\n\nAdditional array types follow the same basic template as the final example in the table and can contain any of the previous types. The unsized version of arrays with more than one dimension is specified by using commas, e.g. array[ , ] is a 2-D array.\nFor more on how function arguments and return types are declared, consult the User’s Guide chapter on functions.\n\n\n\nStan allows assignable variables to be declared and defined in a single statement. Assignable variables are\n\nlocal variables, and\nvariables declared in the transformed data, transformed parameters, or generated quantities blocks.\n\nFor example, the statement\nint N = 5;\ndeclares the variable N to be an integer scalar type and at the same time defines it to be the value of the expression 5.\n\n\nThe type of the expression on the right-hand side of the assignment must be assignable to the type of the variable being declared. For example, it is legal to have\nreal sum = 0;\neven though 0 is of type int and sum is of type real, because integer-typed scalar expressions can be assigned to real-valued scalar variables. In all other cases, the type of the expression on the right-hand side of the assignment must be identical to the type of the variable being declared.\nVariables of any type may have values assigned to them. For example,\nmatrix[3, 2] a = b;\ndeclares a \\(3 \\times 2\\) matrix variable a and assigns a copy of the value of b to the variable a. The variable b must be of type matrix for the statement to be well formed. For the code to execute successfully, b must be the same shape as a, but this cannot be validated until run time. Because a copy is assigned, subsequent changes to a do not affect b and subsequent changes to b do not affect a.\n\n\n\nThe right-hand side may be any expression which has a type which is assignable to the variable being declared. For example,\nmatrix[3, 2] a = 0.5 * (b + c);\nassigns the matrix variable a to half of the sum of b and c. The only requirement on b and c is that the expression b + c be of type matrix. For example, b could be of type matrix and c of type real, because adding a matrix to a scalar produces a matrix, and the multiplying by a scalar produces another matrix.\nSimilarly,\ncomplex z = 2 + 3i;\nassigns the the complex number \\(2 + 3i\\) to the complex scalar z. The right-hand side expression can be a call to a user defined function, allowing general algorithms to be applied that might not be otherwise expressible as simple expressions (e.g., iterative or recursive algorithms).\n\n\n\nAny variable that is in scope and any function that is available in the block in which the compound declaration and definition appears may be used in the expression on the right-hand side of the compound declaration and definition statement.\n\n\n\n\nStan will interpret multiple comma-separated variable names following a single type as declaring multiple new variables. This is available for all variable declarations in all blocks.\n\n\nThe code:\nreal x, y;\nis equivalent to\nreal x;\nreal y;\nAs a result, all declarations on the same line must be of the same type.\n\n\n\nThe ability to declare multiple variables can be combined with assignments whenever a declare-define is valid, as documented in the section introducing compound declarations and definitions :\nreal x = 3, y = 5.6;\nConstrained data types can also be declared together, so long as the constraint for each variable is the same:\nreal&lt;lower=0&gt; x, y;",
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-    "text": "Arguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.",
+    "text": "Arguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the Removed Features chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.",
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-    "text": "Unfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.",
+    "text": "Unfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.",
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     "title": "Data Types and Declarations",
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-    "text": "All variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be read as data in the dump format.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.",
+    "text": "All variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be supplied as data.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.",
     "crumbs": [
       "Reference Manual",
       "Language",
@@ -1678,7 +1678,7 @@
     "href": "reference-manual/syntax.html",
     "title": "Language Syntax",
     "section": "",
-    "text": "This chapter defines the basic syntax of the Stan modeling language using a Backus-Naur form (BNF) grammar plus extra-grammatical constraints on function typing and operator precedence and associativity.\n\n\n\n\nIn the following BNF grammars, tokens are represented in ALLCAPS. Grammar non-terminals are surrounded by &lt; and &gt;. A square brackets ([A]) indicates optionality of A. A postfixed Kleene star (A*) indicates zero or more occurrences of A. Parenthesis can be used to group symbols together in productions.\nFinally, this grammar uses the concept of “parameterized nonterminals” as used in the parsing library Menhir. A rule like &lt;list(x)&gt; ::= x (COMMA x)* declares a generic list rule, which can later be applied to others by the symbol &lt;list(&lt;expression&gt;)&gt;.\nThe following representation is constructed directly from the OCaml reference parser using a tool called Obelisk. The raw output is available here.\n\n\n\n\n&lt;program&gt; ::= [&lt;function_block&gt;] [&lt;data_block&gt;] [&lt;transformed_data_block&gt;]\n              [&lt;parameters_block&gt;] [&lt;transformed_parameters_block&gt;]\n              [&lt;model_block&gt;] [&lt;generated_quantities_block&gt;] EOF\n\n&lt;functions_only&gt; ::= &lt;function_def&gt;* EOF\n\n&lt;function_block&gt; ::= FUNCTIONBLOCK LBRACE &lt;function_def&gt;* RBRACE\n\n&lt;data_block&gt; ::= DATABLOCK LBRACE &lt;top_var_decl_no_assign&gt;* RBRACE\n\n&lt;transformed_data_block&gt; ::= TRANSFORMEDDATABLOCK LBRACE\n                             &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;parameters_block&gt; ::= PARAMETERSBLOCK LBRACE &lt;top_var_decl_no_assign&gt;*\n                       RBRACE\n\n&lt;transformed_parameters_block&gt; ::= TRANSFORMEDPARAMETERSBLOCK LBRACE\n                                   &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;model_block&gt; ::= MODELBLOCK LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;generated_quantities_block&gt; ::= GENERATEDQUANTITIESBLOCK LBRACE\n                                 &lt;top_vardecl_or_statement&gt;* RBRACE\n\n\n\n&lt;function_def&gt; ::= &lt;return_type&gt; &lt;decl_identifier&gt; LPAREN [&lt;arg_decl&gt; (COMMA\n                   &lt;arg_decl&gt;)*] RPAREN &lt;statement&gt;\n\n&lt;return_type&gt; ::= VOID\n                | &lt;unsized_type&gt;\n\n&lt;arg_decl&gt; ::= [DATABLOCK] &lt;unsized_type&gt; &lt;decl_identifier&gt;\n\n&lt;unsized_type&gt; ::= ARRAY &lt;unsized_dims&gt; &lt;basic_type&gt;\n                 | ARRAY &lt;unsized_dims&gt; &lt;unsized_tuple_type&gt;\n                 | &lt;basic_type&gt;\n                 | &lt;unsized_tuple_type&gt;\n\n&lt;unsized_tuple_type&gt; ::= TUPLE LPAREN &lt;unsized_type&gt; COMMA &lt;unsized_type&gt;\n                         (COMMA &lt;unsized_type&gt;)* RPAREN\n\n&lt;basic_type&gt; ::= INT\n               | REAL\n               | COMPLEX\n               | VECTOR\n               | ROWVECTOR\n               | MATRIX\n               | COMPLEXVECTOR\n               | COMPLEXROWVECTOR\n               | COMPLEXMATRIX\n\n&lt;unsized_dims&gt; ::= LBRACK COMMA* RBRACK\n\n\n\n\n&lt;identifier&gt; ::= IDENTIFIER\n               | TRUNCATE\n\n&lt;decl_identifier&gt; ::= &lt;identifier&gt;\n\n&lt;no_assign&gt; ::= UNREACHABLE\n\n&lt;optional_assignment(rhs)&gt; ::= [ASSIGN rhs]\n\n&lt;id_and_optional_assignment(rhs)&gt; ::= &lt;decl_identifier&gt;\n                                      &lt;optional_assignment(rhs)&gt;\n\n&lt;decl(type_rule, rhs)&gt; ::= type_rule &lt;decl_identifier&gt; &lt;dims&gt;\n                           &lt;optional_assignment(rhs)&gt; SEMICOLON\n                         | &lt;higher_type(type_rule)&gt;\n                           &lt;id_and_optional_assignment(rhs)&gt; (COMMA\n                           &lt;id_and_optional_assignment(rhs)&gt;)* SEMICOLON\n\n&lt;higher_type(type_rule)&gt; ::= &lt;array_type(type_rule)&gt;\n                           | &lt;tuple_type(type_rule)&gt;\n                           | type_rule\n\n&lt;array_type(type_rule)&gt; ::= &lt;arr_dims&gt; type_rule\n                          | &lt;arr_dims&gt; &lt;tuple_type(type_rule)&gt;\n\n&lt;tuple_type(type_rule)&gt; ::= TUPLE LPAREN &lt;higher_type(type_rule)&gt; COMMA\n                            &lt;higher_type(type_rule)&gt; (COMMA\n                            &lt;higher_type(type_rule)&gt;)* RPAREN\n\n&lt;var_decl&gt; ::= &lt;decl(&lt;sized_basic_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl_no_assign&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;no_assign&gt;)&gt;\n                           | SEMICOLON\n\n&lt;sized_basic_type&gt; ::= INT\n                     | REAL\n                     | COMPLEX\n                     | VECTOR LBRACK &lt;expression&gt; RBRACK\n                     | ROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | MATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt; RBRACK\n                     | COMPLEXVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXMATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt;\n                       RBRACK\n\n&lt;top_var_type&gt; ::= INT [LABRACK &lt;range&gt; RABRACK]\n                 | REAL &lt;type_constraint&gt;                  | TUPLE\n\n                 | COMPLEX &lt;type_constraint&gt;\n                 | VECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | ROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | MATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | COMPLEXVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | COMPLEXROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt;\n                   RBRACK\n                 | COMPLEXMATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | ORDERED LBRACK &lt;expression&gt; RBRACK\n                 | POSITIVEORDERED LBRACK &lt;expression&gt; RBRACK\n                 | SIMPLEX LBRACK &lt;expression&gt; RBRACK\n                 | UNITVECTOR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCORR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCOV LBRACK &lt;expression&gt; [COMMA &lt;expression&gt;]\n                   RBRACK\n                 | CORRMATRIX LBRACK &lt;expression&gt; RBRACK\n                 | COVMATRIX LBRACK &lt;expression&gt; RBRACK\n\n&lt;type_constraint&gt; ::= [LABRACK &lt;range&gt; RABRACK]\n                    | LABRACK &lt;offset_mult&gt; RABRACK\n\n&lt;range&gt; ::= LOWER ASSIGN &lt;constr_expression&gt; COMMA UPPER ASSIGN\n            &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt; COMMA LOWER ASSIGN\n            &lt;constr_expression&gt;\n          | LOWER ASSIGN &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt;\n\n&lt;offset_mult&gt; ::= OFFSET ASSIGN &lt;constr_expression&gt; COMMA MULTIPLIER ASSIGN\n                  &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt; COMMA OFFSET ASSIGN\n                  &lt;constr_expression&gt;\n                | OFFSET ASSIGN &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt;\n\n&lt;arr_dims&gt; ::= ARRAY LBRACK &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACK\n\n\n\n\n&lt;expression&gt; ::= &lt;expression&gt; QMARK &lt;expression&gt; COLON &lt;expression&gt;\n               | &lt;expression&gt; &lt;infixOp&gt; &lt;expression&gt;\n               | &lt;prefixOp&gt; &lt;expression&gt;\n               | &lt;expression&gt; &lt;postfixOp&gt;\n               | &lt;common_expression&gt;\n\n&lt;constr_expression&gt; ::= &lt;constr_expression&gt; &lt;arithmeticBinOp&gt;\n                        &lt;constr_expression&gt;\n                      | &lt;prefixOp&gt; &lt;constr_expression&gt;\n                      | &lt;constr_expression&gt; &lt;postfixOp&gt;\n                      | &lt;common_expression&gt;\n\n&lt;common_expression&gt; ::= &lt;identifier&gt;\n                      | INTNUMERAL\n                      | REALNUMERAL\n                      | DOTNUMERAL\n                      | IMAGNUMERAL\n                      | LBRACE &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACE\n                      | LBRACK [&lt;expression&gt; (COMMA &lt;expression&gt;)*] RBRACK\n                      | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                        &lt;expression&gt;)*] RPAREN\n                      | TARGET LPAREN RPAREN\n                      | &lt;identifier&gt; LPAREN &lt;expression&gt; BAR [&lt;expression&gt;\n                        (COMMA &lt;expression&gt;)*] RPAREN\n                      | LPAREN &lt;expression&gt; COMMA &lt;expression&gt; (COMMA\n                        &lt;expression&gt;)* RPAREN\n                      | &lt;common_expression&gt; DOTNUMERAL\n                      | &lt;common_expression&gt; LBRACK &lt;indexes&gt; RBRACK\n                      | LPAREN &lt;expression&gt; RPAREN\n\n&lt;prefixOp&gt; ::= BANG\n             | MINUS\n             | PLUS\n\n&lt;postfixOp&gt; ::= TRANSPOSE\n\n&lt;infixOp&gt; ::= &lt;arithmeticBinOp&gt;\n            | &lt;logicalBinOp&gt;\n\n&lt;arithmeticBinOp&gt; ::= PLUS\n                    | MINUS\n                    | TIMES\n                    | DIVIDE\n                    | IDIVIDE\n                    | MODULO\n                    | LDIVIDE\n                    | ELTTIMES\n                    | ELTDIVIDE\n                    | HAT\n                    | ELTPOW\n\n&lt;logicalBinOp&gt; ::= OR\n                 | AND\n                 | EQUALS\n                 | NEQUALS\n                 | LABRACK\n                 | LEQ\n                 | RABRACK\n                 | GEQ\n\n&lt;indexes&gt; ::= epsilon\n            | COLON\n            | &lt;expression&gt;\n            | &lt;expression&gt; COLON\n            | COLON &lt;expression&gt;\n            | &lt;expression&gt; COLON &lt;expression&gt;\n            | &lt;indexes&gt; COMMA &lt;indexes&gt;\n\n&lt;printables&gt; ::= &lt;expression&gt;\n               | &lt;string_literal&gt;\n               | &lt;printables&gt; COMMA &lt;printables&gt;\n\n\n\n&lt;statement&gt; ::= &lt;atomic_statement&gt;\n              | &lt;nested_statement&gt;\n\n&lt;atomic_statement&gt; ::= &lt;common_expression&gt; &lt;assignment_op&gt; &lt;expression&gt;\n                       SEMICOLON\n                     | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                       &lt;expression&gt;)*] RPAREN SEMICOLON\n                     | &lt;expression&gt; TILDE &lt;identifier&gt; LPAREN [&lt;expression&gt;\n                       (COMMA &lt;expression&gt;)*] RPAREN [&lt;truncation&gt;] SEMICOLON\n                     | TARGET PLUSASSIGN &lt;expression&gt; SEMICOLON\n                     | BREAK SEMICOLON\n                     | CONTINUE SEMICOLON\n                     | PRINT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | REJECT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | RETURN &lt;expression&gt; SEMICOLON\n                     | RETURN SEMICOLON\n                     | SEMICOLON\n\n&lt;assignment_op&gt; ::= ASSIGN\n                  | PLUSASSIGN\n                  | MINUSASSIGN\n                  | TIMESASSIGN\n                  | DIVIDEASSIGN\n                  | ELTTIMESASSIGN\n                  | ELTDIVIDEASSIGN\n\n&lt;string_literal&gt; ::= STRINGLITERAL\n\n&lt;truncation&gt; ::= TRUNCATE LBRACK [&lt;expression&gt;] COMMA [&lt;expression&gt;] RBRACK\n\n&lt;nested_statement&gt; ::= IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                       ELSE &lt;vardecl_or_statement&gt;\n                     | IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | WHILE LPAREN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; COLON\n                       &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | PROFILE LPAREN &lt;string_literal&gt; RPAREN LBRACE\n                       &lt;vardecl_or_statement&gt;* RBRACE\n                     | LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;vardecl_or_statement&gt; ::= &lt;statement&gt;\n                         | &lt;var_decl&gt;\n\n&lt;top_vardecl_or_statement&gt; ::= &lt;statement&gt;\n                             | &lt;top_var_decl&gt;\n\n\n\n\nMany of the tokens used in the BNF grammars follow obviously from their names: DATABLOCK is the literal string ‘data’, COMMA is a single ‘,’ character, etc. The literal representation of each operator is additionally provided in the operator precedence table.\nA few tokens are not so obvious, and are defined here in regular expressions:\nIDENTIFIER = [a-zA-Z] [a-zA-Z0-9_]*\n\nSTRINGLITERAL = \".*\"\n\nINTNUMERAL = [0-9]+ (_ [0-9]+)*\n\nEXPLITERAL = [eE] [+-]? INTNUMERAL\n\nREALNUMERAL = INTNUMERAL \\. INTNUMERAL? EXPLITERAL?\n            | \\. INTNUMERAL EXPLITERAL\n            | INTNUMERAL EXPLITERAL\n\nIMAGNUMERAL = (REALNUMERAL | INTNUMERAL) i\n\nDOTNUMERAL = \\. INTNUMERAL\n\n\n\n\n\nA well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type matrix[ , , ], a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
+    "text": "This chapter defines the basic syntax of the Stan modeling language using a Backus-Naur form (BNF) grammar plus extra-grammatical constraints on function typing and operator precedence and associativity.\n\n\n\n\nIn the following BNF grammars, tokens are represented in ALLCAPS. Grammar non-terminals are surrounded by &lt; and &gt;. A square brackets ([A]) indicates optionality of A. A postfixed Kleene star (A*) indicates zero or more occurrences of A. Parenthesis can be used to group symbols together in productions.\nFinally, this grammar uses the concept of “parameterized nonterminals” as used in the parsing library Menhir. A rule like &lt;list(x)&gt; ::= x (COMMA x)* declares a generic list rule, which can later be applied to others by the symbol &lt;list(&lt;expression&gt;)&gt;.\nThe following representation is constructed directly from the OCaml reference parser using a tool called Obelisk. The raw output is available here.\n\n\n\n\n&lt;program&gt; ::= [&lt;function_block&gt;] [&lt;data_block&gt;] [&lt;transformed_data_block&gt;]\n              [&lt;parameters_block&gt;] [&lt;transformed_parameters_block&gt;]\n              [&lt;model_block&gt;] [&lt;generated_quantities_block&gt;] EOF\n\n&lt;functions_only&gt; ::= &lt;function_def&gt;* EOF\n\n&lt;function_block&gt; ::= FUNCTIONBLOCK LBRACE &lt;function_def&gt;* RBRACE\n\n&lt;data_block&gt; ::= DATABLOCK LBRACE &lt;top_var_decl_no_assign&gt;* RBRACE\n\n&lt;transformed_data_block&gt; ::= TRANSFORMEDDATABLOCK LBRACE\n                             &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;parameters_block&gt; ::= PARAMETERSBLOCK LBRACE &lt;top_var_decl_no_assign&gt;*\n                       RBRACE\n\n&lt;transformed_parameters_block&gt; ::= TRANSFORMEDPARAMETERSBLOCK LBRACE\n                                   &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;model_block&gt; ::= MODELBLOCK LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;generated_quantities_block&gt; ::= GENERATEDQUANTITIESBLOCK LBRACE\n                                 &lt;top_vardecl_or_statement&gt;* RBRACE\n\n\n\n&lt;function_def&gt; ::= &lt;return_type&gt; &lt;decl_identifier&gt; LPAREN [&lt;arg_decl&gt; (COMMA\n                   &lt;arg_decl&gt;)*] RPAREN &lt;statement&gt;\n\n&lt;return_type&gt; ::= VOID\n                | &lt;unsized_type&gt;\n\n&lt;arg_decl&gt; ::= [DATABLOCK] &lt;unsized_type&gt; &lt;decl_identifier&gt;\n\n&lt;unsized_type&gt; ::= ARRAY &lt;unsized_dims&gt; &lt;basic_type&gt;\n                 | ARRAY &lt;unsized_dims&gt; &lt;unsized_tuple_type&gt;\n                 | &lt;basic_type&gt;\n                 | &lt;unsized_tuple_type&gt;\n\n&lt;unsized_tuple_type&gt; ::= TUPLE LPAREN &lt;unsized_type&gt; COMMA &lt;unsized_type&gt;\n                         (COMMA &lt;unsized_type&gt;)* RPAREN\n\n&lt;basic_type&gt; ::= INT\n               | REAL\n               | COMPLEX\n               | VECTOR\n               | ROWVECTOR\n               | MATRIX\n               | COMPLEXVECTOR\n               | COMPLEXROWVECTOR\n               | COMPLEXMATRIX\n\n&lt;unsized_dims&gt; ::= LBRACK COMMA* RBRACK\n\n\n\n\n&lt;identifier&gt; ::= IDENTIFIER\n               | TRUNCATE\n\n&lt;decl_identifier&gt; ::= &lt;identifier&gt;\n\n&lt;no_assign&gt; ::= UNREACHABLE\n\n&lt;optional_assignment(rhs)&gt; ::= [ASSIGN rhs]\n\n&lt;id_and_optional_assignment(rhs)&gt; ::= &lt;decl_identifier&gt;\n                                      &lt;optional_assignment(rhs)&gt;\n\n&lt;decl(type_rule, rhs)&gt; ::= type_rule &lt;decl_identifier&gt; &lt;dims&gt;\n                           &lt;optional_assignment(rhs)&gt; SEMICOLON\n                         | &lt;higher_type(type_rule)&gt;\n                           &lt;id_and_optional_assignment(rhs)&gt; (COMMA\n                           &lt;id_and_optional_assignment(rhs)&gt;)* SEMICOLON\n\n&lt;higher_type(type_rule)&gt; ::= &lt;array_type(type_rule)&gt;\n                           | &lt;tuple_type(type_rule)&gt;\n                           | type_rule\n\n&lt;array_type(type_rule)&gt; ::= &lt;arr_dims&gt; type_rule\n                          | &lt;arr_dims&gt; &lt;tuple_type(type_rule)&gt;\n\n&lt;tuple_type(type_rule)&gt; ::= TUPLE LPAREN &lt;higher_type(type_rule)&gt; COMMA\n                            &lt;higher_type(type_rule)&gt; (COMMA\n                            &lt;higher_type(type_rule)&gt;)* RPAREN\n\n&lt;var_decl&gt; ::= &lt;decl(&lt;sized_basic_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl_no_assign&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;no_assign&gt;)&gt;\n                           | SEMICOLON\n\n&lt;sized_basic_type&gt; ::= INT\n                     | REAL\n                     | COMPLEX\n                     | VECTOR LBRACK &lt;expression&gt; RBRACK\n                     | ROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | MATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt; RBRACK\n                     | COMPLEXVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXMATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt;\n                       RBRACK\n\n&lt;top_var_type&gt; ::= INT [LABRACK &lt;range&gt; RABRACK]\n                 | REAL &lt;type_constraint&gt;                  | TUPLE\n\n                 | COMPLEX &lt;type_constraint&gt;\n                 | VECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | ROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | MATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | COMPLEXVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | COMPLEXROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt;\n                   RBRACK\n                 | COMPLEXMATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | ORDERED LBRACK &lt;expression&gt; RBRACK\n                 | POSITIVEORDERED LBRACK &lt;expression&gt; RBRACK\n                 | SIMPLEX LBRACK &lt;expression&gt; RBRACK\n                 | UNITVECTOR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCORR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCOV LBRACK &lt;expression&gt; [COMMA &lt;expression&gt;]\n                   RBRACK\n                 | CORRMATRIX LBRACK &lt;expression&gt; RBRACK\n                 | COVMATRIX LBRACK &lt;expression&gt; RBRACK\n\n&lt;type_constraint&gt; ::= [LABRACK &lt;range&gt; RABRACK]\n                    | LABRACK &lt;offset_mult&gt; RABRACK\n\n&lt;range&gt; ::= LOWER ASSIGN &lt;constr_expression&gt; COMMA UPPER ASSIGN\n            &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt; COMMA LOWER ASSIGN\n            &lt;constr_expression&gt;\n          | LOWER ASSIGN &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt;\n\n&lt;offset_mult&gt; ::= OFFSET ASSIGN &lt;constr_expression&gt; COMMA MULTIPLIER ASSIGN\n                  &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt; COMMA OFFSET ASSIGN\n                  &lt;constr_expression&gt;\n                | OFFSET ASSIGN &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt;\n\n&lt;arr_dims&gt; ::= ARRAY LBRACK &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACK\n\n\n\n\n&lt;expression&gt; ::= &lt;expression&gt; QMARK &lt;expression&gt; COLON &lt;expression&gt;\n               | &lt;expression&gt; &lt;infixOp&gt; &lt;expression&gt;\n               | &lt;prefixOp&gt; &lt;expression&gt;\n               | &lt;expression&gt; &lt;postfixOp&gt;\n               | &lt;common_expression&gt;\n\n&lt;constr_expression&gt; ::= &lt;constr_expression&gt; &lt;arithmeticBinOp&gt;\n                        &lt;constr_expression&gt;\n                      | &lt;prefixOp&gt; &lt;constr_expression&gt;\n                      | &lt;constr_expression&gt; &lt;postfixOp&gt;\n                      | &lt;common_expression&gt;\n\n&lt;common_expression&gt; ::= &lt;identifier&gt;\n                      | INTNUMERAL\n                      | REALNUMERAL\n                      | DOTNUMERAL\n                      | IMAGNUMERAL\n                      | LBRACE &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACE\n                      | LBRACK [&lt;expression&gt; (COMMA &lt;expression&gt;)*] RBRACK\n                      | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                        &lt;expression&gt;)*] RPAREN\n                      | TARGET LPAREN RPAREN\n                      | &lt;identifier&gt; LPAREN &lt;expression&gt; BAR [&lt;expression&gt;\n                        (COMMA &lt;expression&gt;)*] RPAREN\n                      | LPAREN &lt;expression&gt; COMMA &lt;expression&gt; (COMMA\n                        &lt;expression&gt;)* RPAREN\n                      | &lt;common_expression&gt; DOTNUMERAL\n                      | &lt;common_expression&gt; LBRACK &lt;indexes&gt; RBRACK\n                      | LPAREN &lt;expression&gt; RPAREN\n\n&lt;prefixOp&gt; ::= BANG\n             | MINUS\n             | PLUS\n\n&lt;postfixOp&gt; ::= TRANSPOSE\n\n&lt;infixOp&gt; ::= &lt;arithmeticBinOp&gt;\n            | &lt;logicalBinOp&gt;\n\n&lt;arithmeticBinOp&gt; ::= PLUS\n                    | MINUS\n                    | TIMES\n                    | DIVIDE\n                    | IDIVIDE\n                    | MODULO\n                    | LDIVIDE\n                    | ELTTIMES\n                    | ELTDIVIDE\n                    | HAT\n                    | ELTPOW\n\n&lt;logicalBinOp&gt; ::= OR\n                 | AND\n                 | EQUALS\n                 | NEQUALS\n                 | LABRACK\n                 | LEQ\n                 | RABRACK\n                 | GEQ\n\n&lt;indexes&gt; ::= epsilon\n            | COLON\n            | &lt;expression&gt;\n            | &lt;expression&gt; COLON\n            | COLON &lt;expression&gt;\n            | &lt;expression&gt; COLON &lt;expression&gt;\n            | &lt;indexes&gt; COMMA &lt;indexes&gt;\n\n&lt;printables&gt; ::= &lt;expression&gt;\n               | &lt;string_literal&gt;\n               | &lt;printables&gt; COMMA &lt;printables&gt;\n\n\n\n&lt;statement&gt; ::= &lt;atomic_statement&gt;\n              | &lt;nested_statement&gt;\n\n&lt;atomic_statement&gt; ::= &lt;common_expression&gt; &lt;assignment_op&gt; &lt;expression&gt;\n                       SEMICOLON\n                     | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                       &lt;expression&gt;)*] RPAREN SEMICOLON\n                     | &lt;expression&gt; TILDE &lt;identifier&gt; LPAREN [&lt;expression&gt;\n                       (COMMA &lt;expression&gt;)*] RPAREN [&lt;truncation&gt;] SEMICOLON\n                     | TARGET PLUSASSIGN &lt;expression&gt; SEMICOLON\n                     | BREAK SEMICOLON\n                     | CONTINUE SEMICOLON\n                     | PRINT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | REJECT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | RETURN &lt;expression&gt; SEMICOLON\n                     | RETURN SEMICOLON\n                     | SEMICOLON\n\n&lt;assignment_op&gt; ::= ASSIGN\n                  | PLUSASSIGN\n                  | MINUSASSIGN\n                  | TIMESASSIGN\n                  | DIVIDEASSIGN\n                  | ELTTIMESASSIGN\n                  | ELTDIVIDEASSIGN\n\n&lt;string_literal&gt; ::= STRINGLITERAL\n\n&lt;truncation&gt; ::= TRUNCATE LBRACK [&lt;expression&gt;] COMMA [&lt;expression&gt;] RBRACK\n\n&lt;nested_statement&gt; ::= IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                       ELSE &lt;vardecl_or_statement&gt;\n                     | IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | WHILE LPAREN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; COLON\n                       &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | PROFILE LPAREN &lt;string_literal&gt; RPAREN LBRACE\n                       &lt;vardecl_or_statement&gt;* RBRACE\n                     | LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;vardecl_or_statement&gt; ::= &lt;statement&gt;\n                         | &lt;var_decl&gt;\n\n&lt;top_vardecl_or_statement&gt; ::= &lt;statement&gt;\n                             | &lt;top_var_decl&gt;\n\n\n\n\nMany of the tokens used in the BNF grammars follow obviously from their names: DATABLOCK is the literal string ‘data’, COMMA is a single ‘,’ character, etc. The literal representation of each operator is additionally provided in the operator precedence table.\nA few tokens are not so obvious, and are defined here in regular expressions:\nIDENTIFIER = [a-zA-Z] [a-zA-Z0-9_]*\n\nSTRINGLITERAL = \".*\"\n\nINTNUMERAL = [0-9]+ (_ [0-9]+)*\n\nEXPLITERAL = [eE] [+-]? INTNUMERAL\n\nREALNUMERAL = INTNUMERAL \\. INTNUMERAL? EXPLITERAL?\n            | \\. INTNUMERAL EXPLITERAL\n            | INTNUMERAL EXPLITERAL\n\nIMAGNUMERAL = (REALNUMERAL | INTNUMERAL) i\n\nDOTNUMERAL = \\. INTNUMERAL\n\n\n\n\n\nA well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type array[ , , ] matrix, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
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-    "text": "A well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type matrix[ , , ], a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
+    "text": "A well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type array[ , , ] matrix, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
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-    "text": "Stan and its dependent libraries, are distributed under generous, freedom-respecting licenses approved by the Open Source Initiative.\nIn particular, the licenses for Stan and its dependent libraries have no “copyleft” provisions requiring applications of Stan to be open source if they are redistributed.\nThis chapter specifies the licenses for the libraries on which Stan’s math library, language, and algorithms depend. The last tool mentioned, Google Test, is only used for testing and is not needed to run Stan.\n\n\nStan is distributed under\n\nBSD 3-clause license (BSD New)\n\nThe copyright holder of each contribution is the developer or his or her assignee.1\n\n\n\nStan uses the Boost library for template metaprograms, traits programs, the parser, and various numerical libraries for special functions, probability functions, and random number generators. Boost is distributed under the\n\nBoost Software License version 1.0\n\nThe copyright for each Boost package is held by its developers or their assignees.\n\n\n\nStan uses the Eigen library for matrix arithmetic and linear algebra. Eigen is distributed under the\n\nMozilla Public License, version 2\n\nThe copyright of Eigen is owned jointly by its developers or their assignees.\n\n\n\nStan uses the SUNDIALS package for solving differential equations. SUNDIALS is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.\n\n\n\nStan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of Google Test is owned by Google, Inc.",
+    "text": "Stan and its dependent libraries, are distributed under generous, freedom-respecting licenses approved by the Open Source Initiative.\nIn particular, the licenses for Stan and its dependent libraries have no “copyleft” provisions requiring applications of Stan to be open source if they are redistributed.\nThis chapter specifies the licenses for the libraries on which Stan’s math library, language, and algorithms depend. The last tool mentioned, Google Test, is only used for testing and is not needed to run Stan.\n\n\nStan is distributed under\n\nBSD 3-clause license (BSD New)\n\nThe copyright holder of each contribution is the developer or his or her assignee.1\n\n\n\nStan uses the Boost library for template metaprograms, traits programs, the parser, and various numerical libraries for special functions, probability functions, and random number generators. Boost is distributed under the\n\nBoost Software License version 1.0\n\nThe copyright for each Boost package is held by its developers or their assignees.\n\n\n\nStan uses the Eigen library for matrix arithmetic and linear algebra. Eigen is distributed under the\n\nMozilla Public License, version 2\n\nThe copyright of Eigen is owned jointly by its developers or their assignees.\n\n\n\nStan uses the SUNDIALS package for solving differential equations. SUNDIALS is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.\n\n\n\nStan uses the Threaded Building Blocks (TBB) library for parallel computations. TBB is distributed under the\n\nApache License, version 2\n\nThe copyright of TBB is owned by Intel Corporation.\n\n\n\nStan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of Google Test is owned by Google, Inc.",
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+    "text": "Stan uses the Threaded Building Blocks (TBB) library for parallel computations. TBB is distributed under the\n\nApache License, version 2\n\nThe copyright of TBB is owned by Intel Corporation.",
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-    "text": "An expression is the syntactic unit in a Stan program that denotes a value. Every expression in a well-formed Stan program has a type that is determined statically (at compile time), based only on the type of its variables and the types of the functions used in it. If an expressions type cannot be determined statically, the Stan compiler will report the location of the problem.\nThis chapter covers the syntax, typing, and usage of the various forms of expressions in Stan.\n\n\nThe simplest form of expression is a literal that denotes a primitive numerical value.\n\n\nInteger literals represent integers of type int. Integer literals are written in base 10 without any separators. Integer literals may contain a single negative sign. (The expression --1 is interpreted as the negation of the literal -1.)\nThe following list contains well-formed integer literals.\n0, 1, -1, 256, -127098, 24567898765\nInteger literals must have values that fall within the bounds for integer values (see the section on numerical data types).\nInteger literals may not contain decimal points (.). Thus the expressions 1. and 1.0 are of type real and may not be used where a value of type int is required.\n\n\n\nA number written with a period or with scientific notation is assigned to a the continuous numeric type real. Real literals are written in base 10 with a period (.) as a separator and optionally an exponent with optional sign. Examples of well-formed real literals include the following.\n0.0, 1.0, 3.14, -217.9387, 2.7e3, -2E-5, 1.23e+3.\nThe notation e or E followed by a positive or negative integer denotes a power of 10 to multiply. For instance, 2.7e3 and 2.7e+3 denote \\(2.7 \\times 10^3\\), whereas -2E-5 denotes \\(-2 \\times\n10^{-5}\\).\n\n\n\nA number followed by the character i denotes an imaginary number and is assigned to the numeric type complex. The number preceding i may be either a real or integer literal and determines the magnitude of the imaginary number. Examples of well-formed imaginary literals include the following.\n1i, 2i, -325.786i, 1e10i, 2.87e-10i.\nNote that the character i by itself is not a well-formed imaginary literal. The unit imaginary number must be written as 1i.\n\n\n\nStan does not include complex literals directly, but a real or integer literal can be added to an imaginary literal to derive an expression that behaves like a complex literal. Examples include the following.\n1 + 2i, -3.2e9 + 1e10i\nThese will be assigned the type complex, which is the result of adding a real or integer and a complex number. They will also function like literals in the sense that the C++ compiler is able to reduce them to a single complex constant at compile time.\n\n\n\n\nA variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, get_lp, increment_log_prob,\ntarget\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.\n\n\n\n\n\nExpressions for the Stan container objects, namely arrays, vectors, row vectors, matrices, and tuples, can all be constructed using expressions.\n\n\nSquare brackets may be wrapped around a sequence of comma separated primitive expressions to produce a row vector expression. For example, the expression [ 1, 10, 100 ] denotes a row vector of three elements with real values 1.0, 10.0, and 100.0. Applying the transpose operator to a row vector expression produces a vector expression. This syntax provides a way declare and define small vectors a single line, as follows.\nrow_vector[2] rv2 =  [ 1, 2 ];\nvector[3] v3 = [ 3, 4, 5 ]';\nThe vector expression values may be compound expressions or variable names, so it is legal to write [ 2 * 3, 1 + 4] or [ x, y ], providing that x and y are primitive variables.\n\n\n\nA matrix expression consists of square brackets wrapped around a sequence of comma separated row vector expressions. This syntax provides a way declare and define a matrix in a single line, as follows.\nmatrix[3, 2] m1 = [ [ 1, 2 ], [ 3, 4 ], [5, 6 ] ];\nAny expression denoting a row vector can be used in a matrix expression. For example, the following code is valid:\nvector[2] vX = [ 1, 10 ]';\nrow_vector[2] vY = [ 100, 1000 ];\nmatrix[3, 2] m2 = [ vX', vY, [ 1, 2 ]  ];\n\n\n\nComplex vector expressions work the same way as real vector expressions. For example, the following are all legal Stan expressions and assignments.\ncomplex_vector[3] = [1 + 2i, 3 - 1.7i, 0]';\ncomplex_row_vector[2] = [12, -2i];\ncomplex_matrix[2, 3] = [[1 + 2i, 3 - 1.7i, 0],\n                        [3.9 - 1.234i, 176i, 1 + 1i]];\n\n\nThe empty expression [ ] is ambiguous and therefore is not allowed and similarly expressions such as [ [ ] ] or [ [ ], [ ] ] are not allowed.\n\n\n\n\nCurly braces may be wrapped around a sequence of expressions to produce an array expression. For example, the expression { 1, 10, 100 } denotes an integer array of three elements with values 1, 10, and 100. This syntax is particularly convenient to define small arrays in a single line, as follows.\narray[3] int a = { 1, 10, 100 };\nThe values may be compound expressions, so it is legal to write { 2 * 3, 1 + 4 }. It is also possible to write two dimensional arrays directly, as in the following example.\narray[2, 3] int b = { { 1, 2, 3 }, { 4, 5, 6 } };\nThis way, b[1] is { 1, 2, 3 } and b[2] is { 4, 5, 6 }.\nWhitespace is always interchangeable in Stan, so the above can be laid out as follows to more clearly indicate the row and column structure of the resulting two dimensional array.\narray[2, 3] int b = { { 1, 2, 3 },\n                { 4, 5, 6 } };\n\n\n\nAny type of expression may be used within braces to form an array expression. In the simplest case, all of the elements will be of the same type and the result will be an array of elements of that type. For example, the elements of the array can be vectors, in which case the result is an array of vectors.\nvector[3] b;\nvector[3] c;\n// ...\narray[2] vector[3] d = { b, c };\nThe elements may also be a mixture of int and real typed expressions, in which case the result is an array of real values.\narray[2] real b = { 1, 1.9 };\n\n\n\nStan uses parentheses around a comma-separated sequence of expressions to construct a tuple. For example, we can construct a 2-tuple as follows.\ntuple(int, vector[3]) xy = (42, [1, 2.9, -1.3]');\nThe expression 42 is of type int and the expression [1, 2.9, -1.3] is of type row_vector so that [1, 2.9, -1.3]' is of type vector and of size 3. The whole tuple expression (42, [1, 2.9, -1.3]') thus has a sized type of tuple(int, vector[3]) and an unsized type (e.g., for a function argument) of tuple(int, vector).\nStan does not support the Python notation with trailing commas, such as (1, 2, 3, ) for a 3-tuple.\n\n\n\nThere are some restrictions on how array expressions may be used that arise from their types being calculated bottom up and the basic data type and assignment rules of Stan.\n\n\nAlthough it is tempting to try to define a ragged array expression, all Stan data types are rectangular (or boxes or other higher-dimensional generalizations). Thus the following nested array expression will cause an error when it tries to create a non-rectangular array.\n{ { 1, 2, 3 }, { 4, 5 } }  // compile time error: size mismatch\nThis may appear to be OK, because it is creating a two-dimensional integer array (array[,] int) out of two one-dimensional array integer arrays (array[] int). But it is not allowed because the two one-dimensional arrays are not the same size. If the elements are array expressions, this can be diagnosed at compile time. If one or both expressions is a variable, then that won’t be caught until runtime.\n{ { 1, 2, 3 }, m }  // runtime error if m not size 3\n\n\n\nBecause there is no way to infer the type of the result, the empty array expression ({ }) is not allowed. This does not sacrifice expressive power, because a declaration is sufficient to initialize a zero-element array.\narray[0] int a;   // a is fully defined as zero element array\n\n\n\nThere is no way to declare or construct a zero-tuple or one-tuple in Stan. Tuples must be at least two elements long. The expression () does not pick out a zero-tuple—it is ill formed. Similarly, the expression (1) is of type int rather than a tuple.\n\n\n\n\n\nAny expression wrapped in parentheses is also an expression. Like in C++, but unlike in R, only the round parentheses, ( and ), are allowed. The square brackets [ and ] are reserved for array indexing and the curly braces { and } for grouping statements.\nWith parentheses it is possible to explicitly group subexpressions with operators. Without parentheses, the expression 1 + 2 * 3 has a subexpression 2 * 3 and evaluates to 7. With parentheses, this grouping may be made explicit with the expression 1 + (2 * 3). More importantly, the expression (1 + 2) *   3 has 1 + 2 as a subexpression and evaluates to 9.\n\n\n\nFor integer and real-valued expressions, Stan supports the basic binary arithmetic operations of addition (+), subtraction (-), multiplication (*) and division (/) in the usual ways.\nFor integer expressions, Stan supports the modulus (%) binary arithmetic operation. Stan also supports the unary operation of negation for integer and real-valued expressions. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3.0 + 0.14\n-15\n2 * 3 + 1\n(x - y) / 2.0\n(n * (n + 1)) / 2\nx / n\nm % n\nThe negation, addition, subtraction, and multiplication operations are extended to matrices, vectors, and row vectors. The transpose operation, written using an apostrophe (') is also supported for vectors, row vectors, and matrices. Return types for matrix operations are the smallest types that can be statically guaranteed to contain the result. The full set of allowable input types and corresponding return types is detailed in the list of functions.\nFor example, if y and mu are variables of type vector and Sigma is a variable of type matrix, then (y - mu)' * Sigma * (y - mu) is a well-formed expression of type real. The type of the complete expression is inferred working outward from the subexpressions. The subexpression(s) y - mu are of type vector because the variables y and mu are of type vector. The transpose of this expression, the subexpression (y - mu)' is of type row_vector. Multiplication is left associative and transpose has higher precedence than multiplication, so the above expression is equivalent to the following fully specified form (((y - mu)') * Sigma) * (y - mu).\nThe type of subexpression (y - mu)' * Sigma is inferred to be row_vector, being the result of multiplying a row vector by a matrix. The whole expression’s type is thus the type of a row vector multiplied by a (column) vector, which produces a real value.\nStan provides elementwise matrix multiplication (e.g., a .* b) and division (e.g., a ./ b) operations. These provide a shorthand to replace loops, but are not intrinsically more efficient than a version programmed with an elementwise calculations and assignments in a loop. For example, given declarations,\nvector[N] a;\nvector[N] b;\nvector[N] c;\nthe assignment,\nc = a .* b;\nproduces the same result with roughly the same efficiency as the loop\nfor (n in 1:N) {\n  c[n] = a[n] * b[n];\n}\nStan supports exponentiation (^) of integer and real-valued expressions. The return type of exponentiation is always a real-value. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3 ^ 2\n3.0 ^ -2\n3.0 ^ 0.14\nx ^ n\nn ^ x\nn ^ m\nx ^ y\nExponentiation is right associative, so the expression 2 ^ 3 ^ 4 is equivalent to the fully specified form 2 ^ (3 ^ 4).\n\n\nThe precedence and associativity of operators, as well as built-in syntax such as array indexing and function application is given in tabular form in the following table.\n\n\nStan’s unary, binary, and ternary operators, with their precedences, associativities, place in an expression, and a description. The last two lines list the precedence of function application and array, matrix, and vector indexing. The operators are listed in order of precedence, from least tightly binding to most tightly binding. The full set of legal arguments and corresponding result types are provided in the function documentation for the operators (i.e., operator*(int, int):int indicates the application of the multiplication operator to two integers, which returns an integer). Parentheses may be used to group expressions explicitly rather than relying on precedence and associativity.\n\n\n\n\n\n\n\n\n\n\nOp.\nPrec.\nAssoc.\nPlacement\nDescription\n\n\n\n\n? ~ :\n10\nright\nternary infix\nconditional\n\n\n||\n9\nleft\nbinary infix\nlogical or\n\n\n&&\n8\nleft\nbinary infix\nlogical and\n\n\n==\n7\nleft\nbinary infix\nequality\n\n\n!=\n7\nleft\nbinary infix\ninequality\n\n\n&lt;\n6\nleft\nbinary infix\nless than\n\n\n&lt;=\n6\nleft\nbinary infix\nless than or equal\n\n\n&gt;\n6\nleft\nbinary infix\ngreater than\n\n\n&gt;=\n6\nleft\nbinary infix\ngreater than or equal\n\n\n+\n5\nleft\nbinary infix\naddition\n\n\n-\n5\nleft\nbinary infix\nsubtraction\n\n\n*\n4\nleft\nbinary infix\nmultiplication\n\n\n.*\n4\nleft\nbinary infix\nelementwise multiplication\n\n\n/\n4\nleft\nbinary infix\n(right) division\n\n\n./\n4\nleft\nbinary infix\nelementwise division\n\n\n%\n4\nleft\nbinary infix\nmodulus\n\n\n\\\n3\nleft\nbinary infix\nleft division\n\n\n%/%\n3\nleft\nbinary infix\ninteger division\n\n\n!\n2\nn/a\nunary prefix\nlogical negation\n\n\n-\n2\nn/a\nunary prefix\nnegation\n\n\n+\n2\nn/a\nunary prefix\npromotion (no-op in Stan)\n\n\n^\n1\nright\nbinary infix\nexponentiation\n\n\n.^\n1\nright\nbinary infix\nelementwise exponentiation\n\n\n'\n0\nn/a\nunary postfix\ntransposition\n\n\n()\n0\nn/a\nprefix, wrap\nfunction application\n\n\n[]\n0\nleft\nprefix, wrap\narray, matrix indexing\n\n\n\nOther expression-forming operations, such as function application and subscripting bind more tightly than any of the arithmetic operations.\nThe precedence and associativity determine how expressions are interpreted. Because addition is left associative, the expression a + b + c is interpreted as (a + b) + c. Similarly, a / b * c is interpreted as (a / b) * c.\nBecause multiplication has higher precedence than addition, the expression a * b + c is interpreted as (a * b) + c and the expression a + b * c is interpreted as a + (b * c). Similarly, 2 * x + 3 * - y is interpreted as (2 * x) + (3 * (-y)).\nTransposition and exponentiation bind more tightly than any other arithmetic or logical operation. For vectors, row vectors, and matrices, -u' is interpreted as -(u'), u * v' as u* (v'), and u' * v as (u') * v. For integer and reals, -n ^ 3 is interpreted as -(n ^ 3).\n\n\n\n\n\n\n\nThe ternary conditional operator is unique in that it takes three arguments and uses a mixed syntax. If a is an expression of type int and b and c are expressions that can be converted to one another (e.g., compared with ==), then\na ? b : c\nis an expression of the promoted type of b and c. The only promotion allowed in Stan is integer -&gt; real -&gt; complex; e.g. if one argument is of type int and the other of type real, the conditional expression as a whole is of type real. In other cases, the arguments have to be of the same underlying Stan type (i.e., constraints don’t count, only the shape) and the conditional expression is of that type.\n\n\nThe conditional operator is the most loosely binding operator, so its arguments rarely require parentheses for disambiguation. For example,\na &gt; 0 || b &lt; 0 ? c + d : e - f\nis equivalent to the explicitly grouped version\n(a &gt; 0 || b &lt; 0) ? (c + d) : (e - f)\nThe latter is easier to read even if the parentheses are not strictly necessary.\n\n\n\nThe conditional operator is right associative, so that\na ? b : c ? d : e\nparses as if explicitly grouped as\na ? b : (c ? d : e)\nAgain, the explicitly grouped version is easier to read.\n\n\n\n\nStan’s conditional operator works very much like its C++ analogue. The first argument must be an expression denoting an integer. Typically this is a variable or a relation operator, as in the variable a in the example above. Then there are two resulting arguments, the first being the result returned if the condition evaluates to true (i.e., non-zero) and the second if the condition evaluates to false (i.e., zero). In the example above, the value b is returned if the condition evaluates to a non-zero value and c is returned if the condition evaluates to zero.\n\n\nThe key property of the conditional operator that makes it so useful in high-performance computing is that it only evaluates the returned subexpression, not the alternative expression. In other words, it is not like a typical function that evaluates its argument expressions eagerly in order to pass their values to the function. As usual, the saving is mostly in the derivatives that do not get computed rather than the unnecessary function evaluation itself.\n\n\n\nIf one return expression is a data value (an expression involving only constants and variables defined in the data or transformed data block), and the other is not, then the ternary operator will promote the data value to a parameter value. This can cause needless work calculating derivatives in some cases and be less efficient than a full if-then conditional statement. For example,\ndata {\n  array[10] real x;\n  // ...\n}\nparameters {\n  array[10] real z;\n  // ...\n}\nmodel {\n  y ~ normal(cond ? x : z, sigma);\n  // ...\n}\nwould be more efficiently (if not more transparently) coded as\nif (cond) {\n  y ~ normal(x, sigma);\n} else {\n  y ~ normal(z, sigma);\n}\nThe conditional statement, like the conditional operator, only evaluates one of the result statements. In this case, the variable x will not be promoted to a parameter and thus not cause any needless work to be carried out when propagating the chain rule during derivative calculations.\n\n\n\n\n\nStan arrays, matrices, vectors, and row vectors are all accessed using the same array-like notation. For instance, if x is a variable of type array [] real (a one-dimensional array of reals) then x[1] is the value of the first element of the array.\nSubscripting has higher precedence than any of the arithmetic operations. For example, alpha * x[1] is equivalent to alpha * (x[1]).\nMultiple subscripts may be provided within a single pair of square brackets. If x is of type array[,] real, a two-dimensional array, then x[2, 501] is of type real.\n\n\nThe subscripting operator also returns subarrays of arrays. For example, if x is of type array[,,] real, then x[2] is of type array[,] real, and x[2, 3] is of type array[] real. As a result, the expressions x[2, 3] and x[2][3] have the same meaning.\n\n\n\nIf Sigma is a variable of type matrix, then Sigma[1] denotes the first row of Sigma and has the type row_vector.\n\n\n\nStan supports mixed indexing of arrays and their vector, row vector or matrix values. For example, if m is of type matrix[ , ], a two-dimensional array of matrices, then m[1] refers to the first row of the array, which is a one-dimensional array of matrices. More than one index may be used, so that m[1, 2] is of type matrix and denotes the matrix in the first row and second column of the array. Continuing to add indices, m[1, 2, 3] is of type row_vector and denotes the third row of the matrix denoted by m[1, 2]. Finally, m[1, 2, 3, 4] is of type real and denotes the value in the third row and fourth column of the matrix that is found at the first row and second column of the array m.\n\n\n\n\nIn addition to single integer indexes, as described in the language indexing section, Stan supports multiple indexing. Multiple indexes can be integer arrays of indexes, lower bounds, upper bounds, lower and upper bounds, or simply shorthand for all of the indexes. A complete table of index types is given in the following table.\n\n\nTypes of indexes and examples with one-dimensional containers of size N and an integer array ii of type array [] real size K.\n\n\n\nindex type\nexample\nvalue\n\n\n\n\ninteger\na[11]\nvalue of a at index 11\n\n\ninteger array\na[ii]\na[ii[1]], …, a[ii[K]]\n\n\nlower bound\na[3:]\na[3], …, a[N]\n\n\nupper bound\na[:5]\na[1], …, a[5]\n\n\nrange\na[2:7]\na[2], …, a[7]\n\n\nall\na[:]\na[1], …, a[N]\n\n\nall\na[]\na[1], …, a[N]\n\n\n\n\n\n\nThe fundamental semantic rule for dealing with multiple indexes is the following. If idxs is a multiple index, then it produces an indexable position in the result. To evaluate that index position in the result, the index is first passed to the multiple index, and the resulting index used.\na[idxs, ...][i, ...] = a[idxs[i], ...][...]\nOn the other hand, if idx is a single index, it reduces the dimensionality of the output, so that\na[idx, ...] = a[idx][...]\nThe only issue is what happens with matrices and vectors. Vectors work just like arrays. Matrices with multiple row indexes and multiple column indexes produce matrices. Matrices with multiple row indexes and a single column index become (column) vectors. Matrices with a single row index and multiple column indexes become row vectors. The types are summarized in the following table.\n\n\nSpecial rules for reducing matrices based on whether the argument is a single or multiple index. Examples are for a matrix a, with integer single indexes i and j and integer array multiple indexes is and js. The same typing rules apply for all multiple indexes.\n\n\n\nexample\nrow index\ncolumn index\nresult type\n\n\n\n\na[i]\nsingle\nn/a\nrow vector\n\n\na[is]\nmultiple\nn/a\nmatrix\n\n\na[i, j]\nsingle\nsingle\nreal\n\n\na[i, js]\nsingle\nmultiple\nrow vector\n\n\na[is, j]\nmultiple\nsingle\nvector\n\n\na[is, js]\nmultiple\nmultiple\nmatrix\n\n\n\nEvaluation of matrices with multiple indexes is defined to respect the following distributivity conditions.\nm[idxs1, idxs2][i, j] = m[idxs1[i], idxs2[j]]\nm[idxs, idx][j] = m[idxs[j], idx]\nm[idx, idxs][j] = m[idx, idxs[j]]\nEvaluation of arrays of matrices and arrays of vectors or row vectors is defined recursively, beginning with the array dimensions.\n\n\n\n\n\nStan provides a range of built in mathematical and statistical functions, which are documented in the built-in function documentation.\nExpressions in Stan may consist of the name of function followed by a sequence of zero or more argument expressions. For instance, log(2.0) is the expression of type real denoting the result of applying the natural logarithm to the value of the real literal 2.0.\nSyntactically, function application has higher precedence than any of the other operators, so that y + log(x) is interpreted as y + (log(x)).\n\n\nEach function has a type signature which determines the allowable type of its arguments and its return type. For instance, the function signature for the logarithm function can be expressed as\nreal log(real);\nand the signature for the lmultiply function is\nreal lmultiply(real, real);\nA function is uniquely determined by its name and its sequence of argument types. For instance, the following two functions are different functions.\nreal mean(array [] real);\nreal mean(vector);\nThe first applies to a one-dimensional array of real values and the second to a vector.\nThe identity conditions for functions explicitly forbids having two functions with the same name and argument types but different return types. This restriction also makes it possible to infer the type of a function expression compositionally by only examining the type of its subexpressions.\n\n\n\nConstants in Stan are nothing more than nullary (no-argument) functions. For instance, the mathematical constants \\(\\pi\\) and \\(e\\) are represented as nullary functions named pi() and e(). See the Stan Functions Reference built-in constants section for a list of built-in constants.\n\n\n\nBecause of integer to real type promotion, rules must be established for which function is called given a sequence of argument types. The scheme employed by Stan is the same as that used by C++, which resolves a function call to the function requiring the minimum number of type promotions.\nFor example, consider a situation in which the following two function signatures have been registered for foo.\nreal foo(real, real);\nint foo(int, int);\nThe use of foo in the expression foo(1.0, 1.0) resolves to foo(real, real), and thus the expression foo(1.0, 1.0) itself is assigned a type of real.\nBecause integers may be promoted to real values, the expression foo(1, 1) could potentially match either foo(real, real) or foo(int, int). The former requires two type promotions and the latter requires none, so foo(1, 1) is resolved to function foo(int, int) and is thus assigned the type int.\nThe expression foo(1, 1.0) has argument types (int, real) and thus does not explicitly match either function signature. By promoting the integer expression 1 to type real, it is able to match foo(real, real), and hence the type of the function expression foo(1, 1.0) is real.\nIn some cases (though not for any built-in Stan functions), a situation may arise in which the function referred to by an expression remains ambiguous. For example, consider a situation in which there are exactly two functions named bar with the following signatures.\nreal bar(real, int);\nreal bar(int, real);\nWith these signatures, the expression bar(1.0, 1) and bar(1, 1.0) resolve to the first and second of the above functions, respectively. The expression bar(1.0, 1.0) is illegal because real values may not be demoted to integers. The expression bar(1, 1) is illegal for a different reason. If the first argument is promoted to a real value, it matches the first signature, whereas if the second argument is promoted to a real value, it matches the second signature. The problem is that these both require one promotion, so the function name bar is ambiguous. If there is not a unique function requiring fewer promotions than all others, as with bar(1, 1) given the two declarations above, the Stan compiler will flag the expression as illegal.\n\n\n\nFor most of the distributions supported by Stan, there is a corresponding random-number generating function. These random number generators are named by the distribution with the suffix _rng. For example, a univariate normal random number can be generated by normal_rng(0, 1); only the parameters of the distribution, here a location (0) and scale (1) are specified because the variate is generated.\n\n\nThe use of random-number generating functions is restricted to the transformed data and generated quantities blocks; attempts to use them elsewhere will result in a parsing error with a diagnostic message. They may also be used in the bodies of user-defined functions whose names end in _rng.\nThis allows the random number generating functions to be used for simulation in general, and for Bayesian posterior predictive checking in particular.\n\n\n\nPosterior predictive checks typically use the parameters of the model to generate simulated data (at the individual and optionally at the group level for hierarchical models), which can then be compared informally using plots and formally by means of test statistics, to the actual data in order to assess the suitability of the model; see Chapter 6 of (Gelman et al. 2013) for more information on posterior predictive checks.\n\n\n\n\n\nStan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).\n\n\n\n\n\nThere are several expression constructions in Stan that act as higher-order functions.2\nThe higher-order functions and the signature of their argument functions are listed in the following pair of tables.\n\n\nHigher-order functions in Stan with their argument function types. The first group of arguments can be a function of parameters or data. The second group of arguments, consisting of a real and integer array in all cases, must be expressions involving only data and literals.\n\n\n\n\n\n\n\n\n\nfunction\nparameter or data args\ndata args\nreturn type\n\n\n\n\nalgebra_solver\nvector, vector\narray [] real, array [] real\nvector\n\n\nalgebra_solver_newton\nvector, vector\narray [] real, array [] real\nvector\n\n\nintegrate_1d,\nreal, real, array [] real\narray [] real, array [] real\nreal\n\n\nintegrate_ode_X,\nreal, array [] real, array [] real\narray [] real, array [] real\narray [] real\n\n\nmap_rect\nvector, vector\narray [] real, array [] real\nvector\n\n\n\nFor example, the integrate_ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  array [] real foo(real t,\n                    array [] real y,\n                    array [] real theta,\n                    array [] real x_r,\n                    array [] real x_i) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\narray[2] real y0;\nreal t0;\narray[T] real ts;\narray[1] real theta;\narray[0] real x_r;\narray[0] int x_i;\n// ...\narray[T, 2] real y_hat = integrate_ode_rk45(foo, y0, t0,\n                                              ts, theta, x_r, x_i);\nThe function argument is foo, the name of the user-defined function; as shown in the higher-order functions table, integrate_ode_rk45 takes a real array, a real, three more real arrays, and an integer array as arguments and returns 2D real array.\n\n\n\nVariadic Higher-order functions in Stan with their argument function types. The first group of arguments are restricted in type. The sequence of trailing arguments can be of any length with any types.\n\n\n\nfunction\nrestricted args\nreturn type\n\n\n\n\nsolve_X\nvector\nvector\n\n\node_X,\nvector, real, array [] real\nvector[]\n\n\nreduce_sum\narray[] T, T1, T2\nreal\n\n\n\nT, T1, and T2 can be any Stan type.\nFor example, the ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  vector foo(real t, vector y, real theta, vector beta,\n            array [] real x_i, int index) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\nvector[2] y0;\nreal t0;\narray[T] real ts;\nreal theta;\nvector[7] beta;\narray[10] int x_i;\nint index;\n// ...\nvector[2] y_hat[T] = ode_rk45(foo, y0, t0, ts, theta,\n                              beta, x_i, index);\nThe function argument is foo, the name of the user-defined function. As shown in the variadic higher-order functions table, ode_rk45 takes a real, a vector, a real, a real array, and a sequence of arguments whose types match those at the end of foo and returns an array of vectors.\n\n\n\nThe function argument to higher-order functions is always passed as the first argument. This function argument must be provided as the name of a user-defined or built-in function. No quotes are necessary.\n\n\n\nSome of the arguments to higher-order functions are restricted to data. This means they must be expressions containing only data variables, transformed data variables, or literals; the may contain arbitrary functions applied to data variables or literals, but must not contain parameters, transformed parameters, or local variables from any block other than transformed data.\nFor user-defined functions the qualifier data may be prepended to the type to restrict the argument to data-only variables.\n\n\n\n\nDerivatives of the log probability function defined by a model are used in several ways by Stan. The Hamiltonian Monte Carlo samplers, including NUTS, use gradients to guide updates. The BFGS optimizers also use gradients to guide search for posterior modes.\n\n\nUnlike evaluations in pure mathematics, evaluation of derivatives in Stan is done by applying the chain rule on an expression-by-expression basis, evaluating using floating-point arithmetic. As a result, models such as the following are problematic for inference involving derivatives.\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(sqrt(x - x), 1);\n}\nAlgebraically, the sampling statement in the model could be reduced to\n  x ~ normal(0, 1);\nand it would seem the model should produce unit normal draws for x. But rather than canceling, the expression sqrt(x -   x) causes a problem for derivatives. The cause is the mechanistic evaluation of the chain rule,\n\\[\n\\begin{array}{rcl}\n\\frac{d}{dx} \\sqrt{x - x}\n& = &\n\\frac{1}{2 \\sqrt{x - x}} \\times \\frac{d}{dx} (x - x)\n\\\\[4pt]\n& = &\n\\frac{1}{0} \\times (1 - 1)\n\\\\[4pt]\n& = &\n\\infty \\times 0\n\\\\[4pt]\n& = & \\mathrm{NaN}.\n\\end{array}\n\\]\nRather than the \\(x - x\\) canceling out, it introduces a 0 into the numerator and denominator of the chain-rule evaluation.\nThe only way to avoid this kind problem is to be careful to do the necessary algebraic reductions as part of the model and not introduce expressions like sqrt(x - x) for which the chain rule produces not-a-number values.\n\n\n\nThe best way to diagnose whether something is going wrong with the derivatives is to use the test-gradient option to the sampler or optimizer inputs; this option is available in both Stan and RStan (though it may be slow, because it relies on finite differences to make a comparison to the built-in automatic differentiation).\nFor example, compiling the above model to an executable sqrt-x-minus-x in CmdStan, the test can be run as\n&gt; ./sqrt-x-minus-x diagnose test=gradient\nwhich produces\n...\nTEST GRADIENT MODE\n\n Log probability=-0.393734\n\n param idx           value           model     finite diff           error\n         0       -0.887393             nan               0             nan\nEven though finite differences calculates the right gradient of 0, automatic differentiation follows the chain rule and produces a not-a-number output.",
+    "text": "An expression is the syntactic unit in a Stan program that denotes a value. Every expression in a well-formed Stan program has a type that is determined statically (at compile time), based only on the type of its variables and the types of the functions used in it. If an expressions type cannot be determined statically, the Stan compiler will report the location of the problem.\nThis chapter covers the syntax, typing, and usage of the various forms of expressions in Stan.\n\n\nThe simplest form of expression is a literal that denotes a primitive numerical value.\n\n\nInteger literals represent integers of type int. Integer literals are written in base 10 without any separators. Integer literals may contain a single negative sign. (The expression --1 is interpreted as the negation of the literal -1.)\nThe following list contains well-formed integer literals.\n0, 1, -1, 256, -127098, 24567898765\nInteger literals must have values that fall within the bounds for integer values (see the section on numerical data types).\nInteger literals may not contain decimal points (.). Thus the expressions 1. and 1.0 are of type real and may not be used where a value of type int is required.\n\n\n\nA number written with a period or with scientific notation is assigned to a the continuous numeric type real. Real literals are written in base 10 with a period (.) as a separator and optionally an exponent with optional sign. Examples of well-formed real literals include the following.\n0.0, 1.0, 3.14, -217.9387, 2.7e3, -2E-5, 1.23e+3.\nThe notation e or E followed by a positive or negative integer denotes a power of 10 to multiply. For instance, 2.7e3 and 2.7e+3 denote \\(2.7 \\times 10^3\\), whereas -2E-5 denotes \\(-2 \\times\n10^{-5}\\).\n\n\n\nA number followed by the character i denotes an imaginary number and is assigned to the numeric type complex. The number preceding i may be either a real or integer literal and determines the magnitude of the imaginary number. Examples of well-formed imaginary literals include the following.\n1i, 2i, -325.786i, 1e10i, 2.87e-10i.\nNote that the character i by itself is not a well-formed imaginary literal. The unit imaginary number must be written as 1i.\n\n\n\nStan does not include complex literals directly, but a real or integer literal can be added to an imaginary literal to derive an expression that behaves like a complex literal. Examples include the following.\n1 + 2i, -3.2e9 + 1e10i\nThese will be assigned the type complex, which is the result of adding a real or integer and a complex number. They will also function like literals in the sense that the C++ compiler is able to reduce them to a single complex constant at compile time.\n\n\n\n\nA variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, target\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.\n\n\n\n\n\nExpressions for the Stan container objects, namely arrays, vectors, row vectors, matrices, and tuples, can all be constructed using expressions.\n\n\nSquare brackets may be wrapped around a sequence of comma separated primitive expressions to produce a row vector expression. For example, the expression [ 1, 10, 100 ] denotes a row vector of three elements with real values 1.0, 10.0, and 100.0. Applying the transpose operator to a row vector expression produces a vector expression. This syntax provides a way declare and define small vectors a single line, as follows.\nrow_vector[2] rv2 =  [ 1, 2 ];\nvector[3] v3 = [ 3, 4, 5 ]';\nThe vector expression values may be compound expressions or variable names, so it is legal to write [ 2 * 3, 1 + 4] or [ x, y ], providing that x and y are primitive variables.\n\n\n\nA matrix expression consists of square brackets wrapped around a sequence of comma separated row vector expressions. This syntax provides a way declare and define a matrix in a single line, as follows.\nmatrix[3, 2] m1 = [ [ 1, 2 ], [ 3, 4 ], [5, 6 ] ];\nAny expression denoting a row vector can be used in a matrix expression. For example, the following code is valid:\nvector[2] vX = [ 1, 10 ]';\nrow_vector[2] vY = [ 100, 1000 ];\nmatrix[3, 2] m2 = [ vX', vY, [ 1, 2 ]  ];\n\n\n\nComplex vector expressions work the same way as real vector expressions. For example, the following are all legal Stan expressions and assignments.\ncomplex_vector[3] = [1 + 2i, 3 - 1.7i, 0]';\ncomplex_row_vector[2] = [12, -2i];\ncomplex_matrix[2, 3] = [[1 + 2i, 3 - 1.7i, 0],\n                        [3.9 - 1.234i, 176i, 1 + 1i]];\n\n\nThe empty expression [ ] is ambiguous and therefore is not allowed and similarly expressions such as [ [ ] ] or [ [ ], [ ] ] are not allowed.\n\n\n\n\nCurly braces may be wrapped around a sequence of expressions to produce an array expression. For example, the expression { 1, 10, 100 } denotes an integer array of three elements with values 1, 10, and 100. This syntax is particularly convenient to define small arrays in a single line, as follows.\narray[3] int a = { 1, 10, 100 };\nThe values may be compound expressions, so it is legal to write { 2 * 3, 1 + 4 }. It is also possible to write two dimensional arrays directly, as in the following example.\narray[2, 3] int b = { { 1, 2, 3 }, { 4, 5, 6 } };\nThis way, b[1] is { 1, 2, 3 } and b[2] is { 4, 5, 6 }.\nWhitespace is always interchangeable in Stan, so the above can be laid out as follows to more clearly indicate the row and column structure of the resulting two dimensional array.\narray[2, 3] int b = { { 1, 2, 3 },\n                { 4, 5, 6 } };\n\n\n\nAny type of expression may be used within braces to form an array expression. In the simplest case, all of the elements will be of the same type and the result will be an array of elements of that type. For example, the elements of the array can be vectors, in which case the result is an array of vectors.\nvector[3] b;\nvector[3] c;\n// ...\narray[2] vector[3] d = { b, c };\nThe elements may also be a mixture of int and real typed expressions, in which case the result is an array of real values.\narray[2] real b = { 1, 1.9 };\n\n\n\nStan uses parentheses around a comma-separated sequence of expressions to construct a tuple. For example, we can construct a 2-tuple as follows.\ntuple(int, vector[3]) xy = (42, [1, 2.9, -1.3]');\nThe expression 42 is of type int and the expression [1, 2.9, -1.3] is of type row_vector so that [1, 2.9, -1.3]' is of type vector and of size 3. The whole tuple expression (42, [1, 2.9, -1.3]') thus has a sized type of tuple(int, vector[3]) and an unsized type (e.g., for a function argument) of tuple(int, vector).\nStan does not support the Python notation with trailing commas, such as (1, 2, 3, ) for a 3-tuple.\n\n\n\nThere are some restrictions on how array expressions may be used that arise from their types being calculated bottom up and the basic data type and assignment rules of Stan.\n\n\nAlthough it is tempting to try to define a ragged array expression, all Stan data types are rectangular (or boxes or other higher-dimensional generalizations). Thus the following nested array expression will cause an error when it tries to create a non-rectangular array.\n{ { 1, 2, 3 }, { 4, 5 } }  // compile time error: size mismatch\nThis may appear to be OK, because it is creating a two-dimensional integer array (array[,] int) out of two one-dimensional array integer arrays (array[] int). But it is not allowed because the two one-dimensional arrays are not the same size. If the elements are array expressions, this can be diagnosed at compile time. If one or both expressions is a variable, then that won’t be caught until runtime.\n{ { 1, 2, 3 }, m }  // runtime error if m not size 3\n\n\n\nBecause there is no way to infer the type of the result, the empty array expression ({ }) is not allowed. This does not sacrifice expressive power, because a declaration is sufficient to initialize a zero-element array.\narray[0] int a;   // a is fully defined as zero element array\n\n\n\nThere is no way to declare or construct a zero-tuple or one-tuple in Stan. Tuples must be at least two elements long. The expression () does not pick out a zero-tuple—it is ill formed. Similarly, the expression (1) is of type int rather than a tuple.\n\n\n\n\n\nAny expression wrapped in parentheses is also an expression. Like in C++, but unlike in R, only the round parentheses, ( and ), are allowed. The square brackets [ and ] are reserved for array indexing and the curly braces { and } for grouping statements.\nWith parentheses it is possible to explicitly group subexpressions with operators. Without parentheses, the expression 1 + 2 * 3 has a subexpression 2 * 3 and evaluates to 7. With parentheses, this grouping may be made explicit with the expression 1 + (2 * 3). More importantly, the expression (1 + 2) *   3 has 1 + 2 as a subexpression and evaluates to 9.\n\n\n\nFor integer and real-valued expressions, Stan supports the basic binary arithmetic operations of addition (+), subtraction (-), multiplication (*) and division (/) in the usual ways.\nFor integer expressions, Stan supports the modulus (%) binary arithmetic operation. Stan also supports the unary operation of negation for integer and real-valued expressions. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3.0 + 0.14\n-15\n2 * 3 + 1\n(x - y) / 2.0\n(n * (n + 1)) / 2\nx / n\nm % n\nThe negation, addition, subtraction, and multiplication operations are extended to matrices, vectors, and row vectors. The transpose operation, written using an apostrophe (') is also supported for vectors, row vectors, and matrices. Return types for matrix operations are the smallest types that can be statically guaranteed to contain the result. The full set of allowable input types and corresponding return types is detailed in the list of functions.\nFor example, if y and mu are variables of type vector and Sigma is a variable of type matrix, then (y - mu)' * Sigma * (y - mu) is a well-formed expression of type real. The type of the complete expression is inferred working outward from the subexpressions. The subexpression(s) y - mu are of type vector because the variables y and mu are of type vector. The transpose of this expression, the subexpression (y - mu)' is of type row_vector. Multiplication is left associative and transpose has higher precedence than multiplication, so the above expression is equivalent to the following fully specified form (((y - mu)') * Sigma) * (y - mu).\nThe type of subexpression (y - mu)' * Sigma is inferred to be row_vector, being the result of multiplying a row vector by a matrix. The whole expression’s type is thus the type of a row vector multiplied by a (column) vector, which produces a real value.\nStan provides elementwise matrix multiplication (e.g., a .* b) and division (e.g., a ./ b) operations. These provide a shorthand to replace loops, but are not intrinsically more efficient than a version programmed with an elementwise calculations and assignments in a loop. For example, given declarations,\nvector[N] a;\nvector[N] b;\nvector[N] c;\nthe assignment,\nc = a .* b;\nproduces the same result with roughly the same efficiency as the loop\nfor (n in 1:N) {\n  c[n] = a[n] * b[n];\n}\nStan supports exponentiation (^) of integer and real-valued expressions. The return type of exponentiation is always a real-value. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3 ^ 2\n3.0 ^ -2\n3.0 ^ 0.14\nx ^ n\nn ^ x\nn ^ m\nx ^ y\nExponentiation is right associative, so the expression 2 ^ 3 ^ 4 is equivalent to the fully specified form 2 ^ (3 ^ 4).\n\n\nThe precedence and associativity of operators, as well as built-in syntax such as array indexing and function application is given in tabular form in the following table.\n\n\nStan’s unary, binary, and ternary operators, with their precedences, associativities, place in an expression, and a description. The last two lines list the precedence of function application and array, matrix, and vector indexing. The operators are listed in order of precedence, from least tightly binding to most tightly binding. The full set of legal arguments and corresponding result types are provided in the function documentation for the operators (i.e., operator*(int, int):int indicates the application of the multiplication operator to two integers, which returns an integer). Parentheses may be used to group expressions explicitly rather than relying on precedence and associativity.\n\n\n\n\n\n\n\n\n\n\nOp.\nPrec.\nAssoc.\nPlacement\nDescription\n\n\n\n\n? ~ :\n10\nright\nternary infix\nconditional\n\n\n||\n9\nleft\nbinary infix\nlogical or\n\n\n&&\n8\nleft\nbinary infix\nlogical and\n\n\n==\n7\nleft\nbinary infix\nequality\n\n\n!=\n7\nleft\nbinary infix\ninequality\n\n\n&lt;\n6\nleft\nbinary infix\nless than\n\n\n&lt;=\n6\nleft\nbinary infix\nless than or equal\n\n\n&gt;\n6\nleft\nbinary infix\ngreater than\n\n\n&gt;=\n6\nleft\nbinary infix\ngreater than or equal\n\n\n+\n5\nleft\nbinary infix\naddition\n\n\n-\n5\nleft\nbinary infix\nsubtraction\n\n\n*\n4\nleft\nbinary infix\nmultiplication\n\n\n.*\n4\nleft\nbinary infix\nelementwise multiplication\n\n\n/\n4\nleft\nbinary infix\n(right) division\n\n\n./\n4\nleft\nbinary infix\nelementwise division\n\n\n%\n4\nleft\nbinary infix\nmodulus\n\n\n\\\n3\nleft\nbinary infix\nleft division\n\n\n%/%\n3\nleft\nbinary infix\ninteger division\n\n\n!\n2\nn/a\nunary prefix\nlogical negation\n\n\n-\n2\nn/a\nunary prefix\nnegation\n\n\n+\n2\nn/a\nunary prefix\npromotion (no-op in Stan)\n\n\n^\n1\nright\nbinary infix\nexponentiation\n\n\n.^\n1\nright\nbinary infix\nelementwise exponentiation\n\n\n'\n0\nn/a\nunary postfix\ntransposition\n\n\n()\n0\nn/a\nprefix, wrap\nfunction application\n\n\n[]\n0\nleft\nprefix, wrap\narray, matrix indexing\n\n\n\nOther expression-forming operations, such as function application and subscripting bind more tightly than any of the arithmetic operations.\nThe precedence and associativity determine how expressions are interpreted. Because addition is left associative, the expression a + b + c is interpreted as (a + b) + c. Similarly, a / b * c is interpreted as (a / b) * c.\nBecause multiplication has higher precedence than addition, the expression a * b + c is interpreted as (a * b) + c and the expression a + b * c is interpreted as a + (b * c). Similarly, 2 * x + 3 * - y is interpreted as (2 * x) + (3 * (-y)).\nTransposition and exponentiation bind more tightly than any other arithmetic or logical operation. For vectors, row vectors, and matrices, -u' is interpreted as -(u'), u * v' as u* (v'), and u' * v as (u') * v. For integer and reals, -n ^ 3 is interpreted as -(n ^ 3).\n\n\n\n\n\n\n\nThe ternary conditional operator is unique in that it takes three arguments and uses a mixed syntax. If a is an expression of type int and b and c are expressions that can be converted to one another (e.g., compared with ==), then\na ? b : c\nis an expression of the promoted type of b and c. The only promotion allowed in Stan is integer -&gt; real -&gt; complex; e.g. if one argument is of type int and the other of type real, the conditional expression as a whole is of type real. In other cases, the arguments have to be of the same underlying Stan type (i.e., constraints don’t count, only the shape) and the conditional expression is of that type.\n\n\nThe conditional operator is the most loosely binding operator, so its arguments rarely require parentheses for disambiguation. For example,\na &gt; 0 || b &lt; 0 ? c + d : e - f\nis equivalent to the explicitly grouped version\n(a &gt; 0 || b &lt; 0) ? (c + d) : (e - f)\nThe latter is easier to read even if the parentheses are not strictly necessary.\n\n\n\nThe conditional operator is right associative, so that\na ? b : c ? d : e\nparses as if explicitly grouped as\na ? b : (c ? d : e)\nAgain, the explicitly grouped version is easier to read.\n\n\n\n\nStan’s conditional operator works very much like its C++ analogue. The first argument must be an expression denoting an integer. Typically this is a variable or a relation operator, as in the variable a in the example above. Then there are two resulting arguments, the first being the result returned if the condition evaluates to true (i.e., non-zero) and the second if the condition evaluates to false (i.e., zero). In the example above, the value b is returned if the condition evaluates to a non-zero value and c is returned if the condition evaluates to zero.\n\n\nThe key property of the conditional operator that makes it so useful in high-performance computing is that it only evaluates the returned subexpression, not the alternative expression. In other words, it is not like a typical function that evaluates its argument expressions eagerly in order to pass their values to the function. As usual, the saving is mostly in the derivatives that do not get computed rather than the unnecessary function evaluation itself.\n\n\n\nIf one return expression is a data value (an expression involving only constants and variables defined in the data or transformed data block), and the other is not, then the ternary operator will promote the data value to a parameter value. This can cause needless work calculating derivatives in some cases and be less efficient than a full if-then conditional statement. For example,\ndata {\n  array[10] real x;\n  // ...\n}\nparameters {\n  array[10] real z;\n  // ...\n}\nmodel {\n  y ~ normal(cond ? x : z, sigma);\n  // ...\n}\nwould be more efficiently (if not more transparently) coded as\nif (cond) {\n  y ~ normal(x, sigma);\n} else {\n  y ~ normal(z, sigma);\n}\nThe conditional statement, like the conditional operator, only evaluates one of the result statements. In this case, the variable x will not be promoted to a parameter and thus not cause any needless work to be carried out when propagating the chain rule during derivative calculations.\n\n\n\n\n\nStan arrays, matrices, vectors, and row vectors are all accessed using the same array-like notation. For instance, if x is a variable of type array [] real (a one-dimensional array of reals) then x[1] is the value of the first element of the array.\nSubscripting has higher precedence than any of the arithmetic operations. For example, alpha * x[1] is equivalent to alpha * (x[1]).\nMultiple subscripts may be provided within a single pair of square brackets. If x is of type array[,] real, a two-dimensional array, then x[2, 501] is of type real.\n\n\nThe subscripting operator also returns subarrays of arrays. For example, if x is of type array[,,] real, then x[2] is of type array[,] real, and x[2, 3] is of type array[] real. As a result, the expressions x[2, 3] and x[2][3] have the same meaning.\n\n\n\nIf Sigma is a variable of type matrix, then Sigma[1] denotes the first row of Sigma and has the type row_vector.\n\n\n\nStan supports mixed indexing of arrays and their vector, row vector or matrix values. For example, if m is of type matrix[ , ], a two-dimensional array of matrices, then m[1] refers to the first row of the array, which is a one-dimensional array of matrices. More than one index may be used, so that m[1, 2] is of type matrix and denotes the matrix in the first row and second column of the array. Continuing to add indices, m[1, 2, 3] is of type row_vector and denotes the third row of the matrix denoted by m[1, 2]. Finally, m[1, 2, 3, 4] is of type real and denotes the value in the third row and fourth column of the matrix that is found at the first row and second column of the array m.\n\n\n\n\nIn addition to single integer indexes, as described in the language indexing section, Stan supports multiple indexing. Multiple indexes can be integer arrays of indexes, lower bounds, upper bounds, lower and upper bounds, or simply shorthand for all of the indexes. A complete table of index types is given in the following table.\n\n\nTypes of indexes and examples with one-dimensional containers of size N and an integer array ii of type array [] real size K.\n\n\n\nindex type\nexample\nvalue\n\n\n\n\ninteger\na[11]\nvalue of a at index 11\n\n\ninteger array\na[ii]\na[ii[1]], …, a[ii[K]]\n\n\nlower bound\na[3:]\na[3], …, a[N]\n\n\nupper bound\na[:5]\na[1], …, a[5]\n\n\nrange\na[2:7]\na[2], …, a[7]\n\n\nall\na[:]\na[1], …, a[N]\n\n\nall\na[]\na[1], …, a[N]\n\n\n\n\n\n\nThe fundamental semantic rule for dealing with multiple indexes is the following. If idxs is a multiple index, then it produces an indexable position in the result. To evaluate that index position in the result, the index is first passed to the multiple index, and the resulting index used.\na[idxs, ...][i, ...] = a[idxs[i], ...][...]\nOn the other hand, if idx is a single index, it reduces the dimensionality of the output, so that\na[idx, ...] = a[idx][...]\nThe only issue is what happens with matrices and vectors. Vectors work just like arrays. Matrices with multiple row indexes and multiple column indexes produce matrices. Matrices with multiple row indexes and a single column index become (column) vectors. Matrices with a single row index and multiple column indexes become row vectors. The types are summarized in the following table.\n\n\nSpecial rules for reducing matrices based on whether the argument is a single or multiple index. Examples are for a matrix a, with integer single indexes i and j and integer array multiple indexes is and js. The same typing rules apply for all multiple indexes.\n\n\n\nexample\nrow index\ncolumn index\nresult type\n\n\n\n\na[i]\nsingle\nn/a\nrow vector\n\n\na[is]\nmultiple\nn/a\nmatrix\n\n\na[i, j]\nsingle\nsingle\nreal\n\n\na[i, js]\nsingle\nmultiple\nrow vector\n\n\na[is, j]\nmultiple\nsingle\nvector\n\n\na[is, js]\nmultiple\nmultiple\nmatrix\n\n\n\nEvaluation of matrices with multiple indexes is defined to respect the following distributivity conditions.\nm[idxs1, idxs2][i, j] = m[idxs1[i], idxs2[j]]\nm[idxs, idx][j] = m[idxs[j], idx]\nm[idx, idxs][j] = m[idx, idxs[j]]\nEvaluation of arrays of matrices and arrays of vectors or row vectors is defined recursively, beginning with the array dimensions.\n\n\n\n\n\nStan provides a range of built in mathematical and statistical functions, which are documented in the built-in function documentation.\nExpressions in Stan may consist of the name of function followed by a sequence of zero or more argument expressions. For instance, log(2.0) is the expression of type real denoting the result of applying the natural logarithm to the value of the real literal 2.0.\nSyntactically, function application has higher precedence than any of the other operators, so that y + log(x) is interpreted as y + (log(x)).\n\n\nEach function has a type signature which determines the allowable type of its arguments and its return type. For instance, the function signature for the logarithm function can be expressed as\nreal log(real);\nand the signature for the lmultiply function is\nreal lmultiply(real, real);\nA function is uniquely determined by its name and its sequence of argument types. For instance, the following two functions are different functions.\nreal mean(array [] real);\nreal mean(vector);\nThe first applies to a one-dimensional array of real values and the second to a vector.\nThe identity conditions for functions explicitly forbids having two functions with the same name and argument types but different return types. This restriction also makes it possible to infer the type of a function expression compositionally by only examining the type of its subexpressions.\n\n\n\nConstants in Stan are nothing more than nullary (no-argument) functions. For instance, the mathematical constants \\(\\pi\\) and \\(e\\) are represented as nullary functions named pi() and e(). See the Stan Functions Reference built-in constants section for a list of built-in constants.\n\n\n\nBecause of integer to real type promotion, rules must be established for which function is called given a sequence of argument types. The scheme employed by Stan is the same as that used by C++, which resolves a function call to the function requiring the minimum number of type promotions.\nFor example, consider a situation in which the following two function signatures have been registered for foo.\nreal foo(real, real);\nint foo(int, int);\nThe use of foo in the expression foo(1.0, 1.0) resolves to foo(real, real), and thus the expression foo(1.0, 1.0) itself is assigned a type of real.\nBecause integers may be promoted to real values, the expression foo(1, 1) could potentially match either foo(real, real) or foo(int, int). The former requires two type promotions and the latter requires none, so foo(1, 1) is resolved to function foo(int, int) and is thus assigned the type int.\nThe expression foo(1, 1.0) has argument types (int, real) and thus does not explicitly match either function signature. By promoting the integer expression 1 to type real, it is able to match foo(real, real), and hence the type of the function expression foo(1, 1.0) is real.\nIn some cases (though not for any built-in Stan functions), a situation may arise in which the function referred to by an expression remains ambiguous. For example, consider a situation in which there are exactly two functions named bar with the following signatures.\nreal bar(real, int);\nreal bar(int, real);\nWith these signatures, the expression bar(1.0, 1) and bar(1, 1.0) resolve to the first and second of the above functions, respectively. The expression bar(1.0, 1.0) is illegal because real values may not be demoted to integers. The expression bar(1, 1) is illegal for a different reason. If the first argument is promoted to a real value, it matches the first signature, whereas if the second argument is promoted to a real value, it matches the second signature. The problem is that these both require one promotion, so the function name bar is ambiguous. If there is not a unique function requiring fewer promotions than all others, as with bar(1, 1) given the two declarations above, the Stan compiler will flag the expression as illegal.\n\n\n\nFor most of the distributions supported by Stan, there is a corresponding random-number generating function. These random number generators are named by the distribution with the suffix _rng. For example, a univariate normal random number can be generated by normal_rng(0, 1); only the parameters of the distribution, here a location (0) and scale (1) are specified because the variate is generated.\n\n\nThe use of random-number generating functions is restricted to the transformed data and generated quantities blocks; attempts to use them elsewhere will result in a parsing error with a diagnostic message. They may also be used in the bodies of user-defined functions whose names end in _rng.\nThis allows the random number generating functions to be used for simulation in general, and for Bayesian posterior predictive checking in particular.\n\n\n\nPosterior predictive checks typically use the parameters of the model to generate simulated data (at the individual and optionally at the group level for hierarchical models), which can then be compared informally using plots and formally by means of test statistics, to the actual data in order to assess the suitability of the model; see Chapter 6 of (Gelman et al. 2013) for more information on posterior predictive checks.\n\n\n\n\n\nStan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\nTuples have the natural extension of the above rules, applied to all sub-types at once\n\nA tuple(U1, ..., UN) may be promoted to a tuple(T1, ..., TN) if every Un can be promoted to Tn for n in 1:N\n\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).\n\n\n\n\n\nThere are several expression constructions in Stan that act as higher-order functions.2\nThe higher-order functions and the signature of their argument functions are listed in the following pair of tables.\n\n\nHigher-order functions in Stan with their argument function types. The first group of arguments can be a function of parameters or data. The second group of arguments, consisting of a real and integer array in all cases, must be expressions involving only data and literals.\n\n\n\n\n\n\n\n\n\nfunction\nparameter or data args\ndata args\nreturn type\n\n\n\n\nalgebra_solver\nvector, vector\narray [] real, array [] real\nvector\n\n\nalgebra_solver_newton\nvector, vector\narray [] real, array [] real\nvector\n\n\nintegrate_1d,\nreal, real, array [] real\narray [] real, array [] real\nreal\n\n\nintegrate_ode_X,\nreal, array [] real, array [] real\narray [] real, array [] real\narray [] real\n\n\nmap_rect\nvector, vector\narray [] real, array [] real\nvector\n\n\n\nFor example, the integrate_ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  array [] real foo(real t,\n                    array [] real y,\n                    array [] real theta,\n                    array [] real x_r,\n                    array [] real x_i) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\narray[2] real y0;\nreal t0;\narray[T] real ts;\narray[1] real theta;\narray[0] real x_r;\narray[0] int x_i;\n// ...\narray[T, 2] real y_hat = integrate_ode_rk45(foo, y0, t0,\n                                              ts, theta, x_r, x_i);\nThe function argument is foo, the name of the user-defined function; as shown in the higher-order functions table, integrate_ode_rk45 takes a real array, a real, three more real arrays, and an integer array as arguments and returns 2D real array.\n\n\n\nVariadic Higher-order functions in Stan with their argument function types. The first group of arguments are restricted in type. The sequence of trailing arguments can be of any length with any types.\n\n\n\nfunction\nrestricted args\nreturn type\n\n\n\n\nsolve_X\nvector\nvector\n\n\node_X,\nvector, real, array [] real\nvector[]\n\n\nreduce_sum\narray[] T, T1, T2\nreal\n\n\n\nT, T1, and T2 can be any Stan type.\nFor example, the ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  vector foo(real t, vector y, real theta, vector beta,\n            array [] real x_i, int index) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\nvector[2] y0;\nreal t0;\narray[T] real ts;\nreal theta;\nvector[7] beta;\narray[10] int x_i;\nint index;\n// ...\nvector[2] y_hat[T] = ode_rk45(foo, y0, t0, ts, theta,\n                              beta, x_i, index);\nThe function argument is foo, the name of the user-defined function. As shown in the variadic higher-order functions table, ode_rk45 takes a real, a vector, a real, a real array, and a sequence of arguments whose types match those at the end of foo and returns an array of vectors.\n\n\n\nThe function argument to higher-order functions is always passed as the first argument. This function argument must be provided as the name of a user-defined or built-in function. No quotes are necessary.\n\n\n\nSome of the arguments to higher-order functions are restricted to data. This means they must be expressions containing only data variables, transformed data variables, or literals; the may contain arbitrary functions applied to data variables or literals, but must not contain parameters, transformed parameters, or local variables from any block other than transformed data.\nFor user-defined functions the qualifier data may be prepended to the type to restrict the argument to data-only variables.\n\n\n\n\nDerivatives of the log probability function defined by a model are used in several ways by Stan. The Hamiltonian Monte Carlo samplers, including NUTS, use gradients to guide updates. The BFGS optimizers also use gradients to guide search for posterior modes.\n\n\nUnlike evaluations in pure mathematics, evaluation of derivatives in Stan is done by applying the chain rule on an expression-by-expression basis, evaluating using floating-point arithmetic. As a result, models such as the following are problematic for inference involving derivatives.\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(sqrt(x - x), 1);\n}\nAlgebraically, the sampling statement in the model could be reduced to\n  x ~ normal(0, 1);\nand it would seem the model should produce unit normal draws for x. But rather than canceling, the expression sqrt(x -   x) causes a problem for derivatives. The cause is the mechanistic evaluation of the chain rule,\n\\[\n\\begin{array}{rcl}\n\\frac{d}{dx} \\sqrt{x - x}\n& = &\n\\frac{1}{2 \\sqrt{x - x}} \\times \\frac{d}{dx} (x - x)\n\\\\[4pt]\n& = &\n\\frac{1}{0} \\times (1 - 1)\n\\\\[4pt]\n& = &\n\\infty \\times 0\n\\\\[4pt]\n& = & \\mathrm{NaN}.\n\\end{array}\n\\]\nRather than the \\(x - x\\) canceling out, it introduces a 0 into the numerator and denominator of the chain-rule evaluation.\nThe only way to avoid this kind problem is to be careful to do the necessary algebraic reductions as part of the model and not introduce expressions like sqrt(x - x) for which the chain rule produces not-a-number values.\n\n\n\nThe best way to diagnose whether something is going wrong with the derivatives is to use the test-gradient option to the sampler or optimizer inputs; this option is available in both Stan and RStan (though it may be slow, because it relies on finite differences to make a comparison to the built-in automatic differentiation).\nFor example, compiling the above model to an executable sqrt-x-minus-x in CmdStan, the test can be run as\n&gt; ./sqrt-x-minus-x diagnose test=gradient\nwhich produces\n...\nTEST GRADIENT MODE\n\n Log probability=-0.393734\n\n param idx           value           model     finite diff           error\n         0       -0.887393             nan               0             nan\nEven though finite differences calculates the right gradient of 0, automatic differentiation follows the chain rule and produces a not-a-number output.",
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-    "text": "A variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, get_lp, increment_log_prob,\ntarget\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.",
+    "text": "A variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, target\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.",
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-    "text": "Stan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).",
+    "text": "Stan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\nTuples have the natural extension of the above rules, applied to all sub-types at once\n\nA tuple(U1, ..., UN) may be promoted to a tuple(T1, ..., TN) if every Un can be promoted to Tn for n in 1:N\n\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).",
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     "href": "reference-manual/deprecations.html",
     "title": "Deprecated Features",
     "section": "",
-    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Features]. The Stan compiler can automatically update many of these on the behalf of the user for at least one version after they are removed.\nAny feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).\n\n\nDeprecated:The distribution lkj_cov is deprecated.\nReplacement: Replace lkj_cov_lpdf(...) with an lkj_corr distribution on the correlation matrix and independent lognormal distributions on the scales. That is, replace\ncov_matrix[K] Sigma;\n// ...\nSigma ~ lkj_cov(mu, tau, eta);\nwith\ncorr_matrix[K] Omega;\nvector&lt;lower=0&gt;[K] sigma;\n// ...\nOmega ~ lkj_corr(eta);\nsigma ~ lognormal(mu, tau);\n// ...\ncov_matrix[K] Sigma;\nSigma &lt;- quad_form_diag(Omega, sigma);\nThe variable Sigma may be defined as a local variable in the model block or as a transformed parameter. An even more efficient transform would use Cholesky factors rather than full correlation matrix types.\nScheduled Removal: Stan 3.0 or later.\n\n\n\nSeveral built-in Stan functions have been deprecated. Consult the functions reference for more information.",
+    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in Removed Features. The Stan compiler can automatically update many of these on the behalf of the user for at least one version after they are removed.\nAny feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).\n\n\nDeprecated:The distribution lkj_cov is deprecated.\nReplacement: Replace lkj_cov_lpdf(...) with an lkj_corr distribution on the correlation matrix and independent lognormal distributions on the scales. That is, replace\ncov_matrix[K] Sigma;\n// ...\nSigma ~ lkj_cov(mu, tau, eta);\nwith\ncorr_matrix[K] Omega;\nvector&lt;lower=0&gt;[K] sigma;\n// ...\nOmega ~ lkj_corr(eta);\nsigma ~ lognormal(mu, tau);\n// ...\ncov_matrix[K] Sigma;\nSigma &lt;- quad_form_diag(Omega, sigma);\nThe variable Sigma may be defined as a local variable in the model block or as a transformed parameter. An even more efficient transform would use Cholesky factors rather than full correlation matrix types.\nScheduled Removal: Stan 3.0 or later.\n\n\n\nSeveral built-in Stan functions have been deprecated. Consult the functions reference for more information.",
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     "title": "Program Blocks",
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-    "text": "A Stan program is organized into a sequence of named blocks, the bodies of which consist of variable declarations, followed in the case of some blocks with statements.\n\n\nThe full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n\\slshape (initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.\n\n\n\n\nStatistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y[N] is a modeled data vector. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.\n\n\n\nThe rest of this chapter will lay out the details of each block in order, starting with the data block in this section.\n\n\nThe data block is for the declaration of variables that are read in as data. With the current model executable, each Markov chain of draws will be executed in a different process, and each such process will read the data exactly once.3\nData variables are not transformed in any way. The format for data files or data in memory depends on the interface; see the user’s guides and interface documentation for PyStan, RStan, and CmdStan for details.\n\n\n\nThe data block does not allow statements.\n\n\n\nEach variable’s value is validated against its declaration as it is read. For example, if a variable sigma is declared as real&lt;lower=0&gt;, then trying to assign it a negative value will raise an error. As a result, data type errors will be caught as early as possible. Similarly, attempts to provide data of the wrong size for a compound data structure will also raise an error.\n\n\n\n\nThe transformed data block is for declaring and defining variables that do not need to be changed when running the program.\n\n\nFor the transformed data block, variables are all declared in the variable declarations and defined in the statements. There is no reading from external sources and no transformations performed.\nVariables declared in the data block may be used to declare transformed variables.\n\n\n\nThe statements in a transformed data block are used to define (provide values for) variables declared in the transformed data block. Assignments are only allowed to variables declared in the transformed data block.\nThese statements are executed once, in order, right after the data is read into the data variables. This means they are executed once per chain.\nVariables declared in the data block may be used in statements in the transformed data block.\n\n\nThe statements in the transformed data block are designed to be executed once and have a deterministic result. Therefore, log probability is not accumulated and sampling statements may not be used.\n\n\n\n\nAny constraints on variables declared in the transformed data block are checked after the statements are executed. If any defined variable violates its constraints, Stan will halt with a diagnostic error message.\n\n\n\n\nThe variables declared in the parameters program block correspond directly to the variables being sampled by Stan’s samplers (HMC and NUTS). From a user’s perspective, the parameters in the program block are the parameters being sampled by Stan.\nVariables declared as parameters cannot be directly assigned values. So there is no block of statements in the parameters program block. Variable quantities derived from parameters may be declared in the transformed parameters or generated quantities blocks, or may be defined as local variables in any statement blocks following their declaration.\nThere is a substantial amount of computation involved for parameter variables in a Stan program at each leapfrog step within the HMC or NUTS samplers, and a bit more computation along with writes involved for saving the parameter values corresponding to a sample.\n\n\nStan’s two samplers, standard Hamiltonian Monte Carlo (HMC) and the adaptive No-U-Turn sampler (NUTS), are most easily (and often most effectively) implemented over a multivariate probability density that has support on all of \\(\\mathbb{R}^n\\). To do this, the parameters defined in the parameters block must be transformed so they are unconstrained.\nIn practice, the samplers keep an unconstrained parameter vector in memory representing the current state of the sampler. The model defined by the compiled Stan program defines an (unnormalized) log probability function over the unconstrained parameters. In order to do this, the log probability function must apply the inverse transform to the unconstrained parameters to calculate the constrained parameters defined in Stan’s parameters program block. The log Jacobian of the inverse transform is then added to the accumulated log probability function. This then allows the Stan model to be defined in terms of the constrained parameters.\nIn some cases, the number of parameters is reduced in the unconstrained space. For instance, a \\(K\\)-simplex only requires \\(K-1\\) unconstrained parameters, and a \\(K\\)-correlation matrix only requires \\(\\binom{K}{2}\\) unconstrained parameters. This means that the probability function defined by the compiled Stan program may have fewer parameters than it would appear from looking at the declarations in the parameters program block.\nThe probability function on the unconstrained parameters is defined in such a way that the order of the parameters in the vector corresponds to the order of the variables defined in the parameters program block. The details of the specific transformations are provided in the variable transforms chapter.\n\n\n\nHamiltonian Monte Carlo requires the gradient of the (unnormalized) log probability function with respect to the unconstrained parameters to be evaluated during every leapfrog step. There may be one leapfrog step per sample or hundreds, with more being required for models with complex posterior distribution geometries.\nGradients are calculated behind the scenes using Stan’s algorithmic differentiation library. The time to compute the gradient does not depend directly on the number of parameters, only on the number of subexpressions in the calculation of the log probability. This includes the expressions added from the transforms’ Jacobians.\nThe amount of work done by the sampler does depend on the number of unconstrained parameters, but this is usually dwarfed by the gradient calculations.\n\n\n\nIn the basic Stan compiled program, there is a file to which the values of variables are written for each draw. The constrained versions of the variables are written in the order they are defined in the parameters block. In order to do this, the transformed parameter, model, and generated quantities statements must also be executed.\n\n\n\n\nThe transformed parameters program block consists of optional variable declarations followed by statements. After the statements are executed, the constraints on the transformed parameters are validated. Any variable declared as a transformed parameter is part of the output produced for draws.\nAny variable that is defined wholly in terms of data or transformed data should be declared and defined in the transformed data block. Defining such quantities in the transformed parameters block is legal, but much less efficient than defining them as transformed data.\n\n\nLike the constraints on data, the constraints on transformed parameters is meant to catch programming errors as well as convey programmer intent. They are not automatically transformed in such a way as to be satisfied. What will happen if a transformed parameter does not match its constraint is that the current parameter values will be rejected. This can cause Stan’s algorithms to hang or to devolve to random walks. It is not intended to be a way to enforce ad hoc constraints in Stan programs. See the section on reject statements for further discussion of the behavior of reject statements.\n\n\n\n\nThe model program block consists of optional variable declarations followed by statements. The variables in the model block are local variables and are not written as part of the output.\nLocal variables may not be defined with constraints because there is no well-defined way to have them be both flexible and easy to validate.\nThe statements in the model block typically define the model. This is the block in which probability (sampling notation) statements are allowed. These are typically used when programming in the BUGS idiom to define the probability model.\n\n\n\nThe generated quantities program block is rather different than the other blocks. Nothing in the generated quantities block affects the sampled parameter values. The block is executed only after a sample has been generated.\nAmong the applications of posterior inference that can be coded in the generated quantities block are\n\nforward sampling to generate simulated data for model testing,\ngenerating predictions for new data,\ncalculating posterior event probabilities, including multiple comparisons, sign tests, etc.,\ncalculating posterior expectations,\ntransforming parameters for reporting,\napplying full Bayesian decision theory,\ncalculating log likelihoods, deviances, etc. for model comparison.\n\nParameter estimates, predictions, statistics, and event probabilities calculated directly using plug-in estimates. Stan automatically provides full Bayesian inference by producing draws from the posterior distribution of any calculated event probabilities, predictions, or statistics.\nWithin the generated quantities block, the values of all other variables declared in earlier program blocks (other than local variables) are available for use in the generated quantities block.\nIt is more efficient to define a variable in the generated quantities block instead of the transformed parameters block. Therefore, if a quantity does not play a role in the model, it should be defined in the generated quantities block.\nAfter the generated quantities statements are executed, the constraints on the declared generated quantity variables are validated.\nAll variables declared as generated quantities are printed as part of the output. Variables declared in nested blocks are local variables, not generated quantities, and thus won’t be printed. For example:\ngenerated quantities {\n  int a; // added to the output\n\n  {\n    int b; // not added to the output\n  }\n}",
+    "text": "A Stan program is organized into a sequence of named blocks, the bodies of which consist of variable declarations, followed in the case of some blocks with statements.\n\n\nThe full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n(initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.\n\n\n\n\nStatistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y is an array of modeled data. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.\n\n\n\nThe rest of this chapter will lay out the details of each block in order, starting with the data block in this section.\n\n\nThe data block is for the declaration of variables that are read in as data. With the current model executable, each Markov chain of draws will be executed in a different process, and each such process will read the data exactly once.3\nData variables are not transformed in any way. The format for data files or data in memory depends on the interface; see the user’s guides and interface documentation for PyStan, RStan, and CmdStan for details.\n\n\n\nThe data block does not allow statements.\n\n\n\nEach variable’s value is validated against its declaration as it is read. For example, if a variable sigma is declared as real&lt;lower=0&gt;, then trying to assign it a negative value will raise an error. As a result, data type errors will be caught as early as possible. Similarly, attempts to provide data of the wrong size for a compound data structure will also raise an error.\n\n\n\n\nThe transformed data block is for declaring and defining variables that do not need to be changed when running the program.\n\n\nFor the transformed data block, variables are all declared in the variable declarations and defined in the statements. There is no reading from external sources and no transformations performed.\nVariables declared in the data block may be used to declare transformed variables.\n\n\n\nThe statements in a transformed data block are used to define (provide values for) variables declared in the transformed data block. Assignments are only allowed to variables declared in the transformed data block.\nThese statements are executed once, in order, right after the data is read into the data variables. This means they are executed once per chain.\nVariables declared in the data block may be used in statements in the transformed data block.\n\n\nThe statements in the transformed data block are designed to be executed once and have a deterministic result. Therefore, log probability is not accumulated and sampling statements may not be used.\n\n\n\n\nAny constraints on variables declared in the transformed data block are checked after the statements are executed. If any defined variable violates its constraints, Stan will halt with a diagnostic error message.\n\n\n\n\nThe variables declared in the parameters program block correspond directly to the variables being sampled by Stan’s samplers (HMC and NUTS). From a user’s perspective, the parameters in the program block are the parameters being sampled by Stan.\nVariables declared as parameters cannot be directly assigned values. So there is no block of statements in the parameters program block. Variable quantities derived from parameters may be declared in the transformed parameters or generated quantities blocks, or may be defined as local variables in any statement blocks following their declaration.\nThere is a substantial amount of computation involved for parameter variables in a Stan program at each leapfrog step within the HMC or NUTS samplers, and a bit more computation along with writes involved for saving the parameter values corresponding to a sample.\n\n\nStan’s two samplers, standard Hamiltonian Monte Carlo (HMC) and the adaptive No-U-Turn sampler (NUTS), are most easily (and often most effectively) implemented over a multivariate probability density that has support on all of \\(\\mathbb{R}^n\\). To do this, the parameters defined in the parameters block must be transformed so they are unconstrained.\nIn practice, the samplers keep an unconstrained parameter vector in memory representing the current state of the sampler. The model defined by the compiled Stan program defines an (unnormalized) log probability function over the unconstrained parameters. In order to do this, the log probability function must apply the inverse transform to the unconstrained parameters to calculate the constrained parameters defined in Stan’s parameters program block. The log Jacobian of the inverse transform is then added to the accumulated log probability function. This then allows the Stan model to be defined in terms of the constrained parameters.\nIn some cases, the number of parameters is reduced in the unconstrained space. For instance, a \\(K\\)-simplex only requires \\(K-1\\) unconstrained parameters, and a \\(K\\)-correlation matrix only requires \\(\\binom{K}{2}\\) unconstrained parameters. This means that the probability function defined by the compiled Stan program may have fewer parameters than it would appear from looking at the declarations in the parameters program block.\nThe probability function on the unconstrained parameters is defined in such a way that the order of the parameters in the vector corresponds to the order of the variables defined in the parameters program block. The details of the specific transformations are provided in the variable transforms chapter.\n\n\n\nHamiltonian Monte Carlo requires the gradient of the (unnormalized) log probability function with respect to the unconstrained parameters to be evaluated during every leapfrog step. There may be one leapfrog step per sample or hundreds, with more being required for models with complex posterior distribution geometries.\nGradients are calculated behind the scenes using Stan’s algorithmic differentiation library. The time to compute the gradient does not depend directly on the number of parameters, only on the number of subexpressions in the calculation of the log probability. This includes the expressions added from the transforms’ Jacobians.\nThe amount of work done by the sampler does depend on the number of unconstrained parameters, but this is usually dwarfed by the gradient calculations.\n\n\n\nIn the basic Stan compiled program, there is a file to which the values of variables are written for each draw. The constrained versions of the variables are written in the order they are defined in the parameters block. In order to do this, the transformed parameter, model, and generated quantities statements must also be executed.\n\n\n\n\nThe transformed parameters program block consists of optional variable declarations followed by statements. After the statements are executed, the constraints on the transformed parameters are validated. Any variable declared as a transformed parameter is part of the output produced for draws.\nAny variable that is defined wholly in terms of data or transformed data should be declared and defined in the transformed data block. Defining such quantities in the transformed parameters block is legal, but much less efficient than defining them as transformed data.\n\n\nLike the constraints on data, the constraints on transformed parameters is meant to catch programming errors as well as convey programmer intent. They are not automatically transformed in such a way as to be satisfied. What will happen if a transformed parameter does not match its constraint is that the current parameter values will be rejected. This can cause Stan’s algorithms to hang or to devolve to random walks. It is not intended to be a way to enforce ad hoc constraints in Stan programs. See the section on reject statements for further discussion of the behavior of reject statements.\n\n\n\n\nThe model program block consists of optional variable declarations followed by statements. The variables in the model block are local variables and are not written as part of the output.\nLocal variables may not be defined with constraints because there is no well-defined way to have them be both flexible and easy to validate.\nThe statements in the model block typically define the model. This is the block in which probability (sampling notation) statements are allowed. These are typically used when programming in the BUGS idiom to define the probability model.\n\n\n\nThe generated quantities program block is rather different than the other blocks. Nothing in the generated quantities block affects the sampled parameter values. The block is executed only after a sample has been generated.\nAmong the applications of posterior inference that can be coded in the generated quantities block are\n\nforward sampling to generate simulated data for model testing,\ngenerating predictions for new data,\ncalculating posterior event probabilities, including multiple comparisons, sign tests, etc.,\ncalculating posterior expectations,\ntransforming parameters for reporting,\napplying full Bayesian decision theory,\ncalculating log likelihoods, deviances, etc. for model comparison.\n\nParameter estimates, predictions, statistics, and event probabilities calculated directly using plug-in estimates. Stan automatically provides full Bayesian inference by producing draws from the posterior distribution of any calculated event probabilities, predictions, or statistics.\nWithin the generated quantities block, the values of all other variables declared in earlier program blocks (other than local variables) are available for use in the generated quantities block.\nIt is more efficient to define a variable in the generated quantities block instead of the transformed parameters block. Therefore, if a quantity does not play a role in the model, it should be defined in the generated quantities block.\nAfter the generated quantities statements are executed, the constraints on the declared generated quantity variables are validated.\nAll variables declared as generated quantities are printed as part of the output. Variables declared in nested blocks are local variables, not generated quantities, and thus won’t be printed. For example:\ngenerated quantities {\n  int a; // added to the output\n\n  {\n    int b; // not added to the output\n  }\n}",
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-    "text": "The full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n\\slshape (initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.",
+    "text": "The full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n(initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.",
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-    "text": "Statistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y[N] is a modeled data vector. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.",
+    "text": "Statistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y is an array of modeled data. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.",
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-    "text": "This chapter describes Stan’s built-in function that take various types of arguments and return integer values.\n\n\nStan’s arithmetic is based on standard double-precision C++ integer and floating-point arithmetic. If the arguments to an arithmetic operator are both integers, as in 2 + 2, integer arithmetic is used. If one argument is an integer and the other a floating-point value, as in 2.0 + 2 and 2 + 2.0, then the integer is promoted to a floating point value and floating-point arithmetic is used.\nInteger arithmetic behaves slightly differently than floating point arithmetic. The first difference is how overflow is treated. If the sum or product of two integers overflows the maximum integer representable, the result is an undesirable wraparound behavior at the bit level. If the integers were first promoted to real numbers, they would not overflow a floating-point representation. There are no extra checks in Stan to flag overflows, so it is up to the user to make sure it does not occur.\nSecondly, because the set of integers is not closed under division and there is no special infinite value for integers, integer division implicitly rounds the result. If both arguments are positive, the result is rounded down. For example, 1 / 2 evaluates to 0 and 5 / 3 evaluates to 1.\nIf one of the integer arguments to division is negative, the latest C++ specification ( C++11), requires rounding toward zero. This would have 1 / 2 and -1 / 2 evaluate to 0, -7 / 2 evaluate to -3, and 7 / 2 evaluate to 3. Before the C++11 specification, the behavior was platform dependent, allowing rounding up or down. All compilers recent enough to be able to deal with Stan’s templating should follow the C++11 specification, but it may be worth testing if you are not sure and plan to use integer division with negative values.\nUnlike floating point division, where 1.0 / 0.0 produces the special positive infinite value, integer division by zero, as in 1 / 0, has undefined behavior in the C++ standard. For example, the clang++ compiler on Mac OS X returns 3764, whereas the g++ compiler throws an exception and aborts the program with a warning. As with overflow, it is up to the user to make sure integer divide-by-zero does not occur.\n\n\nOperators are described using the C++ syntax. For instance, the binary operator of addition, written X + Y, would have the Stan signature int operator+(int, int) indicating it takes two real arguments and returns a real value. As noted previously, the value of integer division is platform-dependent when rounding is platform dependent before C++11; the descriptions below provide the C++11 definition.\n \n\nint operator+(int x, int y) The sum of the addends x and y \\[\\begin{equation*} \\text{operator+}(x,y) = (x + y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator-(int x, int y) The difference between the minuend x and subtrahend y \\[\\begin{equation*}\n\\text{operator-}(x,y) = (x - y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator*(int x, int y) The product of the factors x and y \\[\\begin{equation*} \\text{operator*}(x,y) = (x\n\\times y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator/(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator/}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\] deprecated; - use operator%/% instead.\nAvailable since 2.0, deprecated in 2.24\n \n\nint operator%/%(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator\\%/\\%}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\]\nAvailable since 2.24\n \n\nint operator%(int x, int y) x modulo y, which is the positive remainder after dividing x by y. If both x and y are non-negative, so is the result; otherwise, the sign of the result is platform dependent. \\[\\begin{equation*} \\mathrm{operator\\%}(x, y) \\ =\n\\ x \\ \\text{mod} \\ y \\ = \\ x - y * \\lfloor x / y \\rfloor \\end{equation*}\\]\nAvailable since 2.13\n\n\n\n \n\nint operator-(int x) The negation of the subtrahend x \\[\\begin{equation*} \\text{operator-}(x) = -x \\end{equation*}\\]\nAvailable since 2.0\n \n\nT operator-(T x) Vectorized version of operator-. If T x is a (possibly nested) array of integers, -x is the same shape array where each individual integer is negated.\nAvailable since 2.31\n \n\nint operator+(int x) This is a no-op. \\[\\begin{equation*} \\text{operator+}(x) = x \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n\n \n\nT abs(T x) The absolute value of x.\nThis function works elementwise over containers such as vectors. Given a type T which is int, or an array of ints, abs returns the same type where each element has had its absolute value taken.\nAvailable since 2.0, vectorized in 2.30\n \n\nint int_step(int x)\n \n\nint int_step(real x) Return the step function of x as an integer, \\[\\begin{equation*} \\mathrm{int\\_step}(x)\n= \\begin{cases} 1 & \\text{if } x &gt; 0 \\\\ 0 & \\text{if } x \\leq 0 \\text{\nor } x \\text{ is } NaN \\end{cases} \\end{equation*}\\] Warning: int_step(0) and int_step(NaN) return 0 whereas step(0) and step(NaN) return 1.\nSee the warning in section step functions about the dangers of step functions applied to anything other than data.\nAvailable since 2.0\n\n\n\n \n\nint min(int x, int y) Return the minimum of x and y. \\[\\begin{equation*} \\text{min}(x, y) = \\begin{cases} x &\n\\text{if } x &lt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n \n\nint max(int x, int y) Return the maximum of x and y. \\[\\begin{equation*} \\text{max}(x, y) = \\begin{cases} x &\n\\text{if } x &gt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n \n\nint size(int x)\n \n\nint size(real x)\nReturn the size of x which for scalar-valued x is 1\nAvailable since 2.26\n\n\n\nIt is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see [Rounding functions]) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
+    "text": "This chapter describes Stan’s built-in function that take various types of arguments and return integer values.\n\n\nStan’s arithmetic is based on standard double-precision C++ integer and floating-point arithmetic. If the arguments to an arithmetic operator are both integers, as in 2 + 2, integer arithmetic is used. If one argument is an integer and the other a floating-point value, as in 2.0 + 2 and 2 + 2.0, then the integer is promoted to a floating point value and floating-point arithmetic is used.\nInteger arithmetic behaves slightly differently than floating point arithmetic. The first difference is how overflow is treated. If the sum or product of two integers overflows the maximum integer representable, the result is an undesirable wraparound behavior at the bit level. If the integers were first promoted to real numbers, they would not overflow a floating-point representation. There are no extra checks in Stan to flag overflows, so it is up to the user to make sure it does not occur.\nSecondly, because the set of integers is not closed under division and there is no special infinite value for integers, integer division implicitly rounds the result. If both arguments are positive, the result is rounded down. For example, 1 / 2 evaluates to 0 and 5 / 3 evaluates to 1.\nIf one of the integer arguments to division is negative, the latest C++ specification ( C++11), requires rounding toward zero. This would have 1 / 2 and -1 / 2 evaluate to 0, -7 / 2 evaluate to -3, and 7 / 2 evaluate to 3. Before the C++11 specification, the behavior was platform dependent, allowing rounding up or down. All compilers recent enough to be able to deal with Stan’s templating should follow the C++11 specification, but it may be worth testing if you are not sure and plan to use integer division with negative values.\nUnlike floating point division, where 1.0 / 0.0 produces the special positive infinite value, integer division by zero, as in 1 / 0, has undefined behavior in the C++ standard. For example, the clang++ compiler on Mac OS X returns 3764, whereas the g++ compiler throws an exception and aborts the program with a warning. As with overflow, it is up to the user to make sure integer divide-by-zero does not occur.\n\n\nOperators are described using the C++ syntax. For instance, the binary operator of addition, written X + Y, would have the Stan signature int operator+(int, int) indicating it takes two real arguments and returns a real value. As noted previously, the value of integer division is platform-dependent when rounding is platform dependent before C++11; the descriptions below provide the C++11 definition.\n \n\nint operator+(int x, int y) The sum of the addends x and y \\[\\begin{equation*} \\text{operator+}(x,y) = (x + y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator-(int x, int y) The difference between the minuend x and subtrahend y \\[\\begin{equation*}\n\\text{operator-}(x,y) = (x - y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator*(int x, int y) The product of the factors x and y \\[\\begin{equation*} \\text{operator*}(x,y) = (x\n\\times y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator/(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator/}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\] deprecated; - use operator%/% instead.\nAvailable since 2.0, deprecated in 2.24\n \n\nint operator%/%(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator\\%/\\%}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\]\nAvailable since 2.24\n \n\nint operator%(int x, int y) x modulo y, which is the positive remainder after dividing x by y. If both x and y are non-negative, so is the result; otherwise, the sign of the result is platform dependent. \\[\\begin{equation*} \\mathrm{operator\\%}(x, y) \\ =\n\\ x \\ \\text{mod} \\ y \\ = \\ x - y * \\lfloor x / y \\rfloor \\end{equation*}\\]\nAvailable since 2.13\n\n\n\n \n\nint operator-(int x) The negation of the subtrahend x \\[\\begin{equation*} \\text{operator-}(x) = -x \\end{equation*}\\]\nAvailable since 2.0\n \n\nT operator-(T x) Vectorized version of operator-. If T x is a (possibly nested) array of integers, -x is the same shape array where each individual integer is negated.\nAvailable since 2.31\n \n\nint operator+(int x) This is a no-op. \\[\\begin{equation*} \\text{operator+}(x) = x \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n\n \n\nT abs(T x) The absolute value of x.\nThis function works elementwise over containers such as vectors. Given a type T which is int, or an array of ints, abs returns the same type where each element has had its absolute value taken.\nAvailable since 2.0, vectorized in 2.30\n \n\nint int_step(int x)\n \n\nint int_step(real x) Return the step function of x as an integer, \\[\\begin{equation*} \\mathrm{int\\_step}(x)\n= \\begin{cases} 1 & \\text{if } x &gt; 0 \\\\ 0 & \\text{if } x \\leq 0 \\text{\nor } x \\text{ is } NaN \\end{cases} \\end{equation*}\\] Warning: int_step(0) and int_step(NaN) return 0 whereas step(0) and step(NaN) return 1.\nSee the warning in section step functions about the dangers of step functions applied to anything other than data.\nAvailable since 2.0\n\n\n\n \n\nint min(int x, int y) Return the minimum of x and y. \\[\\begin{equation*} \\text{min}(x, y) = \\begin{cases} x &\n\\text{if } x &lt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n \n\nint max(int x, int y) Return the maximum of x and y. \\[\\begin{equation*} \\text{max}(x, y) = \\begin{cases} x &\n\\text{if } x &gt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n \n\nint size(int x)\n \n\nint size(real x)\nReturn the size of x which for scalar-valued x is 1\nAvailable since 2.26\n\n\n\nIt is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see Rounding functions) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
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-    "text": "It is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see [Rounding functions]) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
+    "text": "It is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see Rounding functions) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
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-    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of get_lp or multiply_log) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Functions]. The Stan compiler can automatically update these on the behalf of the user for the entire deprecation window and at least one version following the removal.\n\n\nDeprecated: Using / with two integer arguments is interpreted as integer floor division, such that\n\\[ 1 / 2 = 0 \\]\nThis is deprecated due to its confusion with real-valued division, where\n\\[ 1.0 / 2.0 = 0.5 \\]\nReplacement: Use the integer division operator operator%/% instead.\n\n\n\nThese ODE integrator functions have been replaced by those described in Ordinary Differential Equation (ODE) Solvers.\n\n\nA system of ODEs is specified as an ordinary function in Stan within the functions block. The ODE system function must have this function signature:\narray[] real ode(real time, array[] real state, array[] real theta,\n                 array[] real x_r, array[] int x_i);\nThe ODE system function should return the derivative of the state with respect to time at the time provided. The length of the returned real array must match the length of the state input into the function.\nThe arguments to this function are:\n\ntime, the time to evaluate the ODE system\nstate, the state of the ODE system at the time specified\ntheta, parameter values used to evaluate the ODE system\nx_r, data values used to evaluate the ODE system\nx_i, integer data values used to evaluate the ODE system.\n\nThe ODE system function separates parameter values, theta, from data values, x_r, for efficiency in computing the gradients of the ODE.\n\n\n\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i, real rel_tol, real abs_tol, int max_num_steps) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the Adams-Moulton method.\nAvailable since 2.23, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the Adams-Moulton method with additional control parameters for the solver.\nAvailable since 2.23, deprecated in 2.24\n\n\n\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n\n\n\nThe arguments to the ODE solvers in both the stiff and non-stiff cases are as follows.\n\node: function literal referring to a function specifying the system of differential equations with signature:\n\n(real, array[] real, array[] real, data array[] real, data array[] int):array[] real\nThe arguments represent (1) time, (2) system state, (3) parameters, (4) real data, and (5) integer data, and the return value contains the derivatives with respect to time of the state,\n\ninitial_state: initial state, type array[] real,\ninitial_time: initial time, type int or real,\ntimes: solution times, type array[] real,\ntheta: parameters, type array[] real,\ndata x_r: real data, type array[] real, data only, and\ndata x_i: integer data, type array[] int, data only.\n\nFor more fine-grained control of the ODE solvers, these parameters can also be provided:\n\ndata rel_tol: relative tolerance for the ODE solver, type real, data only,\ndata abs_tol: absolute tolerance for the ODE solver, type real, data only, and\ndata max_num_steps: maximum number of steps to take in the ODE solver, type int, data only.\n\n\n\nThe return value for the ODE solvers is an array of type array[,] real, with values consisting of solutions at the specified times.\n\n\n\nThe sizes must match, and in particular, the following groups are of the same size:\n\nstate variables passed into the system function, derivatives returned by the system function, initial state passed into the solver, and rows of the return value of the solver,\nsolution times and number of rows of the return value of the solver,\nparameters, real data and integer data passed to the solver will be passed to the system function\n\n\n\n\n\n\nThese algebraic solver functions have been replaced by those described in Algebraic Equation Solvers..\n\n\nAn algebraic system is specified as an ordinary function in Stan within the function block. The algebraic system function must have this signature:\n vector algebra_system(vector y, vector theta,\n                              data array[] real x_r, array[] int x_i)\nThe algebraic system function should return the value of the algebraic function which goes to 0, when we plug in the solution to the algebraic system.\nThe argument of this function are:\n\ny, the unknowns we wish to solve for\ntheta, parameter values used to evaluate the algebraic system\nx_r, data values used to evaluate the algebraic system\nx_i, integer data used to evaluate the algebraic system\n\nThe algebraic system function separates parameter values, theta, from data values, x_r, for efficiency in propagating the derivatives through the algebraic system.\n\n\n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm.\nAvailable since 2.17, deprecated in 2.31\n \n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm with additional control parameters for the solver.\nAvailable since 2.17, deprecated in 2.31\nNote: In future releases, the function algebra_solver will be deprecated and replaced with algebra_solver_powell.\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using Newton’s method.\nAvailable since 2.24, deprecated in 2.31\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using Newton’s method with additional control parameters for the solver.\nAvailable since 2.24, deprecated in 2.31\n\n\nThe arguments to the algebraic solvers are as follows:\n\nalgebra_system: function literal referring to a function specifying the system of algebraic equations with signature (vector, vector, array[] real, array[] int):vector. The arguments represent (1) unknowns, (2) parameters, (3) real data, and (4) integer data, and the return value contains the value of the algebraic function, which goes to 0 when we plug in the solution to the algebraic system,\ny_guess: initial guess for the solution, type vector,\ntheta: parameters only, type vector,\nx_r: real data only, type array[] real, and\nx_i: integer data only, type array[] int.\n\nFor more fine-grained control of the algebraic solver, these parameters can also be provided:\n\nrel_tol: relative tolerance for the algebraic solver, type real, data only,\nfunction_tol: function tolerance for the algebraic solver, type real, data only,\nmax_num_steps: maximum number of steps to take in the algebraic solver, type int, data only.\n\n\n\n\nThe return value for the algebraic solver is an object of type vector, with values which, when plugged in as y make the algebraic function go to 0.\n\n\n\nCertain sizes have to be consistent. The initial guess, return value of the solver, and return value of the algebraic function must all be the same size.\nThe parameters, real data, and integer data will be passed from the solver directly to the system function.",
+    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of get_lp or multiply_log) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in Removed Functions. The Stan compiler can automatically update these on the behalf of the user for the entire deprecation window and at least one version following the removal.\n\n\nDeprecated: Using / with two integer arguments is interpreted as integer floor division, such that\n\\[ 1 / 2 = 0 \\]\nThis is deprecated due to its confusion with real-valued division, where\n\\[ 1.0 / 2.0 = 0.5 \\]\nReplacement: Use the integer division operator operator%/% instead.\n\n\n\nThese ODE integrator functions have been replaced by those described in Ordinary Differential Equation (ODE) Solvers.\n\n\nA system of ODEs is specified as an ordinary function in Stan within the functions block. The ODE system function must have this function signature:\narray[] real ode(real time, array[] real state, array[] real theta,\n                 array[] real x_r, array[] int x_i);\nThe ODE system function should return the derivative of the state with respect to time at the time provided. The length of the returned real array must match the length of the state input into the function.\nThe arguments to this function are:\n\ntime, the time to evaluate the ODE system\nstate, the state of the ODE system at the time specified\ntheta, parameter values used to evaluate the ODE system\nx_r, data values used to evaluate the ODE system\nx_i, integer data values used to evaluate the ODE system.\n\nThe ODE system function separates parameter values, theta, from data values, x_r, for efficiency in computing the gradients of the ODE.\n\n\n\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i, real rel_tol, real abs_tol, int max_num_steps) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the Adams-Moulton method.\nAvailable since 2.23, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the Adams-Moulton method with additional control parameters for the solver.\nAvailable since 2.23, deprecated in 2.24\n\n\n\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n\n\n\nThe arguments to the ODE solvers in both the stiff and non-stiff cases are as follows.\n\node: function literal referring to a function specifying the system of differential equations with signature:\n\n(real, array[] real, array[] real, data array[] real, data array[] int):array[] real\nThe arguments represent (1) time, (2) system state, (3) parameters, (4) real data, and (5) integer data, and the return value contains the derivatives with respect to time of the state,\n\ninitial_state: initial state, type array[] real,\ninitial_time: initial time, type int or real,\ntimes: solution times, type array[] real,\ntheta: parameters, type array[] real,\ndata x_r: real data, type array[] real, data only, and\ndata x_i: integer data, type array[] int, data only.\n\nFor more fine-grained control of the ODE solvers, these parameters can also be provided:\n\ndata rel_tol: relative tolerance for the ODE solver, type real, data only,\ndata abs_tol: absolute tolerance for the ODE solver, type real, data only, and\ndata max_num_steps: maximum number of steps to take in the ODE solver, type int, data only.\n\n\n\nThe return value for the ODE solvers is an array of type array[,] real, with values consisting of solutions at the specified times.\n\n\n\nThe sizes must match, and in particular, the following groups are of the same size:\n\nstate variables passed into the system function, derivatives returned by the system function, initial state passed into the solver, and rows of the return value of the solver,\nsolution times and number of rows of the return value of the solver,\nparameters, real data and integer data passed to the solver will be passed to the system function\n\n\n\n\n\n\nThese algebraic solver functions have been replaced by those described in Algebraic Equation Solvers..\n\n\nAn algebraic system is specified as an ordinary function in Stan within the function block. The algebraic system function must have this signature:\n vector algebra_system(vector y, vector theta,\n                              data array[] real x_r, array[] int x_i)\nThe algebraic system function should return the value of the algebraic function which goes to 0, when we plug in the solution to the algebraic system.\nThe argument of this function are:\n\ny, the unknowns we wish to solve for\ntheta, parameter values used to evaluate the algebraic system\nx_r, data values used to evaluate the algebraic system\nx_i, integer data used to evaluate the algebraic system\n\nThe algebraic system function separates parameter values, theta, from data values, x_r, for efficiency in propagating the derivatives through the algebraic system.\n\n\n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm.\nAvailable since 2.17, deprecated in 2.31\n \n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm with additional control parameters for the solver.\nAvailable since 2.17, deprecated in 2.31\nNote: In future releases, the function algebra_solver will be deprecated and replaced with algebra_solver_powell.\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using Newton’s method.\nAvailable since 2.24, deprecated in 2.31\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using Newton’s method with additional control parameters for the solver.\nAvailable since 2.24, deprecated in 2.31\n\n\nThe arguments to the algebraic solvers are as follows:\n\nalgebra_system: function literal referring to a function specifying the system of algebraic equations with signature (vector, vector, array[] real, array[] int):vector. The arguments represent (1) unknowns, (2) parameters, (3) real data, and (4) integer data, and the return value contains the value of the algebraic function, which goes to 0 when we plug in the solution to the algebraic system,\ny_guess: initial guess for the solution, type vector,\ntheta: parameters only, type vector,\nx_r: real data only, type array[] real, and\nx_i: integer data only, type array[] int.\n\nFor more fine-grained control of the algebraic solver, these parameters can also be provided:\n\nrel_tol: relative tolerance for the algebraic solver, type real, data only,\nfunction_tol: function tolerance for the algebraic solver, type real, data only,\nmax_num_steps: maximum number of steps to take in the algebraic solver, type int, data only.\n\n\n\n\nThe return value for the algebraic solver is an object of type vector, with values which, when plugged in as y make the algebraic function go to 0.\n\n\n\nCertain sizes have to be consistent. The initial guess, return value of the solver, and return value of the algebraic function must all be the same size.\nThe parameters, real data, and integer data will be passed from the solver directly to the system function.",
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-    "text": "CmdStan can approximate the posterior distribution using variational inference. The approximation is a Gaussian in the unconstrained variable space.\nStan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) Kucukelbir et al. (2017). ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using stochastic gradient ascent. The measures of convergence are similar to the relative tolerance scheme of Stan’s optimization algorithms.\nThe algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run with method=variational via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\nThe console output includes a notice that this algorithm is considered to be experimental:\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n\n\nStan implements two variational algorithms. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.\n\n\n\n\n\niter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv. Lines 1 - 28 contain configuration information:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nThe column header row is:\nlp__,log_p__,log_g__,theta\nThe stepsize adaptation information is:\n# Stepsize adaptation complete.\n# eta = 1\nThe reported mean variational approximations information is:\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...",
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-    "text": "Stan implements two variational algorithms. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.",
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-    "text": "iter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.",
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-    "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv. Lines 1 - 28 contain configuration information:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nThe column header row is:\nlp__,log_p__,log_g__,theta\nThe stepsize adaptation information is:\n# Stepsize adaptation complete.\n# eta = 1\nThe reported mean variational approximations information is:\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...",
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-    "text": "CmdStan translates Stan programs to C++ using the Stan compiler program which is included in the CmdStan release bin directory as program stanc. One can view the complete stanc documentation in the Stan User’s Guide.\nAs of release 2.22, the CmdStan Stan to C++ compiler is written in OCaml. This compiler is called “stanc3” and has has its own repository https://github.com/stan-dev/stanc3, from which pre-built binaries for Linux, Mac, and Windows can be downloaded.\n\n\nBefore the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.\n\n\n\nThe Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
+    "text": "The CmdStan stansummary program reports statistics for one or more sampler chains over all sampler and model parameters and quantities of interest. The statistics reported include both summary statistics of the estimates and diagnostic statistics on the sampler chains, reported in the following order:\n\nMean - sample mean\nMCSE - Monte Carlo Standard Error, a measure of the amount of noise in the sample\nStdDev - sample standard deviation\nQuantiles - default 5%, 50%, 95%\nN_eff - effective sample size - the number of independent draws in the sample\nN_eff/S - the number of independent draws per second\nR_hat - \\(\\hat{R}\\) statistic, a measure of chain equilibrium, must be within \\(0.05\\) of \\(1.0\\).\n\nWhen reviewing the stansummary output, it is important to check the final three output columns first - these are the diagnostic statistics on chain convergence and number of independent draws in the sample. A \\(\\hat{R}\\) statistic of greater than \\(1.05\\) indicates that the chain has not converged and therefore the sample is not drawn from the posterior, thus the estimates of the mean and all other summary statistics are invalid.\nEstimation by sampling produces an approximate value for the model parameters; the MCSE statistic indicates the amount of noise in the estimate. Therefore MCSE column is placed next to the sample mean column, in order to make it easy to compare this sample with others.\nFor more information, see the Posterior Analysis chapter of the Stan Reference Manual which describes both the theory and practice of MCMC estimation techniques. The summary statistics and the algorithms used to compute them are described in sections Notation for samples and Effective Sample Size.\n\n\nThe CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary\n\n\n\nThe stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].\n\n\n\n\nThe stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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-    "text": "Before the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.",
+    "text": "The CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary",
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-    "text": "The Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
+    "text": "The stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].",
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-    "text": "NOTE: Although the RDump format is still supported, I/O with JSON is faster and recommended. See the chapter on JSON for more details.\nRDump format can be used to represent values for Stan variables. This format was introduced in SPLUS and is used in R, JAGS, and in BUGS (but with a different ordering).\nA dump file is structured as a sequence of variable definitions. Each variable is defined in terms of its dimensionality and its values. There are three kinds of variable declarations: - scalars - sequences - general arrays\n\n\nDump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.\n\n\n\nA simple scalar value can be thought of as having an empty list of dimensions. Its declaration in the dump format follows the SPLUS assignment syntax. For example, the following would constitute a valid dump file defining a single scalar variable y with value \\(17.2\\):\ny &lt;- 17.2\n\n\n\nOne-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)\n\n\n\nFor more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))\n\n\n\nThe dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;\n\n\n\n\nAt this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.\n\n\n\nThere is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.\n\n\n\n\nIn order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}\n\n\n\nThe line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. For example, the following declaration of a \\(2 \\times 2 \\times 3\\) array is valid.\ny &lt;-\nstructure(c(1,2,3,\n4,5,6,7,8,9,10,11,\n12), .Dim = c(2,2,\n3))\nBecause there are no decimal points in the values, the resulting dump file may be used for three-dimensional array variables declared as int or real.\n\n\n\nA more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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-    "text": "One-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)",
+    "text": "The stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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-    "text": "For more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))",
+    "text": "The output from all CmdStan methods is in CSV format. A Stan CSV file is a data table where the columns are the method and model parameters and quantities of interest. Each row contains one record’s worth of data in plain-text format using the comma character (‘,’) as the field delimiter (hence the name).\nFor the Stan CSV files, data is strictly numerical, however, possible values include both positive and negative infinity and “Not-a-Number” which are represented as the strings NaN, inf, +inf, -inf. All other values are written in decimal notation with at most 6 digits of precision.\nStan CSV files have a header row containing the column names. They also make extensive use of CSV comments, i.e., lines which begin with the # character. In addition to initial and final comment rows, some methods also put comment rows in the middle of the data table, which makes it difficult to use many of the commonly used CSV parser packages.\n\n\nThe data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3\n\n\n\nThe sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation\n\n\n\n\n\n\nConfig as comments\nHeader row\nPenalized maximum likelihood estimate\n\n\n\n\n\nConfig as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior\n\n\n\n\n\nHeader row\nQuantities of interest\n\n\n\n\n\nHeader row\nGradients",
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-    "text": "The dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;",
+    "text": "The data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3",
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-    "text": "At this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.",
+    "text": "The sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation",
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-    "text": "There is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.",
+    "text": "Config as comments\nHeader row\nPenalized maximum likelihood estimate",
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-    "text": "In order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}",
+    "text": "Config as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior",
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-    "text": "The line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. For example, the following declaration of a \\(2 \\times 2 \\times 3\\) array is valid.\ny &lt;-\nstructure(c(1,2,3,\n4,5,6,7,8,9,10,11,\n12), .Dim = c(2,2,\n3))\nBecause there are no decimal points in the values, the resulting dump file may be used for three-dimensional array variables declared as int or real.",
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-    "text": "A more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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-    "text": "Variational Inference using Pathfinder\nThe CmdStan method pathfinder uses the Pathfinder algorithm of Zhang et al. (2022). Pathfinder is a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the negative inverse Hessian estimates produced by the L-BFGS optimizer. Pathfinder returns draws from the Gaussian approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior.\nPathfinder differs from the ADVI method in that it uses quasi-Newton optimization on the log posterior instead of stochastic gradient descent (SGD) on the Monte Carlo computation of the evidence lower bound (ELBO). Pathfinder’s approach is both faster and more stable than that of ADVI. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors.\nA single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes. By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws.\nThe following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli pathfinder data file=bernoulli.data.R\nExecuting this command prints both output to the console and csv files.\nThe first part of the console output reports on the configuration used.\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\nid = 1 (Default)\ndata\n  file = examples/bernoulli/bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 1995513073 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n  sig_figs = -1 (Default)\n  profile_file = profile.csv (Default)\nnum_threads = 1 (Default)\nThe rest of the output describes the progression of the algorithm.\nBy default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, the uses importance resampling on the set of returned draws to produce the specified number of draws.\nPath [1] :Initial log joint density = -11.543343\nPath [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00\nPath [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)\nPath [2] :Initial log joint density = -7.443345\nPath [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00\nPath [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)\nPath [3] :Initial log joint density = -18.986308\nPath [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00\nPath [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)\nPath [4] :Initial log joint density = -8.304453\nPath [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00\nPath [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)\nTotal log probability function evaluations:8404\nPathfinder outputs a StanCSV file file which contains the importance resampled draws from multi-path Pathfinder. The initial CSV comment rows contain the complete set of CmdStan configuration options. Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs. See section Pathfinder Method for details.\n\n\n\n\n\n Back to topReferences\n\nZhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. “Pathfinder: Parallel Quasi-Newton Variational Inference.” Journal of Machine Learning Research 23 (306): 1–49. http://jmlr.org/papers/v23/21-0889.html.",
+    "text": "print (deprecated): MCMC Output Analysis\nThe print utility is deprecated, but is still available until CmdStan v3.0. It has been replaced by the stansummary utility.\n\n\n\n\n Back to top",
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     "text": "Stan provides three ways of parallelizing execution of a Stan model:\n\nmulti-threading with Intel Threading Building Blocks (TBB),\nmulti-processing with Message Passing Interface (MPI) and\nmanycore processing with OpenCL.\n\n\n\nIn order to exploit multi-threading in a Stan model, the models must be rewritten to use the reduce_sum and map_rect functions. For instructions on how to rewrite Stan models to use these functions see Stan’s User guide chapter on parallelization, the reduce_sum case study or the Multithreading and Map-Reduce tutorial.\n\n\nOnce a model is rewritten to use the above-mentioned functions, the model must be compiled with the STAN_THREADS makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable threading with TBB:\nSTAN_THREADS=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nBefore running a multi-threaded model, we need to specify the maximum number of threads the program can run (total threads for all chains). This is done by setting the num_threads argument. Valid values for num_threads are positive integers and -1. If num_threads is set to -1, all available cores will be used.\nGenerally, this number should not exceed the number of available cores for best performance.\nExample:\n./model sample data file=data.json num_threads=4 ...\nWhen the model is compiled with STAN_THREADS we can sample with multiple chains with a single executable (see section running multiple chains for cases when this is available). When running multiple chains num_threads is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:\n./model sample num_chains=2 data file=data.json num_threads=8 ...\n\n\n\n\nIn order to use multi-processing with MPI in a Stan model, the models must be rewritten to use the map_rect function. By using MPI, the model can be parallelized across multiple cores or a cluster. MPI with Stan is supported on MacOS and Linux.\n\n\nCompiling and running Stan models with MPI requires that the system has an MPI implementation installed. For Unix systems the most commonly used implementations are MPICH and OpenMPI.\n\n\n\nOnce a model is rewritten to use map_rect, additional makefile flags must be written to the make/local. These are:\n\nSTAN_MPI: Enables the use of MPI with Stan if true.\nCXX: The name of the MPI C++ compiler wrapper. Typically mpicxx.\nTBB_CXX_TYPE: The C++ compiler the MPI wrapper wraps. Typically gcc on Linux and clang on macOS.\n\nAn example of make/local on Linux:\nSTAN_MPI=true\nCXX=mpicxx\nTBB_CXX_TYPE=gcc\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_MPI is run using an MPI launcher. The MPI standard suggests using mpiexec, but a vendor wrapper for the launcher like mpirun can also be used. The launcher is supplied the path to the built executable and the number of processes to start: -n X for mpiexec or -np X for mpirun where X is replaced by the integer representing the number of processes.\nExample for running a model with six processes:\nmpiexec -n 6 path/to/model sample data file=data.json ...\n\n\n\n\n\n\nOpenCL is supported on most modern CPUs and GPUs. In order to run OpenCL-enabled Stan models, an OpenCL runtime for the target device must be installed. This subsection lists installation instructions for OpenCL runtimes of the commonly-found devices.\nIn order to check if any OpenCL-enabled device and its runtime is already present use the clinfo tool. On Linux, clinfo can typically be installed with the default package manager (for example sudo apt-get install clinfo on Ubuntu). For Windows, pre-built clinfo binary can be found here.\nAlso use clinfo to verify successful installation of OpenCL runtimes.\n\n\n\nLinux:\nInstall the NVIDIA GPU driver and the NVIDIA CUDA Toolkit. On Ubuntu the commands to install both is:\nsudo apt update\nsudo apt install nvidia-driver-460 nvidia-cuda-toolkit\nReplace the driver version (460 in the above case) with the lastest number at the time of installation.\nWindows:\nInstall the NVIDIA GPU Driver and CUDA Toolkit.\n\n\n\n\n\nLinux:\nInstall Radeon Software for Linux available here.\nWindows:\nWe recommend installing the open source OCL-SDK.\n\n\n\n\nInstall the open source PoCL.\n\n\n\nFollow Intel’s install instructions given here (requires registration).\n\n\n\n\nIn order to enable the OpenCL backend the model must be compiled with the STAN_OPENCL makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable parallelization with OpenCL:\nSTAN_OPENCL=true\nIf you are using OpenCL with an integrated GPU you also need to add the INTEGRATED_OPENCL flag, as the sharing of memory between CPU and GPU is slightly different with integrated graphics:\nINTEGRATED_OPENCL=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_OPENCL can also be supplied the OpenCL platform and device IDs of the target device. These IDs determine the device on which to run the OpenCL-supported functions on. You can list the devices on your system using the clinfo program. If the system has one GPU and no OpenCL CPU runtime, the platform and device IDs of the GPU are typically 0. In that case you can also omit the OpenCL IDs as the default 0 IDs are used in that case.\nWe supply these IDs when starting the executable as shown below:\npath/to/model sample data file=data.json opencl platform=0 device=1",
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     "section": "",
     "text": "In order to exploit multi-threading in a Stan model, the models must be rewritten to use the reduce_sum and map_rect functions. For instructions on how to rewrite Stan models to use these functions see Stan’s User guide chapter on parallelization, the reduce_sum case study or the Multithreading and Map-Reduce tutorial.\n\n\nOnce a model is rewritten to use the above-mentioned functions, the model must be compiled with the STAN_THREADS makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable threading with TBB:\nSTAN_THREADS=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nBefore running a multi-threaded model, we need to specify the maximum number of threads the program can run (total threads for all chains). This is done by setting the num_threads argument. Valid values for num_threads are positive integers and -1. If num_threads is set to -1, all available cores will be used.\nGenerally, this number should not exceed the number of available cores for best performance.\nExample:\n./model sample data file=data.json num_threads=4 ...\nWhen the model is compiled with STAN_THREADS we can sample with multiple chains with a single executable (see section running multiple chains for cases when this is available). When running multiple chains num_threads is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:\n./model sample num_chains=2 data file=data.json num_threads=8 ...",
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     "section": "",
     "text": "In order to use multi-processing with MPI in a Stan model, the models must be rewritten to use the map_rect function. By using MPI, the model can be parallelized across multiple cores or a cluster. MPI with Stan is supported on MacOS and Linux.\n\n\nCompiling and running Stan models with MPI requires that the system has an MPI implementation installed. For Unix systems the most commonly used implementations are MPICH and OpenMPI.\n\n\n\nOnce a model is rewritten to use map_rect, additional makefile flags must be written to the make/local. These are:\n\nSTAN_MPI: Enables the use of MPI with Stan if true.\nCXX: The name of the MPI C++ compiler wrapper. Typically mpicxx.\nTBB_CXX_TYPE: The C++ compiler the MPI wrapper wraps. Typically gcc on Linux and clang on macOS.\n\nAn example of make/local on Linux:\nSTAN_MPI=true\nCXX=mpicxx\nTBB_CXX_TYPE=gcc\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_MPI is run using an MPI launcher. The MPI standard suggests using mpiexec, but a vendor wrapper for the launcher like mpirun can also be used. The launcher is supplied the path to the built executable and the number of processes to start: -n X for mpiexec or -np X for mpirun where X is replaced by the integer representing the number of processes.\nExample for running a model with six processes:\nmpiexec -n 6 path/to/model sample data file=data.json ...",
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     "section": "",
     "text": "OpenCL is supported on most modern CPUs and GPUs. In order to run OpenCL-enabled Stan models, an OpenCL runtime for the target device must be installed. This subsection lists installation instructions for OpenCL runtimes of the commonly-found devices.\nIn order to check if any OpenCL-enabled device and its runtime is already present use the clinfo tool. On Linux, clinfo can typically be installed with the default package manager (for example sudo apt-get install clinfo on Ubuntu). For Windows, pre-built clinfo binary can be found here.\nAlso use clinfo to verify successful installation of OpenCL runtimes.\n\n\n\nLinux:\nInstall the NVIDIA GPU driver and the NVIDIA CUDA Toolkit. On Ubuntu the commands to install both is:\nsudo apt update\nsudo apt install nvidia-driver-460 nvidia-cuda-toolkit\nReplace the driver version (460 in the above case) with the lastest number at the time of installation.\nWindows:\nInstall the NVIDIA GPU Driver and CUDA Toolkit.\n\n\n\n\n\nLinux:\nInstall Radeon Software for Linux available here.\nWindows:\nWe recommend installing the open source OCL-SDK.\n\n\n\n\nInstall the open source PoCL.\n\n\n\nFollow Intel’s install instructions given here (requires registration).\n\n\n\n\nIn order to enable the OpenCL backend the model must be compiled with the STAN_OPENCL makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable parallelization with OpenCL:\nSTAN_OPENCL=true\nIf you are using OpenCL with an integrated GPU you also need to add the INTEGRATED_OPENCL flag, as the sharing of memory between CPU and GPU is slightly different with integrated graphics:\nINTEGRATED_OPENCL=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_OPENCL can also be supplied the OpenCL platform and device IDs of the target device. These IDs determine the device on which to run the OpenCL-supported functions on. You can list the devices on your system using the clinfo program. If the system has one GPU and no OpenCL CPU runtime, the platform and device IDs of the GPU are typically 0. In that case you can also omit the OpenCL IDs as the default 0 IDs are used in that case.\nWe supply these IDs when starting the executable as shown below:\npath/to/model sample data file=data.json opencl platform=0 device=1",
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-    "text": "Optimization\nThe CmdStan executable can run Stan’s optimization algorithms which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability.\nThe executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.\n&gt; ./bernoulli optimize data file=bernoulli.data.json\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:\n ./bernoulli optimize data file=bernoulli.data.json\nmethod = optimize\n  optimize\n    algorithm = lbfgs (Default)\n      lbfgs\n        init_alpha = 0.001 (Default)\n        tol_obj = 1e-12 (Default)\n        tol_rel_obj = 10000 (Default)\n        tol_grad = 1e-08 (Default)\n        tol_rel_grad = 10000000 (Default)\n        tol_param = 1e-08 (Default)\n        history_size = 5 (Default)\n    iter = 2000 (Default)\n    save_iterations = 0 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 87122538 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nThe second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The alpha value is for step size used. In the final state the change in parameters was roughly \\(0.002\\) and the length of the gradient roughly 3e-05 (\\(0.00003\\)).\nInitial log joint probability = -6.85653\n    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes \n       5      -5.00402    0.00184936   3.35074e-05           1           1        8   \nOptimization terminated normally: \n  Convergence detected: relative gradient magnitude is below tolerance\nThe output from optimization is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n...\nFollowing the config information, are two lines of output: the CSV headers and the recorded values:\nlp__,theta\n-5.00402,0.200003\nNote that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with lp__ and the model parameter theta. The maximum log probability is -5.0 and the posterior mode for theta is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file. Running the optimizer with save_iterations=1 writes both the initial log joint probability and values for all iterations to the output CSV file.\nRunning the example model with option save_iterations=1, i.e., the command\n&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json\nproduces CSV file output rows:\nlp__,theta\n-6.85653,0.493689\n-6.10128,0.420936\n-5.02953,0.22956\n-5.00517,0.206107\n-5.00403,0.200299\n-5.00402,0.200003\n\n\n\n\n Back to top",
+    "text": "The sample method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on MCMC Sampling.\n\n\nTo generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory.\nThe full set of configuration options available for the sample method is available by using the sample help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.\nHere, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\n&gt; ls\n  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n...\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)\n\n\n\nEach execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\nWhen the example model bernoulli.stan is run via the command line with all default arguments, the following configuration is displayed:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)\n\n\n\nAt every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.\n\n\n\nThe adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.\n\n\n\n\n\nThe algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).\n\n\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.\n\n\n\nA Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThe preferred way of using multiple chains is to run them all from the same executable using the num_chains argument. There is also the option to use the Unix or DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with STAN_THREADS=true, these will either run in parallel or one after the other.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\nFor example, this will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nThis will produce samples in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially.\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\n\nThe stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)\n\n\n\nNote: Many of these examples can be simplified by using the num_chains argument.\nWhen the num_chains argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following extended examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
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-    "text": "The sample method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on MCMC Sampling.\nThe full set of configuration options available for the sample method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default sample method configuration options:\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n\n\nAt every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.\n\n\n\nThe adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.\n\n\n\n\n\nThe algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).\n\n\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.\n\n\n\nAs described in the quickstart section on parallelism, the preferred way to run multiple chains is to use the num_chains argument.\nThis will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with STAN_THREADS=true.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\n\n\n\nNote: Many of these examples can be simplified by using the num_chains argument.\nThe Quickstart Guide MCMC Sampling chapter section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.\nThis creates multiple copies of the model process which will all load the data.\nTo run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=my_model.data.json \\\n                  output file=output_${i}.csv &\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. The ampersand (&) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows the corresponding loop is:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=my_model.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
+    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory.\nThe full set of configuration options available for the sample method is available by using the sample help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.\nHere, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\n&gt; ls\n  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n...\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)",
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+    "text": "Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\nWhen the example model bernoulli.stan is run via the command line with all default arguments, the following configuration is displayed:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)",
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     "text": "At every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.",
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     "text": "The adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.",
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     "text": "The algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).",
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     "text": "The output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.",
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-    "text": "As described in the quickstart section on parallelism, the preferred way to run multiple chains is to use the num_chains argument.\nThis will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with STAN_THREADS=true.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.",
+    "text": "A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThe preferred way of using multiple chains is to run them all from the same executable using the num_chains argument. There is also the option to use the Unix or DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with STAN_THREADS=true, these will either run in parallel or one after the other.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\nFor example, this will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nThis will produce samples in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially.\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16",
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-    "text": "Note: Many of these examples can be simplified by using the num_chains argument.\nThe Quickstart Guide MCMC Sampling chapter section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.\nThis creates multiple copies of the model process which will all load the data.\nTo run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=my_model.data.json \\\n                  output file=output_${i}.csv &\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. The ampersand (&) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows the corresponding loop is:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=my_model.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
+    "text": "The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)",
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+    "text": "Note: Many of these examples can be simplified by using the num_chains argument.\nWhen the num_chains argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following extended examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
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@@ -4014,7 +3971,7 @@
     "section": "",
     "text": "The laplace method produces a sample from a normal approximation centered at the mode of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See this case study for a visual illustration.)\nThis is computationally inexpensive compared to exact Bayesian inference with MCMC. The goodness of this estimate depends on both the estimate of the mode and how much the true posterior in the unconstrained space resembles a Gaussian.\n\n\nThis method takes 2 arguments:\n\njacobian - Whether or not the Jacobian adjustment should be included in the gradient. The default value is 1 (include adjustment). (Note: in optimization, the default value is 0, for historical reasons.)\nmode - Input file of parameters values on the constrained scale. When Stan’s optimize method is used to estimate the modal values, the value of boolean argument jacobian should be 0 if optimize was run with default settings, i.e., the input is the MLE estimate; if optimize was run with argument jacobian=1, then the laplace method default setting, jacobian=1, should be used.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the laplace method is reported at the beginning of the output file as CSV comments.\nOutput columns log_p__ and log_q__, the unnormalized log density and the unnormalized density of the Laplace approximation, respectively. These can be used for diagnostics and importance sampling.\nOutput columns for all model parameters on the constrained scale.\n\n\n\n\nTo get an approximate estimate of the mode and standard deviation of the example Bernoulli model given the example dataset:\n\nfind the MAP estimate by running optimization with argument jacobian=1\nrun the Laplace estimator using the MAP estimate as the mode argument.\n\nBecause the default output file name from all methods is output.csv, a more informative name is used for the output of optimization. We run the commands from the CmdStan home directory. This results in a sample with mean 2.7 and standard deviation 0.12. In comparison, running the NUTS-HMC sampler results in mean 2.6 and standard deviation 0.12.\n./examples/bernoulli/bernoulli optimize jacobian=1 \\\n  data file=examples/bernoulli/bernoulli.data.json \\\n  output file=bernoulli_optimize_lbfgs.csv random seed=1234\n\n\n./examples/bernoulli/bernoulli laplace mode=bernoulli_optimize_lbfgs.csv \\\n data file=examples/bernoulli/bernoulli.data.json random seed=1234\nThe header and first few data rows of the output sample are shown below.\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-12-20 01:01:14 UTC\n# method = laplace\n#   laplace\n#     mode = bernoulli_lbfgs.csv\n#     jacobian = 1 (Default)\n#     draws = 1000 (Default)\n# id = 1 (Default)\n# data\n#   file = examples/bernoulli/bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 875960551 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 v2.31.0-7-g20444266\n# stancflags =\nlog_p__,log_q__,theta\n-9.4562,-2.33997,0.0498545\n-6.9144,-0.0117349,0.182898\n-7.18171,-0.746034,0.376428\n...",
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     "text": "This method takes 2 arguments:\n\njacobian - Whether or not the Jacobian adjustment should be included in the gradient. The default value is 1 (include adjustment). (Note: in optimization, the default value is 0, for historical reasons.)\nmode - Input file of parameters values on the constrained scale. When Stan’s optimize method is used to estimate the modal values, the value of boolean argument jacobian should be 0 if optimize was run with default settings, i.e., the input is the MLE estimate; if optimize was run with argument jacobian=1, then the laplace method default setting, jacobian=1, should be used.",
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     "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the laplace method is reported at the beginning of the output file as CSV comments.\nOutput columns log_p__ and log_q__, the unnormalized log density and the unnormalized density of the Laplace approximation, respectively. These can be used for diagnostics and importance sampling.\nOutput columns for all model parameters on the constrained scale.",
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     "text": "To get an approximate estimate of the mode and standard deviation of the example Bernoulli model given the example dataset:\n\nfind the MAP estimate by running optimization with argument jacobian=1\nrun the Laplace estimator using the MAP estimate as the mode argument.\n\nBecause the default output file name from all methods is output.csv, a more informative name is used for the output of optimization. We run the commands from the CmdStan home directory. This results in a sample with mean 2.7 and standard deviation 0.12. In comparison, running the NUTS-HMC sampler results in mean 2.6 and standard deviation 0.12.\n./examples/bernoulli/bernoulli optimize jacobian=1 \\\n  data file=examples/bernoulli/bernoulli.data.json \\\n  output file=bernoulli_optimize_lbfgs.csv random seed=1234\n\n\n./examples/bernoulli/bernoulli laplace mode=bernoulli_optimize_lbfgs.csv \\\n data file=examples/bernoulli/bernoulli.data.json random seed=1234\nThe header and first few data rows of the output sample are shown below.\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-12-20 01:01:14 UTC\n# method = laplace\n#   laplace\n#     mode = bernoulli_lbfgs.csv\n#     jacobian = 1 (Default)\n#     draws = 1000 (Default)\n# id = 1 (Default)\n# data\n#   file = examples/bernoulli/bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 875960551 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 v2.31.0-7-g20444266\n# stancflags =\nlog_p__,log_q__,theta\n-9.4562,-2.33997,0.0498545\n-6.9144,-0.0117349,0.182898\n-7.18171,-0.746034,0.376428\n...",
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-    "text": "There are a few ways that you can install CmdStan. Depending on your operating system and your level of expertise, you can either:\n\nUse the conda package management system to install a pre-built version of CmdStan along with the required dependencies. Recommended for Windows users.\nInstall the source code from GitHub CmdStan repository. This requires a modern C++ compiler and toolchain. See the C++ Toolchain section for further details.\n\n\n\nWith conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.\n\n\n\n\nInstallation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)\n\n\n\n\nTo check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\ \n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.\n\n\n\nUpdates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.\n\n\n\n\nCompiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.\n\n\n\n\n\n\nCmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g., \n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
+    "text": "There are a few ways that you can install CmdStan. Depending on your operating system and your level of expertise, you can either:\n\nUse the conda package management system to install a pre-built version of CmdStan along with the required dependencies. Recommended for Windows users.\nInstall the source code from GitHub CmdStan repository. This requires a modern C++ compiler and toolchain. See the C++ Toolchain section for further details.\n\n\n\nWith conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.\n\n\n\n\nInstallation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)\n\n\n\n\nTo check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\\n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.\n\n\n\nUpdates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.\n\n\n\n\nCompiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.\n\n\n\n\n\n\nCmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g.,\n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
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-    "text": "With conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.",
+    "text": "With conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.",
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     "text": "Installation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)",
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-    "text": "To check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\ \n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.",
+    "text": "To check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\\n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.",
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     "section": "",
     "text": "Updates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.",
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     "section": "",
     "text": "Compiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.",
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-    "text": "CmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g., \n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
+    "text": "CmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g.,\n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
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     "text": "Footnotes\n\n\nTo open a Windows command shell, first open the Start Menu, (usually in the lower left of the screen), select option All Programs, then option Accessories, then program Command Prompt. Alternatively, enter [Windows+r] (both keys together on the keyboard), and enter cmd into the text field that pops up in the Run window, then press [Return] on the keyboard to run.↩︎",
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-    "text": "Generating Quantities of Interest from a Fitted Model\nThe generated quantities block computes quantities of interest (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:\n\ngenerate simulated data for model testing by forward sampling\ngenerate predictions for new data\ncalculate posterior event probabilities, including multiple comparisons, sign tests, etc.\ncalculating posterior expectations\ntransform parameters for reporting\napply full Bayesian decision theory\ncalculate log likelihoods, deviances, etc. for model comparison\n\nThe generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the generate_quantities method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block. See the Stan User’s Guide section Stand-alone generated quantities and ongoing prediction.\nTo illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; theta_rep;\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter estimates from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv consists of just the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2135140492 (Default)\n# output\n#   file = bernoulli_ppc.csv\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\ntheta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10\n0.2,0,0,1,0,0,0,0,0,1,0\n0.3,1,0,0,1,0,1,0,0,0,0\n0.8,1,0,1,1,1,1,1,1,1,0\n0.1,0,0,0,0,0,1,0,0,0,0\n0.3,0,0,0,0,0,0,1,1,1,0\nNote: the only relevant analysis of the resulting CSV output is computing per-column statistics; this can easily be done in Python, R, Excel or similar, or you can use the CmdStanPy and CmdStanR interfaces which provide a better user experience for this workflow.\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  real&lt;lower=0, upper=1&gt; theta_rep;\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ntheta_cp,theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10\n0.102391,0,0,0,0,0,0,0,0,0,0,0\n0.519567,0.2,0,1,0,0,1,0,0,0,0,0\n0.544634,0.6,1,0,0,0,0,1,1,1,1,1\n0.167651,0,0,0,0,0,0,0,0,0,0,0\n0.167651,0.1,1,0,0,0,0,0,0,0,0,0\n\n\n\n\n Back to top",
+    "text": "The generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the generate_quantities method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block.\nThis method requires sub-argument fitted_params which takes as its value an existing Stan CSV file that contains a parameter values from an equivalent model, i.e., a model with the same parameters block, conditioned on the same data.\nThe generated quantities block computes quantities of interest (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:\n\ngenerate simulated data for model testing by forward sampling\ngenerate predictions for new data\ncalculate posterior event probabilities, including multiple comparisons, sign tests, etc.\ncalculate posterior expectations\ntransform parameters for reporting\napply full Bayesian decision theory\ncalculate log likelihoods, deviances, etc. for model comparison\n\nFor an overview of the uses of this feature, see the Stan User’s Guide section on Stand-alone generated quantities and ongoing prediction.\n\n\nTo illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter values from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv contains only the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2983956445 (Default)\n# output\n#   file = output.csv (Default)\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep\n1,1,1,0,0,0,1,1,0,1,0.6\n1,1,0,1,0,0,1,0,1,0,0.5\n1,0,1,1,1,1,1,1,0,1,0.8\n0,1,0,1,0,1,0,1,0,0,0.4\n1,0,0,0,0,0,0,0,0,0,0.1\n0,0,0,0,0,1,1,1,0,0,0.3\n0,0,1,0,1,0,0,0,0,0,0.2\n1,0,1,0,1,1,0,1,1,0,0.6\n...\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep\n0,1,1,0,1,0,0,1,1,0,0.545679,0.5\n1,1,1,1,1,1,0,1,1,0,0.527164,0.8\n1,1,1,1,0,1,1,1,1,0,0.529116,0.8\n1,0,1,1,1,1,0,0,1,0,0.478844,0.6\n0,1,0,0,0,0,1,0,1,0,0.238793,0.3\n0,0,0,0,0,1,1,0,0,0,0.258294,0.2\n1,1,1,0,0,0,0,0,0,0,0.258465,0.3\n\n\n\nThe fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \\\n(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)",
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-    "text": "The --allow-undefined flag can be passed to the call to stanc, which will allow undefined functions in the Stan language to be parsed without an error. We can then include a definition of the function in a C++ header file.\nThis requires specifying two makefile variables:\n\nSTANCFLAGS=--allow-undefined\nUSER_HEADER=&lt;header_file.hpp&gt;, where &lt;header_file.hpp&gt; is the name of a header file that defines a function with the same name and a compatible signature. This function can appear in the global namespace or in the model namespace, which is defined as the name of the model (either the file name, or the --name argument to stanc) followed by _namespace.\n\nThis is an advanced feature which is only recommended to users familiar with the internals of Stan’s Math library. Most existing C++ code will need to be modified to work with Stan, to varying degrees.\nAs an example, consider the following variant of the Bernoulli example\nfunctions {\n  real make_odds(data real theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n}\nmodel {\n  theta ~ beta(1, 1); // uniform prior on interval 0, 1\n  y ~ bernoulli(theta);\n}\ngenerated quantities {\n  real odds;\n  odds = make_odds(theta);\n}\nHere the make_odds function is declared but not defined, which would ordinarily result in a parser error. However, if you put STANCFLAGS = --allow-undefined into the make/local file or into the stanc call, then the stanc compiler will translate this program to C++, but the generated C++ code will not compile unless you write a file such as examples/bernoulli/make_odds.hpp with the following lines\n#include &lt;ostream&gt;\n\ndouble make_odds(const double& theta, std::ostream *pstream__) {\n  return theta / (1 - theta);\n}\nThe signature for this function needs to fulfill all the usages in the C++ class emitted by stanc. The pstream__ argument is mandatory in the signature but need not be used if your function does not print any output. Because make_odds was declared with a data argument and only used in generated quantites, a signature which accepts and returns double is acceptable. Functions which will have parameters passed as input in the transformed parameters or model blocks will require the ability to accept Stan’s autodiff types. If you wish to autodiff through this function, the simplest option is to make it a template, like\ntemplate &lt;typename T&gt;\nT make_odds(const T &theta, std::ostream *pstream__)\n{\n    return theta / (1 - theta);\n}\nGiven the above, the following make invocation should work\n&gt; make STANCFLAGS=--allow-undefined USER_HEADER=examples/bernoulli/make_odds.hpp examples/bernoulli/bernoulli # on Windows add .exe\nAlternatively, you could put STANCFLAGS and USER_HEADER into the make/local file instead of specifying them on the command-line.\nIf the function were more complicated and involved functions in the Stan Math Library, then you would need to add #include &lt;stan/model/model_header.hpp&gt; and prefix the function calls with stan::math::.\n\n\nExternal C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.\n\n\n\nSome functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
+    "text": "To illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter values from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv contains only the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2983956445 (Default)\n# output\n#   file = output.csv (Default)\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep\n1,1,1,0,0,0,1,1,0,1,0.6\n1,1,0,1,0,0,1,0,1,0,0.5\n1,0,1,1,1,1,1,1,0,1,0.8\n0,1,0,1,0,1,0,1,0,0,0.4\n1,0,0,0,0,0,0,0,0,0,0.1\n0,0,0,0,0,1,1,1,0,0,0.3\n0,0,1,0,1,0,0,0,0,0,0.2\n1,0,1,0,1,1,0,1,1,0,0.6\n...\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep\n0,1,1,0,1,0,0,1,1,0,0.545679,0.5\n1,1,1,1,1,1,0,1,1,0,0.527164,0.8\n1,1,1,1,0,1,1,1,1,0,0.529116,0.8\n1,0,1,1,1,1,0,0,1,0,0.478844,0.6\n0,1,0,0,0,0,1,0,1,0,0.238793,0.3\n0,0,0,0,0,1,1,0,0,0,0.258294,0.2\n1,1,1,0,0,0,0,0,0,0,0.258465,0.3",
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-    "text": "External C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.",
+    "text": "The fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \\\n(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)",
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-    "text": "Some functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
+    "text": "The following is a simple, complete Stan program for a Bernoulli model of binary data.1 The model assumes the binary observed data y[1],...,y[N] are i.i.d. with Bernoulli chance-of-success theta.\ndata { \n  int&lt;lower=0&gt; N; \n  array[N] int&lt;lower=0, upper=1&gt; y;\n} \nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n} \nmodel {\n  theta ~ beta(1, 1);  // uniform prior on interval 0,1\n  y ~ bernoulli(theta);\n}\nThe input data file contains definitions for the two variables N and y which are specified in the data block of program bernoulli.stan (above).\nA data set of N=10 observations is included in the example Bernoulli model directory in both JSON notation and Rdump data format where 8 out of 10 trials had outcome 0 (failure) and 2 trials had outcome 1 (success). In JSON, this data is:\n{\n    \"N\" : 10,\n    \"y\" : [0,1,0,0,0,0,0,0,0,1]\n}",
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-    "text": "Footnotes\n\n\nDetails of programming in the Stan Math style are omitted from this section, it is presented only as an example↩︎",
+    "text": "Footnotes\n\n\nThe model is available with the CmdStan distribution at the path &lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan.↩︎",
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-    "text": "Error messages and return codes\nCmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. The signal number is retcode - 128.\n\nCommon sources of program error are:\n\nMissing or ill-formed input data. All variables declared in the data block must be supplied in the input data file.\nModel parameters cannot be initialized due to model misspecification.\nIndexing errors. The Stan language provides indexing and slicing operations. Indexing errors are easy to make and difficult to debug. The compiler and runtime detect index out of bounds errors; they cannot detect the semantic errors such as indexing into the wrong variables or incorrect index updates.\n\n\n\n\n\n Back to top"
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     "text": "Diagnosing HMC by Comparison of Gradients\nCmdStan has a basic diagnostic feature that will calculate the gradients of the initial state and compare them with gradients calculated by finite differences. Discrepancies between the two indicate that there is a problem with the model or initial states or else there is a bug in Stan.\nTo allow for the possibility of adding other kinds of diagnostic tests, the diagnose method argument configuration has subargument test which currently only takes value gradient. There are two available gradient test configuration arguments:\n\nepsilon - The finite difference step size. Must be a positive real number. Default value is \\(1^{-6}\\)\nerror - The error threshold. Must be a positive real number. Default value is \\(1^{-6}\\)\n\nTo run on the different platforms with the default configuration, use one of the following.\nMac OS and Linux\n&gt; ./my_model diagnose data file=my_data\nWindows\n&gt; my_model diagnose data file=my_data\nTo relax the test threshold, specify the error argument as follows:\n&gt; ./my_model diagnose test=gradient error=0.0001 data file=my_data\nTo see how this works, we run diagnostics on the example bernoulli model:\n&gt; ./bernoulli diagnose data file=bernoulli.data.R\nExecuting this command prints output to the console and as a series of comment lines to the output csv file. The console output is:\nmethod = diagnose\n  diagnose\n    test = gradient (Default)\n      gradient\n        epsilon = 9.9999999999999995e-07 (Default)\n        error = 9.9999999999999995e-07 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 2152196153 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n\nTEST GRADIENT MODE\n\n Log probability=-8.42814\n\n param idx           value           model     finite diff           error\n         0       0.0361376         -3.1084         -3.1084    -2.37554e-10\nThe same information is printed to the output file as csv comments, i.e., each line is prefixed with a pound sign #.\n\n\n\n\n Back to top",
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     "text": "A CmdStan executable is built from the Stan model concept and the CmdStan command line parser. The command line argument syntax consists of sets of keywords and keyword-value pairs. Arguments are grouped by the following keywords:\n\nmethod - specifies the kind of inference done on the model. Each kind of inference requires further configuration via sub-arguments. The method argument is required. It can be specified overtly as the a keyword-value pair method=&lt;inference&gt; or implicitly as one of the following:\n\nsample - obtain a sample from the posterior using HMC\noptimize - penalized maximum likelihood estimation\nvariational - automatic variational inference\ngenerate_quantities - run model’s generated quantities block on existing sample to obtain new quantities of interest.\nlog_prob - compute the log probability and gradient of the model for one set of parameters.\ndiagnose - compute and compare sampler gradient calculations to finite differences.\n\ndata - specifies the input data file, if any.\noutput - specifies program outputs, both disk files and terminal window outputs.\ninit - specifies initial values for the model parameters, if any.\nrandom - specifies the seed for the pseudo-random number.\n\nThe remainder of this chapter covers the general configuration options used for all processing. The following chapters cover the per-inference configuration options.\n\n\nThe values for all variables declared in the data block of the model are read in from an input data file in either JSON or Rdump format. The syntax for the input data argument is:\ndata file=&lt;filepath&gt;\nThe keyword data must be followed directly by the keyword-value pair file=&lt;filepath&gt;. If the model doesn’t declare any data variables, this argument is ignored.\nThe input data file must contain definitions for all data variables declared in the data block. If one or more data block variables are missing from the input data file, the program prints an error message to stderr and returns a non-zero return code. For example, the model bernoulli.stan defines two data variables N and y. If the input data file doesn’t include both variables, or if the data variable doesn’t match the declared type and dimensions, the program will exit with an error message at the point where it first encounters missing data.\nFor example if the input data file doesn’t include the definition for variable y, the executable exits with the following message:\nException: variable does not exist; processing stage=data initialization; variable name=y; base type=int (in 'examples/bernoulli/bernoulli.stan', line 3, column 2 to column 28)\n\n\n\nThe output keyword is used to specify non-default options for output files and messages written to the terminal window. The output keyword takes several keyword-value pair sub-arguments.\nThe keyword value pair file=&lt;filepath&gt; specifies the location of the Stan CSV output file. If unspecified, the output file is written to a file named output.csv in the current working directory.\nThe keyword value pair diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file. By default, no auxiliary output file is produced. This option is only valid for the iterative algorithms sample and variational.\nThe keyword value pair refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is 100 iterations.\nThe keyword value pair sig_figs=&lt;int&gt; specifies the number of significant digits for all numerical values in the output files. Allowable values are between 1 and 18, which is the maximum amount of precision available for 64-bit floating point arithmetic. The default value is 6.   Note: increasing sig_figs above the default will increase the size of the output CSV files accordingly.\nThe keyword value pair profile_file=&lt;filepath&gt; specifies the location of the output file for profiling data. If the model uses no profiling, the output profile file is not produced. If the model uses profiling and profile_file is unspecified, the profiling data is written to a file named profile.csv in the current working directory.\nThe keyword value pair save_cmdstan_config=&lt;int&gt; specifies whether to save the configuration options used to run the program to a file named &lt;output file&gt;_config.json alongside the other output files. The default value is 0, which means the configuration file is not saved. The contents of this file are similar to the comments in the Stan CSV file, but should be more portable across versions and easier to parse.\n\n\n\nInitialization is only applied to parameters defined in the parameters block. By default, all parameters are initialized to random draws from a uniform distribution over the range \\([-2, 2]\\). These values are on the unconstrained scale, so must be inverse transformed back to satisfy the constraints declared for parameters. Because zero is chosen to be a reasonable default initial value for most parameters, the interval around zero provides a fairly diffuse starting point. For instance, unconstrained variables are initialized randomly in \\((-2, 2)\\), variables constrained to be positive are initialized roughly in \\((0.14, 7.4)\\), variables constrained to fall between 0 and 1 are initialized with values roughly in \\((0.12, 0.88)\\).\nThe initialization argument is specified as keyword-value pair with keyword init. The value can be one of the following:\n\npositive real number \\(x\\). All parameters will be initialized to random draws from a uniform distribution over the range \\([-x, x]\\).\n\\(0\\) - All parameters will be initialized to zero values on the unconstrained scale. The transforms are arranged in such a way that zero initialization provides reasonable variable initializations: \\(0\\) for unconstrained parameters; \\(1\\) for parameters constrained to be positive; \\(0.5\\) for variables to constrained to lie between \\(0\\) and \\(1\\); a symmetric (uniform) vector for simplexes; unit matrices for both correlation and covariance matrices; and so on.\nfilepath - A data file in JSON or Rdump format containing initial parameters values for some or all of the model parameters. User specified initial values must satisfy the constraints declared in the model (i.e., they are on the constrained scale). Parameters which aren’t explicitly initialized will be initialized randomly over the range \\([-2, 2]\\).\n\n\n\n\nThe random-number generator’s behavior is determined by the unsigned seed (positive integer) it is started with. If a seed is not specified, or a seed of 0 or less is specified, the system time is used to generate a seed. The seed is recorded and included with Stan’s output regardless of whether it was specified or generated randomly from the system time.\nThe syntax for the random seed argument is:\nrandom seed=&lt;int&gt;\nThe keyword random must be followed directly by the keyword-value pair seed=&lt;int&gt;.\n\n\n\nThe chain identifier argument is used in conjunction with the random seed argument when running multiple Markov chains for sampling. The chain identifier is used to advance the random number generator a very large number of random variates so that two chains with the same seed and different identifiers draw from non-overlapping subsequences of the random-number sequence determined by the seed. Together, the seed and chain identifier determine the behavior of the random number generator.\nThe syntax for the random seed argument is:\nid=&lt;int&gt;\nThe default value is 1.\nWhen running a set of chains from the command line with a specified seed, this argument should be set to the chain index. E.g., when running 4 chains, the value should be 1,..,4, successively. When running multiple chains from a single command, Stan’s interfaces manage the chain identifier arguments automatically.\nFor complete reproducibility, every aspect of the environment needs to be locked down from the OS and version to the C++ compiler and version to the version of Stan and all dependent libraries. See the Stan Reference Manual Reproducibility chapter for further details.\n\n\n\nCmdStan provides a help and help-all mechanism that displays either the available top-level or keyword-specific key-value argument pairs. To display top-level help, call the CmdStan executable with keyword help:\n./bernoulli help\n\n\n\nCmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively. Some methods also generate warning messages when the algorithm detects potential problems with the inference. Depending on the method, these messages are sent to either standard out or standard error.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. To determine the signal number, subtract 128 from the return value, e.g. return code 139 results from termination signal 11 (segmentation violation).\n\nA non-zero return code or outputs sent to stderr indicate problems with the inference. However, a return code of zero and absence of error messages doesn’t necessarily mean that the inference is valid, it is still necessary to validate the inferences using all available summary and diagnostic techniques.",
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@@ -4230,7 +4180,7 @@
     "section": "",
     "text": "The values for all variables declared in the data block of the model are read in from an input data file in either JSON or Rdump format. The syntax for the input data argument is:\ndata file=&lt;filepath&gt;\nThe keyword data must be followed directly by the keyword-value pair file=&lt;filepath&gt;. If the model doesn’t declare any data variables, this argument is ignored.\nThe input data file must contain definitions for all data variables declared in the data block. If one or more data block variables are missing from the input data file, the program prints an error message to stderr and returns a non-zero return code. For example, the model bernoulli.stan defines two data variables N and y. If the input data file doesn’t include both variables, or if the data variable doesn’t match the declared type and dimensions, the program will exit with an error message at the point where it first encounters missing data.\nFor example if the input data file doesn’t include the definition for variable y, the executable exits with the following message:\nException: variable does not exist; processing stage=data initialization; variable name=y; base type=int (in 'examples/bernoulli/bernoulli.stan', line 3, column 2 to column 28)",
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@@ -4241,7 +4191,7 @@
     "section": "",
     "text": "The output keyword is used to specify non-default options for output files and messages written to the terminal window. The output keyword takes several keyword-value pair sub-arguments.\nThe keyword value pair file=&lt;filepath&gt; specifies the location of the Stan CSV output file. If unspecified, the output file is written to a file named output.csv in the current working directory.\nThe keyword value pair diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file. By default, no auxiliary output file is produced. This option is only valid for the iterative algorithms sample and variational.\nThe keyword value pair refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is 100 iterations.\nThe keyword value pair sig_figs=&lt;int&gt; specifies the number of significant digits for all numerical values in the output files. Allowable values are between 1 and 18, which is the maximum amount of precision available for 64-bit floating point arithmetic. The default value is 6.   Note: increasing sig_figs above the default will increase the size of the output CSV files accordingly.\nThe keyword value pair profile_file=&lt;filepath&gt; specifies the location of the output file for profiling data. If the model uses no profiling, the output profile file is not produced. If the model uses profiling and profile_file is unspecified, the profiling data is written to a file named profile.csv in the current working directory.\nThe keyword value pair save_cmdstan_config=&lt;int&gt; specifies whether to save the configuration options used to run the program to a file named &lt;output file&gt;_config.json alongside the other output files. The default value is 0, which means the configuration file is not saved. The contents of this file are similar to the comments in the Stan CSV file, but should be more portable across versions and easier to parse.",
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     "section": "",
     "text": "Initialization is only applied to parameters defined in the parameters block. By default, all parameters are initialized to random draws from a uniform distribution over the range \\([-2, 2]\\). These values are on the unconstrained scale, so must be inverse transformed back to satisfy the constraints declared for parameters. Because zero is chosen to be a reasonable default initial value for most parameters, the interval around zero provides a fairly diffuse starting point. For instance, unconstrained variables are initialized randomly in \\((-2, 2)\\), variables constrained to be positive are initialized roughly in \\((0.14, 7.4)\\), variables constrained to fall between 0 and 1 are initialized with values roughly in \\((0.12, 0.88)\\).\nThe initialization argument is specified as keyword-value pair with keyword init. The value can be one of the following:\n\npositive real number \\(x\\). All parameters will be initialized to random draws from a uniform distribution over the range \\([-x, x]\\).\n\\(0\\) - All parameters will be initialized to zero values on the unconstrained scale. The transforms are arranged in such a way that zero initialization provides reasonable variable initializations: \\(0\\) for unconstrained parameters; \\(1\\) for parameters constrained to be positive; \\(0.5\\) for variables to constrained to lie between \\(0\\) and \\(1\\); a symmetric (uniform) vector for simplexes; unit matrices for both correlation and covariance matrices; and so on.\nfilepath - A data file in JSON or Rdump format containing initial parameters values for some or all of the model parameters. User specified initial values must satisfy the constraints declared in the model (i.e., they are on the constrained scale). Parameters which aren’t explicitly initialized will be initialized randomly over the range \\([-2, 2]\\).",
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     "section": "",
     "text": "The random-number generator’s behavior is determined by the unsigned seed (positive integer) it is started with. If a seed is not specified, or a seed of 0 or less is specified, the system time is used to generate a seed. The seed is recorded and included with Stan’s output regardless of whether it was specified or generated randomly from the system time.\nThe syntax for the random seed argument is:\nrandom seed=&lt;int&gt;\nThe keyword random must be followed directly by the keyword-value pair seed=&lt;int&gt;.",
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     "section": "",
     "text": "The chain identifier argument is used in conjunction with the random seed argument when running multiple Markov chains for sampling. The chain identifier is used to advance the random number generator a very large number of random variates so that two chains with the same seed and different identifiers draw from non-overlapping subsequences of the random-number sequence determined by the seed. Together, the seed and chain identifier determine the behavior of the random number generator.\nThe syntax for the random seed argument is:\nid=&lt;int&gt;\nThe default value is 1.\nWhen running a set of chains from the command line with a specified seed, this argument should be set to the chain index. E.g., when running 4 chains, the value should be 1,..,4, successively. When running multiple chains from a single command, Stan’s interfaces manage the chain identifier arguments automatically.\nFor complete reproducibility, every aspect of the environment needs to be locked down from the OS and version to the C++ compiler and version to the version of Stan and all dependent libraries. See the Stan Reference Manual Reproducibility chapter for further details.",
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     "section": "",
     "text": "CmdStan provides a help and help-all mechanism that displays either the available top-level or keyword-specific key-value argument pairs. To display top-level help, call the CmdStan executable with keyword help:\n./bernoulli help",
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     "section": "",
     "text": "CmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively. Some methods also generate warning messages when the algorithm detects potential problems with the inference. Depending on the method, these messages are sent to either standard out or standard error.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. To determine the signal number, subtract 128 from the return value, e.g. return code 139 results from termination signal 11 (segmentation violation).\n\nA non-zero return code or outputs sent to stderr indicate problems with the inference. However, a return code of zero and absence of error messages doesn’t necessarily mean that the inference is valid, it is still necessary to validate the inferences using all available summary and diagnostic techniques.",
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@@ -4321,7 +4271,7 @@
     "section": "",
     "text": "A Stan program must be in a file with extension .stan. The CmdStan makefile rules specify all necessary steps to translate files with suffix .stan to a CmdStan executable program. This is a two-stage process:\n\nfirst the Stan program is translated to C++ by the stanc compiler\nthen the C++ compiler compiles all C++ sources and links them together with the CmdStan interface program and the Stan and math libraries.\n\n\n\nTo compile Stan programs, you must invoke the Make program from the &lt;cmdstan-home&gt; directory. The Stan program can be in a different directory, but the directory path names cannot contain spaces - this limitation is imposed by Make.\n&gt; cd &lt;cmdstan_home&gt;\nIn the call to the Make program, the target is name of the CmdStan executable corresponding to the Stan program file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. To build the Bernoulli example, on Mac and Linux:\n&gt; make examples/bernoulli/bernoulli\nOn Windows, the command is the same with the addition of .exe at the end of the target (note the use of forward slashes):\n&gt; make examples/bernoulli/bernoulli.exe\nThe generated C++ code (bernoulli.hpp), object file (bernoulli.o) and the compiled executable will be placed in the same directory as the Stan program.\nThe compiled executable consists of the Stan model and the CmdStan command line interface which provides inference algorithms to do MCMC sampling, optimization, and variational inference. The following sections provide examples of doing inference using each method on the example model and data file.\n\n\n\nWhen executing a Make target, all its dependencies are checked to see if they are up to date, and if they are not, they are rebuilt. If the you call Make with target bernoulli twice in a row, without any editing bernoulli.stan or otherwise changing the system, on the second invocation, Make will determine that the executable is already newer than the Stan source file and will not recompile the program:\n&gt; make examples/bernoulli/bernoulli\nmake: `examples/bernoulli/bernoulli' is up to date.\nIf the file containing the Stan program is updated, the next call to make will rebuild the CmdStan executable.\n\n\n\nThe Stan probabilistic programming language is a programming language with a rich syntax, as such, it is often the case that a carefully written program contains errors.\nThe simplest class of errors are simple syntax errors such as forgetting the semi-colon statement termination marker at the end of a line, or typos such as a misspelled variable name. For example, if in the bernoulli.stan program, we introduce a typo on line \\(9\\) by writing thata instead of theta, the Make command fails with the following\n--- Translating Stan model to C++ code ---\nbin/stanc  --o=bernoulli.hpp bernoulli.stan\n\nSemantic error in 'bernoulli.stan', line 9, column 2 to column 7:\n   -------------------------------------------------\n     7:  }\n     8:  model {\n     9:    thata ~ beta(1, 1);  // uniform prior on interval 0, 1\n           ^\n    10:    y ~ bernoulli(theta);\n    11:  }\n   -------------------------------------------------\n\nIdentifier 'thata' not in scope.\n\nmake: *** [bernoulli.hpp] Error 1\nStan is a strongly-typed language; and the compiler will throw an error if statements or expressions violate the type rules. The following trivial program foo.stan contains an illegal assignment statement:\ndata {\n  real x;\n}\ntransformed data {\n  int y = x;\n}\nThe Make command fails with the following:\nSemantic error in 'foo.stan', line 5, column 2 to column 12:\n   -------------------------------------------------\n     3:  }\n     4:  transformed data {\n     5:    int y = x;\n           ^\n     6:  }\n   -------------------------------------------------\n\nIll-typed arguments supplied to assignment operator =:\n lhs has type int and rhs has type real\nThe Stan Reference Manual provides a complete specification of the Stan programming language. The Stan User’s Guide also contains a full description of the errors and warnings stanc can emit.\n\n\n\nIf the stanc compiler successfully translates a Stan program to C++, the resulting C++ code should be valid C++ which can be compiled into an executable. The stanc compiler is also a program, and while it has been extensively tested, it may still contain errors such that the generated C++ code fails to compile.\nThe Make command prints the following message to the terminal at the point when it compiles and links the C++ file:\n--- Compiling, linking C++ code ---\nIf the program fails to compile for any reason, the C++ compiler and linker will most likely print a long series of error messages to the console.\nIf this happens, please report the error, together with the Stan program on either the Stan Forums or on the Stan compiler GitHub issues tracker.\n\n\n\nUsers can set flags for the C++ compiler and linker and compiler to optimize their executables. We advise users to only do this once they are sure their basic setup of Cmdstan without flags works.\nThe CXXFLAGS and LDFLAGS makefile variables can be used to set compiler and linker flags respectively. We recommend setting these in the make/local file.\nFor example:\nCXXFLAGS = -O2\nA recommend a set of CXXFLAGS and LDFLAGS flags can be turned on by setting STAN_CPP_OPTIMS=true in the make/local file. These are tested compiler and link-time optimizations that can speed up execution of certain models. We have observed speedups up to 15 percent, but this depends on the model, operating system and hardware used. The use of these flags does considerably slow down compilation, so they are not used by default.\n\n\nWhen assigning or reading from with vectors, row_vectors, matrices or arrays using indexing, Stan checks that a supplied index is valid (not out of range), which avoids segmentation faults and other difficult-to-debug runtime errors.\nFor some models these checks can represent a significant part of the models execution time. By setting the STAN_NO_RANGE_CHECKS=true makefile flag in the make/local file the range checks can be removed. Use this flag with caution (only once the indexing has been validated). In case of any unexpected behavior remove the flag for easier debugging.",
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     "section": "",
     "text": "To compile Stan programs, you must invoke the Make program from the &lt;cmdstan-home&gt; directory. The Stan program can be in a different directory, but the directory path names cannot contain spaces - this limitation is imposed by Make.\n&gt; cd &lt;cmdstan_home&gt;\nIn the call to the Make program, the target is name of the CmdStan executable corresponding to the Stan program file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. To build the Bernoulli example, on Mac and Linux:\n&gt; make examples/bernoulli/bernoulli\nOn Windows, the command is the same with the addition of .exe at the end of the target (note the use of forward slashes):\n&gt; make examples/bernoulli/bernoulli.exe\nThe generated C++ code (bernoulli.hpp), object file (bernoulli.o) and the compiled executable will be placed in the same directory as the Stan program.\nThe compiled executable consists of the Stan model and the CmdStan command line interface which provides inference algorithms to do MCMC sampling, optimization, and variational inference. The following sections provide examples of doing inference using each method on the example model and data file.",
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     "section": "",
     "text": "When executing a Make target, all its dependencies are checked to see if they are up to date, and if they are not, they are rebuilt. If the you call Make with target bernoulli twice in a row, without any editing bernoulli.stan or otherwise changing the system, on the second invocation, Make will determine that the executable is already newer than the Stan source file and will not recompile the program:\n&gt; make examples/bernoulli/bernoulli\nmake: `examples/bernoulli/bernoulli' is up to date.\nIf the file containing the Stan program is updated, the next call to make will rebuild the CmdStan executable.",
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     "section": "",
     "text": "The Stan probabilistic programming language is a programming language with a rich syntax, as such, it is often the case that a carefully written program contains errors.\nThe simplest class of errors are simple syntax errors such as forgetting the semi-colon statement termination marker at the end of a line, or typos such as a misspelled variable name. For example, if in the bernoulli.stan program, we introduce a typo on line \\(9\\) by writing thata instead of theta, the Make command fails with the following\n--- Translating Stan model to C++ code ---\nbin/stanc  --o=bernoulli.hpp bernoulli.stan\n\nSemantic error in 'bernoulli.stan', line 9, column 2 to column 7:\n   -------------------------------------------------\n     7:  }\n     8:  model {\n     9:    thata ~ beta(1, 1);  // uniform prior on interval 0, 1\n           ^\n    10:    y ~ bernoulli(theta);\n    11:  }\n   -------------------------------------------------\n\nIdentifier 'thata' not in scope.\n\nmake: *** [bernoulli.hpp] Error 1\nStan is a strongly-typed language; and the compiler will throw an error if statements or expressions violate the type rules. The following trivial program foo.stan contains an illegal assignment statement:\ndata {\n  real x;\n}\ntransformed data {\n  int y = x;\n}\nThe Make command fails with the following:\nSemantic error in 'foo.stan', line 5, column 2 to column 12:\n   -------------------------------------------------\n     3:  }\n     4:  transformed data {\n     5:    int y = x;\n           ^\n     6:  }\n   -------------------------------------------------\n\nIll-typed arguments supplied to assignment operator =:\n lhs has type int and rhs has type real\nThe Stan Reference Manual provides a complete specification of the Stan programming language. The Stan User’s Guide also contains a full description of the errors and warnings stanc can emit.",
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     "text": "If the stanc compiler successfully translates a Stan program to C++, the resulting C++ code should be valid C++ which can be compiled into an executable. The stanc compiler is also a program, and while it has been extensively tested, it may still contain errors such that the generated C++ code fails to compile.\nThe Make command prints the following message to the terminal at the point when it compiles and links the C++ file:\n--- Compiling, linking C++ code ---\nIf the program fails to compile for any reason, the C++ compiler and linker will most likely print a long series of error messages to the console.\nIf this happens, please report the error, together with the Stan program on either the Stan Forums or on the Stan compiler GitHub issues tracker.",
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     "text": "Users can set flags for the C++ compiler and linker and compiler to optimize their executables. We advise users to only do this once they are sure their basic setup of Cmdstan without flags works.\nThe CXXFLAGS and LDFLAGS makefile variables can be used to set compiler and linker flags respectively. We recommend setting these in the make/local file.\nFor example:\nCXXFLAGS = -O2\nA recommend a set of CXXFLAGS and LDFLAGS flags can be turned on by setting STAN_CPP_OPTIMS=true in the make/local file. These are tested compiler and link-time optimizations that can speed up execution of certain models. We have observed speedups up to 15 percent, but this depends on the model, operating system and hardware used. The use of these flags does considerably slow down compilation, so they are not used by default.\n\n\nWhen assigning or reading from with vectors, row_vectors, matrices or arrays using indexing, Stan checks that a supplied index is valid (not out of range), which avoids segmentation faults and other difficult-to-debug runtime errors.\nFor some models these checks can represent a significant part of the models execution time. By setting the STAN_NO_RANGE_CHECKS=true makefile flag in the make/local file the range checks can be removed. Use this flag with caution (only once the indexing has been validated). In case of any unexpected behavior remove the flag for easier debugging.",
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-    "text": "The following is a simple, complete Stan program for a Bernoulli model of binary data.1 The model assumes the binary observed data y[1],...,y[N] are i.i.d. with Bernoulli chance-of-success theta.\ndata { \n  int&lt;lower=0&gt; N; \n  array[N] int&lt;lower=0, upper=1&gt; y;\n} \nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n} \nmodel {\n  theta ~ beta(1, 1);  // uniform prior on interval 0,1\n  y ~ bernoulli(theta);\n}\nThe input data file contains definitions for the two variables N and y which are specified in the data block of program bernoulli.stan (above).\nA data set of N=10 observations is included in the example Bernoulli model directory in both JSON notation and Rdump data format where 8 out of 10 trials had outcome 0 (failure) and 2 trials had outcome 1 (success). In JSON, this data is:\n{\n    \"N\" : 10,\n    \"y\" : [0,1,0,0,0,0,0,0,0,1]\n}",
+    "text": "The --allow-undefined flag can be passed to the call to stanc, which will allow undefined functions in the Stan language to be parsed without an error. We can then include a definition of the function in a C++ header file.\nThis requires specifying two makefile variables:\n\nSTANCFLAGS=--allow-undefined\nUSER_HEADER=&lt;header_file.hpp&gt;, where &lt;header_file.hpp&gt; is the name of a header file that defines a function with the same name and a compatible signature. This function can appear in the global namespace or in the model namespace, which is defined as the name of the model (either the file name, or the --name argument to stanc) followed by _namespace.\n\nThis is an advanced feature which is only recommended to users familiar with the internals of Stan’s Math library. Most existing C++ code will need to be modified to work with Stan, to varying degrees.\nAs an example, consider the following variant of the Bernoulli example\nfunctions {\n  real make_odds(data real theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n}\nmodel {\n  theta ~ beta(1, 1); // uniform prior on interval 0, 1\n  y ~ bernoulli(theta);\n}\ngenerated quantities {\n  real odds;\n  odds = make_odds(theta);\n}\nHere the make_odds function is declared but not defined, which would ordinarily result in a parser error. However, if you put STANCFLAGS = --allow-undefined into the make/local file or into the stanc call, then the stanc compiler will translate this program to C++, but the generated C++ code will not compile unless you write a file such as examples/bernoulli/make_odds.hpp with the following lines\n#include &lt;ostream&gt;\n\ndouble make_odds(const double& theta, std::ostream *pstream__) {\n  return theta / (1 - theta);\n}\nThe signature for this function needs to fulfill all the usages in the C++ class emitted by stanc. The pstream__ argument is mandatory in the signature but need not be used if your function does not print any output. Because make_odds was declared with a data argument and only used in generated quantites, a signature which accepts and returns double is acceptable. Functions which will have parameters passed as input in the transformed parameters or model blocks will require the ability to accept Stan’s autodiff types. If you wish to autodiff through this function, the simplest option is to make it a template, like\ntemplate &lt;typename T&gt;\nT make_odds(const T &theta, std::ostream *pstream__)\n{\n    return theta / (1 - theta);\n}\nGiven the above, the following make invocation should work\n&gt; make STANCFLAGS=--allow-undefined USER_HEADER=examples/bernoulli/make_odds.hpp examples/bernoulli/bernoulli # on Windows add .exe\nAlternatively, you could put STANCFLAGS and USER_HEADER into the make/local file instead of specifying them on the command-line.\nIf the function were more complicated and involved functions in the Stan Math Library, then you would need to add #include &lt;stan/model/model_header.hpp&gt; and prefix the function calls with stan::math::.\n\n\nExternal C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.\n\n\n\nSome functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
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-    "text": "Footnotes\n\n\nThe model is available with the CmdStan distribution at the path &lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan.↩︎",
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+    "text": "External C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.",
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-    "text": "Standalone Generate Quantities\nThe generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. For an overview of the uses of this feature, see the QuickStart Guide section and the Stan User’s Guide section on Stand-alone generated quantities and ongoing prediction.\nThis method requires sub-argument fitted_params which takes as its value an existing Stan CSV file that contains a sample from an equivalent model, i.e., a model with the same parameters, transformed parameters, and model blocks, conditioned on the same data.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contencts of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] (in 'bernoulli_ppc.stan', line 5, column 2 to column 30)\n\n\n\n\n Back to top",
+    "text": "Some functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
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+    "text": "Footnotes\n\n\nDetails of programming in the Stan Math style are omitted from this section, it is presented only as an example↩︎",
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     "text": "CmdStan can return the computed log probability and the gradient with respect to a set of parameters.\nThis is similar to the diagnose subcommand, but the output format differs and the results here are not compared with those from finite differences.\nNote: Startup and data initialization costs mean that this method is not an efficient way to calculate these quantities. It is provided only for convenience and should not be used for serious computation.\n\n\nThis method takes 3 arguments:\n\njacobian - Whether or not the Jacobian adjustment for constrained parameters should be included in the gradient. Default value is 1 (include adjustment).\nconstrained_params - Input file of parameters values on the constrained scale. A single set of constrained parameters can be specified using JSON format. Alternatively, the input file can be set of draws in StanCSV format.\nunconstrained_params - Input file (JSON or R dump) of parameter values on unconstrained scale. These files should contain a single variable, called params_r, which is a flattened vector of all unconstrained parameters. If this object is two dimensional, each entry should be a vector of the same form and the output will feature multiple rows.\n\nOnly one of constrained_params and unconstrained_params can be specified.\nFor more on the differences between constrained and unconstrained parameters, see the Stan reference manual section on variable transforms.\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the log_prob method is reported at the beginning of the output file as CSV comments.\nColumn headers, the first column is labelled lp__, and the rest are named after parameters. These will be the unconstrained parameters, regardless of whether constrained or unconstrained parameters were supplied as input.\nValues which correspond to the value of the log density (column 1) and the gradient with respect to each parameter (remaining columns).\n\nFor example, if we have a file called params.json:\n{\n    \"theta\" : 0.1\n}\nWe can run the example model:\n/bernoulli log_prob constrained_params=params.json data file=bernoulli.data.json\nThis yields\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-11-17 20:46:06 UTC\n# method = log_prob\n#   log_prob\n#     unconstrained_params =  (Default)\n#     constrained_params = params.json\n#     jacobian = 1 (Default)\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2390820139 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 2.31.0 (Linux)\n# stancflags =\nlp_,theta\n-7.856,1.8",
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     "text": "This method takes 3 arguments:\n\njacobian - Whether or not the Jacobian adjustment for constrained parameters should be included in the gradient. Default value is 1 (include adjustment).\nconstrained_params - Input file of parameters values on the constrained scale. A single set of constrained parameters can be specified using JSON format. Alternatively, the input file can be set of draws in StanCSV format.\nunconstrained_params - Input file (JSON or R dump) of parameter values on unconstrained scale. These files should contain a single variable, called params_r, which is a flattened vector of all unconstrained parameters. If this object is two dimensional, each entry should be a vector of the same form and the output will feature multiple rows.\n\nOnly one of constrained_params and unconstrained_params can be specified.\nFor more on the differences between constrained and unconstrained parameters, see the Stan reference manual section on variable transforms.",
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     "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the log_prob method is reported at the beginning of the output file as CSV comments.\nColumn headers, the first column is labelled lp__, and the rest are named after parameters. These will be the unconstrained parameters, regardless of whether constrained or unconstrained parameters were supplied as input.\nValues which correspond to the value of the log density (column 1) and the gradient with respect to each parameter (remaining columns).\n\nFor example, if we have a file called params.json:\n{\n    \"theta\" : 0.1\n}\nWe can run the example model:\n/bernoulli log_prob constrained_params=params.json data file=bernoulli.data.json\nThis yields\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-11-17 20:46:06 UTC\n# method = log_prob\n#   log_prob\n#     unconstrained_params =  (Default)\n#     constrained_params = params.json\n#     jacobian = 1 (Default)\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2390820139 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 2.31.0 (Linux)\n# stancflags =\nlp_,theta\n-7.856,1.8",
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-    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n....\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)\n\n\n\nA Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThere are two different ways of running multiple chains, with the num_chains argument using a single executable and by using the Unix and DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine.\nExample that will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially and will produce the sample in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS.\n\n\n\n\nEach execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n...\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)\n\n\n\nThe stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. 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-    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n....\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)",
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-    "text": "A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThere are two different ways of running multiple chains, with the num_chains argument using a single executable and by using the Unix and DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine.\nExample that will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially and will produce the sample in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS.",
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-    "text": "Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n...\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)",
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-    "text": "The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)",
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-    "text": "The optimize method finds the mode of the posterior distribution, assuming that there is one. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability. For optimization, the mode is calculated without the Jacobian adjustment for constrained variables, which shifts the mode due to the change of variables. Thus modes correspond to modes of the model as written.\nThe full set of configuration options available for the optimize method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run with method=optimize via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n#       lbfgs\n#         init_alpha = 0.001 (Default)\n#         tol_obj = 9.9999999999999998e-13 (Default)\n#         tol_rel_obj = 10000 (Default)\n#         tol_grad = 1e-08 (Default)\n#         tol_rel_grad = 10000000 (Default)\n#         tol_param = 1e-08 (Default)\n#         history_size = 5 (Default)\n#     jacobian = 0 (Default)\n#     iter = 2000 (Default)\n#     save_iterations = 0 (Default)\n\n\nThe jacobian argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms. Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\), the value which maximizes the likelihood of the data given the parameters, (including prior terms). Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\,p(\\theta)\\). By default this value is 0 (false), do not include the Jacobian adjustment.\n\n\n\nThe algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.\n\n\n\nFor both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. Being too big or too small just means that the first iteration will take longer (i.e., require more gradient evaluations) before the line search finds a good step length.\nIn addition to the above, the L-BFGS algorithm has argument history_size which controls the size of the history it uses to approximate the Hessian. The value should be less than the dimensionality of the parameter space and, in general, relatively small values (\\(5\\)-\\(10\\)) are sufficient; the default value is \\(5\\).\nIf L-BFGS performs poorly but BFGS performs well, consider increasing the history size. Increasing history size will increase the memory usage, although this is unlikely to be an issue for typical Stan models.\n\n\n\nThere are no configuration parameters for the Newton optimizer. It is not recommended because of the slow Hessian calculation involving finite differences.",
+    "text": "The CmdStan executable can run Stan’s optimization algorithms, which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global optimum as opposed to a local optimum of log probability.\nThe full set of configuration options available for the optimize method is available by using the optimize help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the optimizer console output and in the output CSV file’s comments.\nThe executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.\n&gt; ./bernoulli optimize data file=bernoulli.data.json\nExecuting this command prints both output to the console and to a CSV file.\nThe first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:\n ./bernoulli optimize data file=bernoulli.data.json\nmethod = optimize\n  optimize\n    algorithm = lbfgs (Default)\n      lbfgs\n        init_alpha = 0.001 (Default)\n        tol_obj = 1e-12 (Default)\n        tol_rel_obj = 10000 (Default)\n        tol_grad = 1e-08 (Default)\n        tol_rel_grad = 10000000 (Default)\n        tol_param = 1e-08 (Default)\n        history_size = 5 (Default)\n    iter = 2000 (Default)\n    save_iterations = 0 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 87122538 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nThe second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The alpha value is for step size used. In the final state the change in parameters was roughly \\(0.002\\) and the length of the gradient roughly 3e-05 (\\(0.00003\\)).\nInitial log joint probability = -6.85653\n    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes\n       5      -5.00402    0.00184936   3.35074e-05           1           1        8\nOptimization terminated normally:\n  Convergence detected: relative gradient magnitude is below tolerance\nThe output from optimization is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n#       lbfgs\n#         init_alpha = 0.001 (Default)\n#         tol_obj = 9.9999999999999998e-13 (Default)\n#         tol_rel_obj = 10000 (Default)\n#         tol_grad = 1e-08 (Default)\n#         tol_rel_grad = 10000000 (Default)\n#         tol_param = 1e-08 (Default)\n#         history_size = 5 (Default)\n#     jacobian = 0 (Default)\n#     iter = 2000 (Default)\n#     save_iterations = 0 (Default)\n...\nFollowing the config information are two lines of output, the CSV headers and the recorded values:\nlp__,theta\n-5.00402,0.200003\nNote that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with lp__ and the model parameter theta. The maximum log probability is -5.0 and the posterior mode for theta is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file. Running the optimizer with save_iterations=1 writes both the initial log joint probability and values for all iterations to the output CSV file.\nRunning the example model with option save_iterations=1, i.e., the command\n&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json\nproduces CSV file output rows:\nlp__,theta\n-6.85653,0.493689\n-6.10128,0.420936\n-5.02953,0.22956\n-5.00517,0.206107\n-5.00403,0.200299\n-5.00402,0.200003\n\n\nThe jacobian argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms.\nWithout the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\), the value which maximizes the likelihood of the data given the parameters, (including prior terms).\nApplying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\,p(\\theta)\\).\nBy default this value is 0 (false), do not include the Jacobian adjustment.\n\n\n\nThe algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.\n\n\n\nFor both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. 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     "text": "The algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.",
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     "text": "For both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. Being too big or too small just means that the first iteration will take longer (i.e., require more gradient evaluations) before the line search finds a good step length.\nIn addition to the above, the L-BFGS algorithm has argument history_size which controls the size of the history it uses to approximate the Hessian. The value should be less than the dimensionality of the parameter space and, in general, relatively small values (\\(5\\)-\\(10\\)) are sufficient; the default value is \\(5\\).\nIf L-BFGS performs poorly but BFGS performs well, consider increasing the history size. Increasing history size will increase the memory usage, although this is unlikely to be an issue for typical Stan models.",
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     "text": "There are no configuration parameters for the Newton optimizer. It is not recommended because of the slow Hessian calculation involving finite differences.",
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-    "text": "The Pathfinder algorithm is described in section Pathfinder overview.\nThe pathfinder method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument num_paths specifies the number of single-path Pathfinders, the default is \\(4\\). If num_paths is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the pathfinder output file as CSV comments. When the example model bernoulli.stan is run with method=pathfinder via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\n\n\n\nnum_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\n\n\n\n\nArguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.\n\n\n\nBy default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\n\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the sampler output file as CSV comments.\nThe CSV header row consists of columns lp_approx__, lp__, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.\nThe data rows contain the draws from the single- or multi-path run.\nFinal comments containing timing information.\n\n\n\n\nThe boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
+    "text": "The CmdStan method pathfinder uses the Pathfinder algorithm of Zhang et al. (2022), which is further described in the Stan Reference Manual.\nA single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes.\nThe pathfinder method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument num_paths specifies the number of single-path Pathfinders, the default is \\(4\\). If num_paths is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.\nThe full set of configuration options available for the pathfinder method is available by using the pathfinder help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.\nThe following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli pathfinder data file=bernoulli.data.R\nExecuting this command prints both output to the console and csv files.\nThe first part of the console output reports on the configuration used.\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\nid = 1 (Default)\ndata\n  file = examples/bernoulli/bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 1995513073 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n  sig_figs = -1 (Default)\n  profile_file = profile.csv (Default)\nnum_threads = 1 (Default)\nThe rest of the output describes the progression of the algorithm.\nBy default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, then uses importance resampling on the set of returned draws to produce the specified number of draws.\nPath [1] :Initial log joint density = -11.543343\nPath [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00\nPath [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)\nPath [2] :Initial log joint density = -7.443345\nPath [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00\nPath [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)\nPath [3] :Initial log joint density = -18.986308\nPath [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00\nPath [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)\nPath [4] :Initial log joint density = -8.304453\nPath [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00\nPath [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)\nTotal log probability function evaluations:8404\n\n\n\nnum_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\n\n\n\n\nArguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.\n\n\n\nBy default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\nThe initial CSV comment rows contain the complete set of CmdStan configuration options.\n...\n# method = pathfinder\n#   pathfinder\n#     init_alpha = 0.001 (Default)\n#     tol_obj = 9.9999999999999998e-13 (Default)\n#     tol_rel_obj = 10000 (Default)\n#     tol_grad = 1e-08 (Default)\n#     tol_rel_grad = 10000000 (Default)\n#     tol_param = 1e-08 (Default)\n#     history_size = 5 (Default)\n#     num_psis_draws = 1000 (Default)\n#     num_paths = 4 (Default)\n#     psis_resample = 1 (Default)\n#     calculate_lp = 1 (Default)\n#     save_single_paths = 0 (Default)\n#     max_lbfgs_iters = 1000 (Default)\n#     num_draws = 1000 (Default)\n#     num_elbo_draws = 25 (Default)\n...\nNext is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs.\n\n\n\nThe boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
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@@ -4672,7 +4578,7 @@
     "section": "",
     "text": "num_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).",
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@@ -4683,7 +4589,7 @@
     "section": "",
     "text": "Arguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.",
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@@ -4692,9 +4598,9 @@
     "href": "cmdstan-guide/pathfinder_config.html#pathfinder_csv",
     "title": "Pathfinder Method for Approximate Bayesian Inference",
     "section": "",
-    "text": "By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\n\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the sampler output file as CSV comments.\nThe CSV header row consists of columns lp_approx__, lp__, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.\nThe data rows contain the draws from the single- or multi-path run.\nFinal comments containing timing information.",
+    "text": "By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\nThe initial CSV comment rows contain the complete set of CmdStan configuration options.\n...\n# method = pathfinder\n#   pathfinder\n#     init_alpha = 0.001 (Default)\n#     tol_obj = 9.9999999999999998e-13 (Default)\n#     tol_rel_obj = 10000 (Default)\n#     tol_grad = 1e-08 (Default)\n#     tol_rel_grad = 10000000 (Default)\n#     tol_param = 1e-08 (Default)\n#     history_size = 5 (Default)\n#     num_psis_draws = 1000 (Default)\n#     num_paths = 4 (Default)\n#     psis_resample = 1 (Default)\n#     calculate_lp = 1 (Default)\n#     save_single_paths = 0 (Default)\n#     max_lbfgs_iters = 1000 (Default)\n#     num_draws = 1000 (Default)\n#     num_elbo_draws = 25 (Default)\n...\nNext is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs.",
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@@ -4705,150 +4611,205 @@
     "section": "",
     "text": "The boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
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   {
-    "objectID": "cmdstan-guide/print.html",
-    "href": "cmdstan-guide/print.html",
-    "title": "print (deprecated): MCMC Output Analysis",
+    "objectID": "cmdstan-guide/rdump_apdx.html",
+    "href": "cmdstan-guide/rdump_apdx.html",
+    "title": "RDump Format for CmdStan",
     "section": "",
-    "text": "print (deprecated): MCMC Output Analysis\nThe print utility is deprecated, but is still available until CmdStan v3.0. It has been replaced by the stansummary utility.\n\n\n\n\n Back to top",
+    "text": "NOTE: Although the RDump format is still supported, I/O with JSON is faster and recommended. See the chapter on JSON for more details.\nRDump format can be used to represent values for Stan variables. This format was introduced in SPLUS and is used in R, JAGS, and in BUGS (but with a different ordering).\nA dump file is structured as a sequence of variable definitions. Each variable is defined in terms of its dimensionality and its values. There are three kinds of variable declarations: - scalars - sequences - general arrays\n\n\nDump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.\n\n\n\nA simple scalar value can be thought of as having an empty list of dimensions. Its declaration in the dump format follows the SPLUS assignment syntax. For example, the following would constitute a valid dump file defining a single scalar variable y with value \\(17.2\\):\ny &lt;- 17.2\n\n\n\nOne-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)\n\n\n\nFor more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))\n\n\n\nThe dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;\n\n\n\n\nAt this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.\n\n\n\nThere is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.\n\n\n\n\nIn order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}\n\n\n\nThe line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. 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-    "text": "The output from all CmdStan methods is in CSV format. A Stan CSV file is a data table where the columns are the method and model parameters and quantities of interest. Each row contains one record’s worth of data in plain-text format using the comma character (‘,’) as the field delimiter (hence the name).\nFor the Stan CSV files, data is strictly numerical, however, possible values include both positive and negative infinity and “Not-a-Number” which are represented as the strings NaN, inf, +inf, -inf. All other values are written in decimal notation with at most 6 digits of precision.\nStan CSV files have a header row containing the column names. They also make extensive use of CSV comments, i.e., lines which begin with the # character. In addition to initial and final comment rows, some methods also put comment rows in the middle of the data table, which makes it difficult to use many of the commonly used CSV parser packages.\n\n\nThe data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3\n\n\n\nThe sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the Quickstart Guide chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation\n\n\n\n\n\n\nConfig as comments\nHeader row\nPenalized maximum likelihood estimate\n\n\n\n\n\nConfig as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior\n\n\n\n\n\nHeader row\nQuantities of interest\n\n\n\n\n\nHeader row\nGradients",
+    "text": "Dump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.",
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-    "text": "The data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3",
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-    "text": "The sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the Quickstart Guide chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. 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+    "text": "For more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))",
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+    "text": "There is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.",
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-    "text": "The CmdStan stansummary program reports statistics for one or more sampler chains over all sampler and model parameters and quantities of interest. The statistics reported include both summary statistics of the estimates and diagnostic statistics on the sampler chains, reported in the following order:\n\nMean - sample mean\nMCSE - Monte Carlo Standard Error, a measure of the amount of noise in the sample\nStdDev - sample standard deviation\nQuantiles - default 5%, 50%, 95%\nN_eff - effective sample size - the number of independent draws in the sample\nN_eff/S - the number of independent draws per second\nR_hat - \\(\\hat{R}\\) statistic, a measure of chain equilibrium, must be within \\(0.05\\) of \\(1.0\\).\n\nWhen reviewing the stansummary output, it is important to check the final three output columns first - these are the diagnostic statistics on chain convergence and number of independent draws in the sample. A \\(\\hat{R}\\) statistic of greater than \\(1.05\\) indicates that the chain has not converged and therefore the sample is not drawn from the posterior, thus the estimates of the mean and all other summary statistics are invalid.\nEstimation by sampling produces an approximate value for the model parameters; the MCSE statistic indicates the amount of noise in the estimate. Therefore MCSE column is placed next to the sample mean column, in order to make it easy to compare this sample with others.\nFor more information, see the Posterior Analysis chapter of the Stan Reference Manual which describes both the theory and practice of MCMC estimation techniques. The summary statistics and the algorithms used to compute them are described in sections Notation for samples and Effective Sample Size.\n\n\nThe CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary\n\n\n\nThe stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].\n\n\n\n\nThe stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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+    "text": "A more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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+    "text": "CmdStan translates Stan programs to C++ using the Stan compiler program which is included in the CmdStan release bin directory as program stanc. One can view the complete stanc documentation in the Stan User’s Guide.\nAs of release 2.22, the CmdStan Stan to C++ compiler is written in OCaml. This compiler is called “stanc3” and has has its own repository https://github.com/stan-dev/stanc3, from which pre-built binaries for Linux, Mac, and Windows can be downloaded.\n\n\nBefore the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.\n\n\n\nThe Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
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-    "text": "The stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].",
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-    "text": "The stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
+    "text": "The Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
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+    "text": "Stan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) Kucukelbir et al. (2017). ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using stochastic gradient ascent. The measures of convergence are similar to the relative tolerance scheme of Stan’s optimization algorithms.\nThe algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.\nThe full set of configuration options available for the variational method is available by using the variational help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.\nThe following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli variational data file=bernoulli.data.R\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used: the default option algorithm=meanfield and the default tolerances for monitoring the algorithm’s convergence.\nmethod = variational\n  variational\n    algorithm = meanfield (Default)\n      meanfield\n    iter = 10000 (Default)\n    grad_samples = 1 (Default)\n    elbo_samples = 100 (Default)\n    eta = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      iter = 50 (Default)\n    tol_rel_obj = 0.01 (Default)\n    eval_elbo = 100 (Default)\n    output_samples = 1000 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3323783840 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, informational and timing messages are output:\n------------------------------------------------------------\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n------------------------------------------------------------\n\nGradient evaluation took 2.1e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.\nAdjust your expectations accordingly!\nThe rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in \\(\\sim\\) 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.\nBegin eta adaptation.\nIteration:   1 / 250 [  0%]  (Adaptation)\nIteration:  50 / 250 [ 20%]  (Adaptation)\nIteration: 100 / 250 [ 40%]  (Adaptation)\nIteration: 150 / 250 [ 60%]  (Adaptation)\nIteration: 200 / 250 [ 80%]  (Adaptation)\nSuccess! Found best value [eta = 1] earlier than expected.\n\nBegin stochastic gradient ascent.\n  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes\n   100           -6.131             1.000            1.000\n   200           -6.458             0.525            1.000\n   300           -6.300             0.359            0.051\n   400           -6.137             0.276            0.051\n   500           -6.243             0.224            0.027\n   600           -6.305             0.188            0.027\n   700           -6.289             0.162            0.025\n   800           -6.402             0.144            0.025\n   900           -6.103             0.133            0.025\n  1000           -6.314             0.123            0.027\n  1100           -6.348             0.024            0.025\n  1200           -6.244             0.020            0.018\n  1300           -6.293             0.019            0.017\n  1400           -6.250             0.017            0.017\n  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED\n\nDrawing a sample of size 1000 from the approximate posterior...\nCOMPLETED.\n\n\nStan implements two variational algorithms. They differ in the approximating distribution used in the unconstrained variable space. By default, ADVI uses option algorithm=meanfield. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.\n\n\n\n\n\niter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv.\nThe output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext, the column header row:\nlp__,log_p__,log_g__,theta\nAdditional comments provide stepsize adaptation information:\n# Stepsize adaptation complete.\n# eta = 1\nFollowed by the data rows. The first line is special — it is the mean of the variational approximation.\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.",
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+    "text": "iter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.",
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-    "text": "Variational Inference using ADVI\nThe CmdStan method variational uses the Automatic Differentiation Variational Inference (ADVI) algorithm of Kucukelbir et al. (2017) to provide an approximate posterior distribution of the model conditioned on the data. The approximating distribution it uses is a Gaussian in the unconstrained variable space, either a fully factorized Gaussian approximation, specified by argument algorithm=meanfield option, or a Gaussian approximation using a full-rank covariance matrix, specified by argument algorithm=fullrank. By default, ADVI uses option algorithm=meanfield.\nThe following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli variational data file=bernoulli.data.R\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used: the default option algorithm=meanfield and the default tolerances for monitoring the algorithm’s convergence.\nmethod = variational\n  variational\n    algorithm = meanfield (Default)\n      meanfield\n    iter = 10000 (Default)\n    grad_samples = 1 (Default)\n    elbo_samples = 100 (Default)\n    eta = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      iter = 50 (Default)\n    tol_rel_obj = 0.01 (Default)\n    eval_elbo = 100 (Default)\n    output_samples = 1000 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3323783840 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, informational and timing messages are output:\n------------------------------------------------------------\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n------------------------------------------------------------\n\nGradient evaluation took 2.1e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.\nAdjust your expectations accordingly!\nThe rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in \\(\\sim\\) 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.\nBegin eta adaptation.\nIteration:   1 / 250 [  0%]  (Adaptation)\nIteration:  50 / 250 [ 20%]  (Adaptation)\nIteration: 100 / 250 [ 40%]  (Adaptation)\nIteration: 150 / 250 [ 60%]  (Adaptation)\nIteration: 200 / 250 [ 80%]  (Adaptation)\nSuccess! Found best value [eta = 1] earlier than expected.\n\nBegin stochastic gradient ascent.\n  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes \n   100           -6.131             1.000            1.000\n   200           -6.458             0.525            1.000\n   300           -6.300             0.359            0.051\n   400           -6.137             0.276            0.051\n   500           -6.243             0.224            0.027\n   600           -6.305             0.188            0.027\n   700           -6.289             0.162            0.025\n   800           -6.402             0.144            0.025\n   900           -6.103             0.133            0.025\n  1000           -6.314             0.123            0.027\n  1100           -6.348             0.024            0.025\n  1200           -6.244             0.020            0.018\n  1300           -6.293             0.019            0.017\n  1400           -6.250             0.017            0.017\n  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED\n\nDrawing a sample of size 1000 from the approximate posterior... \nCOMPLETED.\nThe output from variational is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext is the column header line, followed more CSV comments reporting the adapted value for the stepsize, followed by the values. The first line is special: it is the mean of the variational approximation. The rest of the output contains output_samples number of samples drawn from the variational approximation.\nlp__,log_p__,log_g__,theta\n# Stepsize adaptation complete.\n# eta = 1\n0,0,0,0.236261\n0,-6.82318,-0.0929121,0.300415\n0,-6.89701,-0.158687,0.321982\n0,-6.99391,-0.23916,0.343643\n0,-7.35801,-0.51787,0.401554\n0,-7.4668,-0.539473,0.123081\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.\n\n\n\n\n\n Back to topReferences\n\nKucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. “Automatic Differentiation Variational Inference.” Journal of Machine Learning Research.",
+    "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv.\nThe output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext, the column header row:\nlp__,log_p__,log_g__,theta\nAdditional comments provide stepsize adaptation information:\n# Stepsize adaptation complete.\n# eta = 1\nFollowed by the data rows. The first line is special — it is the mean of the variational approximation.\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.",
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-    "text": "Stan’s diagnostic mode runs a Stan program with data, initializing parameters either randomly or with user-specified initial values, and then evaluates the log probability and its gradients. The gradients computed by the Stan program are compared to values calculated by finite differences.\nDiagnostic mode may be configured with two parameters.\nDiagnostic Mode Configuration Table. The diagnostic model configuration parameters, constraints, and default values.\n\n\n\n\n\n\n\n\n\nparameter\ndescription\nconstraints\ndefault\n\n\n\n\nepsilon\nfinite difference size\n(0, infty)\n1e–6\n\n\nerror\nerror threshold for matching\n(0, infty)\n1e–6\n\n\n\nIf the difference between the Stan program’s gradient value and that calculated by finite difference is higher than the specified threshold, the argument will be flagged.\n\n\nDiagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.\n\n\n\n\nThe general configuration options for diagnostics are the same as those for MCMC. Initial values may be specified, or they may be drawn at random. Setting the random number generator will only have an effect if a random initialization is specified.\n\n\n\nDue to the application of finite differences, the computation time grows linearly with the number of parameters. This can be require a very long time, especially in models with latent parameters that grow with the data size. It can be helpful to diagnose a model with smaller data sizes in such cases.",
+    "text": "Stan’s diagnostic mode runs a Stan program with data, initializing parameters either randomly or with user-specified initial values, and then evaluates the log probability and its gradients. The gradients computed by the Stan program are compared to values calculated by finite differences.\nDiagnostic mode may be configured with two parameters.\nDiagnostic Mode Configuration Table. The diagnostic model configuration parameters, constraints, and default values.\n\n\n\n\n\n\n\n\n\nparameter\ndescription\nconstraints\ndefault\n\n\n\n\nepsilon\nfinite difference size\n(0, infty)\n1e–6\n\n\nerror\nerror threshold for matching\n(0, infty)\n1e–6\n\n\n\nIf the difference between the Stan program’s gradient value and that calculated by finite difference is higher than the specified threshold, the argument will be flagged.\n\n\nDiagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.\n\n\n\n\nThe general configuration options for diagnostics are the same as those for MCMC. Initial values may be specified, or they may be drawn at random. Setting the random number generator will only have an effect if a random initialization is specified.\n\n\n\nDue to the application of finite differences, the computation time grows linearly with the number of parameters. This can be require a very long time, especially in models with latent parameters that grow with the data size. It can be helpful to diagnose a model with smaller data sizes in such cases.",
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-    "text": "Diagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.",
+    "text": "Diagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.",
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     "title": "Program Execution",
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-    "text": "This chapter provides a sketch of how a compiled Stan model is executed using sampling. Optimization shares the same data reading and initialization steps, but then does optimization rather than sampling.\nThis sketch is elaborated in the following chapters of this part, which cover variable declarations, expressions, statements, and blocks in more detail.\n\n\nThe reading and transforming data steps are the same for sampling, optimization and diagnostics.\n\n\nThe first step of execution is to read data into memory. Data may be read in through file (in CmdStan) or through memory (RStan and PyStan); see their respective manuals for details.1\nAll of the variables declared in the data block will be read. If a variable cannot be read, the program will halt with a message indicating which data variable is missing.\nAfter each variable is read, if it has a declared constraint, the constraint is validated. For example, if a variable N is declared as int&lt;lower=0&gt;, after N is read, it will be tested to make sure it is greater than or equal to zero. If a variable violates its declared constraint, the program will halt with a warning message indicating which variable contains an illegal value, the value that was read, and the constraint that was declared.\n\n\n\nAfter data is read into the model, the transformed data variable statements are executed in order to define the transformed data variables. As the statements execute, declared constraints on variables are not enforced.\nTransformed data variables are initialized with real values set to NaN and integer values set to the smallest integer (large absolute value negative number).\nAfter the statements are executed, all declared constraints on transformed data variables are validated. If the validation fails, execution halts and the variable’s name, value and constraints are displayed.\n\n\n\n\nInitialization is the same for sampling, optimization, and diagnosis\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.\n\n\n\n\nSampling is based on simulating the Hamiltonian of a particle with a starting position equal to the current parameter values and an initial momentum (kinetic energy) generated randomly. The potential energy at work on the particle is taken to be the negative log (unnormalized) total probability function defined by the model. In the usual approach to implementing HMC, the Hamiltonian dynamics of the particle is simulated using the leapfrog integrator, which discretizes the smooth path of the particle into a number of small time steps called leapfrog steps.\n\n\nFor each leapfrog step, the negative log probability function and its gradient need to be evaluated at the position corresponding to the current parameter values (a more detailed sketch is provided in the next section). These are used to update the momentum based on the gradient and the position based on the momentum.\nFor simple models, only a few leapfrog steps with large step sizes are needed. For models with complex posterior geometries, many small leapfrog steps may be needed to accurately model the path of the parameters.\nIf the user specifies the number of leapfrog steps (i.e., chooses to use standard HMC), that number of leapfrog steps are simulated. If the user has not specified the number of leapfrog steps, the No-U-Turn sampler (NUTS) will determine the number of leapfrog steps adaptively (Hoffman and Gelman 2014).\n\n\n\nDuring each leapfrog step, the log probability function and its gradient must be calculated. This is where most of the time in the Stan algorithm is spent. This log probability function, which is used by the sampling algorithm, is defined over the unconstrained parameters.\nThe first step of the calculation requires the inverse transform of the unconstrained parameter values back to the constrained parameters in terms of which the model is defined. There is no error checking required because the inverse transform is a total function on every point in whose range satisfies the constraints.\nBecause the probability statements in the model are defined in terms of constrained parameters, the log Jacobian of the inverse transform must be added to the accumulated log probability.\nNext, the transformed parameter statements are executed. After they complete, any constraints declared for the transformed parameters are checked. If the constraints are violated, the model will halt with a diagnostic error message.\nThe final step in the log probability function calculation is to execute the statements defined in the model block.\nAs the log probability function executes, it accumulates an in-memory representation of the expression tree used to calculate the log probability. This includes all of the transformed parameter operations and all of the Jacobian adjustments. This tree is then used to evaluate the gradients by propagating partial derivatives backward along the expression graph. The gradient calculations account for the majority of the cycles consumed by a Stan program.\n\n\n\nA standard Metropolis accept/reject step is required to retain detailed balance and ensure draws are marginally distributed according to the probability function defined by the model. This Metropolis adjustment is based on comparing log probabilities, here defined by the Hamiltonian, which is the sum of the potential (negative log probability) and kinetic (squared momentum) energies. In theory, the Hamiltonian is invariant over the path of the particle and rejection should never occur. In practice, the probability of rejection is determined by the accuracy of the leapfrog approximation to the true trajectory of the parameters.\nIf step sizes are small, very few updates will be rejected, but many steps will be required to move the same distance. If step sizes are large, more updates will be rejected, but fewer steps will be required to move the same distance. Thus a balance between effort and rejection rate is required. If the user has not specified a step size, Stan will tune the step size during warmup sampling to achieve a desired rejection rate (thus balancing rejection versus number of steps).\nIf the proposal is accepted, the parameters are updated to their new values. Otherwise, the sample is the current set of parameter values.\n\n\n\n\nOptimization runs very much like sampling in that it starts by reading the data and then initializing parameters. Unlike sampling, it produces a deterministic output which requires no further analysis other than to verify that the optimizer itself converged to a posterior mode. The output for optimization is also similar to that for sampling.\n\n\n\nVariational inference also runs similar to sampling. It begins by reading the data and initializing the algorithm. The initial variational approximation is a random draw from the standard normal distribution in the unconstrained (real-coordinate) space. Again, similar to sampling, it outputs draws from the approximate posterior once the algorithm has decided that it has converged. Thus, the tools we use for analyzing the result of Stan’s sampling routines can also be used for variational inference.\n\n\n\nModel diagnostics are like sampling and optimization in that they depend on a model’s data being read and its parameters being initialized. The user’s guides for the interfaces (RStan, PyStan, CmdStan) provide more details on the diagnostics available; as of Stan 2.0, that’s just gradients on the unconstrained scale and log probabilities.\n\n\n\nFor each final draw (not counting draws during warmup or draws that are thinned), there is an output stage of writing the draw.\n\n\nBefore generating any output, the statements in the generated quantities block are executed. This can be used for any forward simulation based on parameters of the model. Or it may be used to transform parameters to an appropriate form for output.\nAfter the generated quantities statements execute, the constraints declared on generated quantities variables are validated. If these constraints are violated, the program will terminate with a diagnostic message.\n\n\n\nThe final step is to write the actual values. The values of all variables declared as parameters, transformed parameters, or generated quantities are written. Local variables are not written, nor is the data or transformed data. All values are written in their constrained forms, that is the form that is used in the model definitions.\nIn the executable form of a Stan models, parameters, transformed parameters, and generated quantities are written to a file in comma-separated value (CSV) notation with a header defining the names of the parameters (including indices for multivariate parameters).2",
+    "text": "This chapter provides a sketch of how a compiled Stan model is executed using sampling. Optimization shares the same data reading and initialization steps, but then does optimization rather than sampling.\nThis sketch is elaborated in the following chapters of this part, which cover variable declarations, expressions, statements, and blocks in more detail.\n\n\nThe reading and transforming data steps are the same for sampling, optimization and diagnostics.\n\n\nThe first step of execution is to read data into memory. Data may be read in through file (in CmdStan) or through memory (RStan and PyStan); see their respective manuals for details.1\nAll of the variables declared in the data block will be read. If a variable cannot be read, the program will halt with a message indicating which data variable is missing.\nAfter each variable is read, if it has a declared constraint, the constraint is validated. For example, if a variable N is declared as int&lt;lower=0&gt;, after N is read, it will be tested to make sure it is greater than or equal to zero. If a variable violates its declared constraint, the program will halt with a warning message indicating which variable contains an illegal value, the value that was read, and the constraint that was declared.\n\n\n\nAfter data is read into the model, the transformed data variable statements are executed in order to define the transformed data variables. As the statements execute, declared constraints on variables are not enforced.\nTransformed data variables are initialized with real values set to NaN and integer values set to the smallest integer (large absolute value negative number).\nAfter the statements are executed, all declared constraints on transformed data variables are validated. If the validation fails, execution halts and the variable’s name, value and constraints are displayed.\n\n\n\n\nInitialization is the same for all of Stan’s algorithms.\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.\n\n\n\n\nSampling is based on simulating the Hamiltonian of a particle with a starting position equal to the current parameter values and an initial momentum (kinetic energy) generated randomly. The potential energy at work on the particle is taken to be the negative log (unnormalized) total probability function defined by the model. In the usual approach to implementing HMC, the Hamiltonian dynamics of the particle is simulated using the leapfrog integrator, which discretizes the smooth path of the particle into a number of small time steps called leapfrog steps.\n\n\nFor each leapfrog step, the negative log probability function and its gradient need to be evaluated at the position corresponding to the current parameter values (a more detailed sketch is provided in the next section). These are used to update the momentum based on the gradient and the position based on the momentum.\nFor simple models, only a few leapfrog steps with large step sizes are needed. For models with complex posterior geometries, many small leapfrog steps may be needed to accurately model the path of the parameters.\nIf the user specifies the number of leapfrog steps (i.e., chooses to use standard HMC), that number of leapfrog steps are simulated. If the user has not specified the number of leapfrog steps, the No-U-Turn sampler (NUTS) will determine the number of leapfrog steps adaptively (Hoffman and Gelman 2014).\n\n\n\nDuring each leapfrog step, the log probability function and its gradient must be calculated. This is where most of the time in the Stan algorithm is spent. This log probability function, which is used by the sampling algorithm, is defined over the unconstrained parameters.\nThe first step of the calculation requires the inverse transform of the unconstrained parameter values back to the constrained parameters in terms of which the model is defined. There is no error checking required because the inverse transform is a total function on every point in whose range satisfies the constraints.\nBecause the probability statements in the model are defined in terms of constrained parameters, the log Jacobian of the inverse transform must be added to the accumulated log probability.\nNext, the transformed parameter statements are executed. After they complete, any constraints declared for the transformed parameters are checked. If the constraints are violated, the model will halt with a diagnostic error message.\nThe final step in the log probability function calculation is to execute the statements defined in the model block.\nAs the log probability function executes, it accumulates an in-memory representation of the expression tree used to calculate the log probability. This includes all of the transformed parameter operations and all of the Jacobian adjustments. This tree is then used to evaluate the gradients by propagating partial derivatives backward along the expression graph. The gradient calculations account for the majority of the cycles consumed by a Stan program.\n\n\n\nA standard Metropolis accept/reject step is required to retain detailed balance and ensure draws are marginally distributed according to the probability function defined by the model. This Metropolis adjustment is based on comparing log probabilities, here defined by the Hamiltonian, which is the sum of the potential (negative log probability) and kinetic (squared momentum) energies. In theory, the Hamiltonian is invariant over the path of the particle and rejection should never occur. In practice, the probability of rejection is determined by the accuracy of the leapfrog approximation to the true trajectory of the parameters.\nIf step sizes are small, very few updates will be rejected, but many steps will be required to move the same distance. If step sizes are large, more updates will be rejected, but fewer steps will be required to move the same distance. Thus a balance between effort and rejection rate is required. If the user has not specified a step size, Stan will tune the step size during warmup sampling to achieve a desired rejection rate (thus balancing rejection versus number of steps).\nIf the proposal is accepted, the parameters are updated to their new values. Otherwise, the sample is the current set of parameter values.\n\n\n\n\nOptimization runs very much like sampling in that it starts by reading the data and then initializing parameters. Unlike sampling, it produces a deterministic output which requires no further analysis other than to verify that the optimizer itself converged to a posterior mode. The output for optimization is also similar to that for sampling.\n\n\n\nVariational inference also runs similar to sampling. It begins by reading the data and initializing the algorithm. The initial variational approximation is a random draw from the standard normal distribution in the unconstrained (real-coordinate) space. Again, similar to sampling, it outputs draws from the approximate posterior once the algorithm has decided that it has converged. Thus, the tools we use for analyzing the result of Stan’s sampling routines can also be used for variational inference.\n\n\n\nModel diagnostics are like sampling and optimization in that they depend on a model’s data being read and its parameters being initialized. The user’s guides for the interfaces (RStan, PyStan, CmdStan) provide more details on the diagnostics available; as of Stan 2.0, that’s just gradients on the unconstrained scale and log probabilities.\n\n\n\nFor each final draw (not counting draws during warmup or draws that are thinned), there is an output stage of writing the draw.\n\n\nBefore generating any output, the statements in the generated quantities block are executed. This can be used for any forward simulation based on parameters of the model. Or it may be used to transform parameters to an appropriate form for output.\nAfter the generated quantities statements execute, the constraints declared on generated quantities variables are validated. If these constraints are violated, the program will terminate with a diagnostic message.\n\n\n\nThe final step is to write the actual values. The values of all variables declared as parameters, transformed parameters, or generated quantities are written. Local variables are not written, nor is the data or transformed data. All values are written in their constrained forms, that is the form that is used in the model definitions.\nIn the executable form of a Stan models, parameters, transformed parameters, and generated quantities are written to a file in comma-separated value (CSV) notation with a header defining the names of the parameters (including indices for multivariate parameters).2",
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-    "text": "Initialization is the same for sampling, optimization, and diagnosis\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.",
+    "text": "Initialization is the same for all of Stan’s algorithms.\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.",
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-    "text": "Stan allows one file to be included within another file using a syntax similar to that from C++. For example, suppose the file my-std-normal.stan defines the standard normal log probability density function (up to an additive constant).\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nSuppose we also have a file containing a Stan program with an include statement.\n#include my-std-normal.stan\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThis Stan program behaves as if the contents of the file my-std-normal.stan replace the line with the #include statement, behaving as if a single Stan program were provided.\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThere are no restrictions on where include statements may be placed within a file or what the contents are of the replaced file.\n\n\nIt is possible to use includes on a line non-initially. For example, the previous example could’ve included space before the # in the include line:\n    #include my-std-normal.stan\nparameters {\n// ...\nIf there is initial space before an include, it will be discarded.\n\n\n\nIt is also possible to include line-based comments after the include. For example, the previous example can be coded as:\n#include my-std-normal.stan  // definition of standard normal\nparameters {\n// ...\nLine comments are discarded when the entire line is replaced with the contents of the included file.\n\n\n\nRecursive includes will be ignored. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThe result of processing this file will be empty, because a.stan will include b.stan, from which the include of a.stan is ignored and a warning printed.\n\n\n\nThe Stan interfaces may provide a mechanism for specifying a sequence of system paths in which to search for include files. The file included is the first one that is found in the sequence.\n\n\nIf there is not a final / or \\ in the path, a / will be appended between the path and the included file name.",
+    "text": "Stan allows one file to be included within another file using a syntax similar to that from C++. For example, suppose the file my-std-normal.stan defines the standard normal log probability density function (up to an additive constant).\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nSuppose we also have a file containing a Stan program with an include statement.\n#include my-std-normal.stan\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThis Stan program behaves as if the contents of the file my-std-normal.stan replace the line with the #include statement, behaving as if a single Stan program were provided.\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThere are no restrictions on where include statements may be placed within a file or what the contents are of the replaced file.\n\n\nIt is possible to use includes on a line non-initially. For example, the previous example could’ve included space before the # in the include line:\n    #include my-std-normal.stan\nparameters {\n// ...\nIf there is initial space before an include, it will be discarded.\n\n\n\nIt is also possible to include line-based comments after the include. For example, the previous example can be coded as:\n#include my-std-normal.stan  // definition of standard normal\nparameters {\n// ...\nLine comments are discarded when the entire line is replaced with the contents of the included file.\n\n\n\nRecursive includes will lead to a compiler error. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThis will result in an error explaining the circular dependency:\nSyntax error in './b.stan', line 1, column 0, included from\n'./a.stan', line 1, column 0, included from\n'./b.stan', line 1, column 0, included from\n'a.stan', line 1, column 0, include error:\n   -------------------------------------------------\n     1:  #include a.stan\n         ^\n   -------------------------------------------------\n\nFile a.stan recursively included itself.\n\n\n\nThe Stan interfaces may provide a mechanism for specifying a sequence of system paths in which to search for include files. The file included is the first one that is found in the sequence.\n\n\nIf there is not a final / or \\ in the path, a / will be appended between the path and the included file name.",
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-    "text": "Recursive includes will be ignored. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThe result of processing this file will be empty, because a.stan will include b.stan, from which the include of a.stan is ignored and a warning printed.",
+    "text": "Recursive includes will lead to a compiler error. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThis will result in an error explaining the circular dependency:\nSyntax error in './b.stan', line 1, column 0, included from\n'./a.stan', line 1, column 0, included from\n'./b.stan', line 1, column 0, included from\n'a.stan', line 1, column 0, include error:\n   -------------------------------------------------\n     1:  #include a.stan\n         ^\n   -------------------------------------------------\n\nFile a.stan recursively included itself.",
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-    "text": "The blocks of a Stan program are made up of variable declarations and statements; see the blocks chapter for details. Unlike programs in BUGS, the declarations and statements making up a Stan program are executed in the order in which they are written. Variables must be defined to have some value (as well as declared to have some type) before they are used — if they do not, the behavior is undefined.\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters. Log probability function can be constructed by using assignment statements. Statements may be grouped into sequences and into for-each loops. In addition, Stan allows local variables to be declared in blocks and also allows an empty statement consisting only of a semicolon.\n\n\nThe data and parameters blocks do not allow statements of any kind because these blocks are solely used to declare the data variables for input and the parameter variables for sampling. All other blocks allow statements. In these blocks, both variable declarations and statements are allowed. All top-level variables in a block are considered block variables. See the blocks chapter for more information about the block structure of Stan programs.\n\n\n\nAn assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,\n\na variable, or\na variable with one or more indices.\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y\n\n\n\n\n\n\n\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.1\ntarget += -0.5 * y * y;\nThe keyword target here is actually not a variable, and may not be accessed as such (though see below on how to access the value of target through a special function).\nIn this example, the unnormalized log probability of a unit normal variable \\(y\\) is added to the total log probability. In the general case, the argument can be any expression.2\nAn entire Stan model can be implemented this way. For instance, the following model will draw a single variable according to a unit normal probability.\nparameters {\n  real y;\n}\nmodel {\n  target += -0.5 * y * y;\n}\nThis model defines a log probability function\n\\[\n\\log p(y) = - \\, \\frac{y^2}{2} - \\log Z\n\\]\nwhere \\(Z\\) is a normalizing constant that does not depend on \\(y\\). The constant \\(Z\\) is conventionally written this way because on the linear scale, \\[\np(y) = \\frac{1}{Z} \\exp\\left(-\\frac{y^2}{2}\\right).\n\\] which is typically written without reference to \\(Z\\) as \\[\np(y) \\propto \\exp\\left(-\\frac{y^2}{2}\\right).\n\\]\nStan only requires models to be defined up to a constant that does not depend on the parameters. This is convenient because often the normalizing constant \\(Z\\) is either time-consuming to compute or intractable to evaluate.\n\n\nThe built in distribution functions in Stan are all available in normalized and unnormalized form. The normalized forms include all of the terms in the log density, and the unnormalized forms drop terms which are not directly or indirectly a function of the model parameters.\nFor instance, the normal_lpdf function returns the log density of a normal distribution:\n\\[\n\\textsf{normal\\_lpdf}(x | \\mu, \\sigma) =\n-\\log \\left( \\sigma \\sqrt{2 \\pi} \\right)\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nThe normal_lupdf function returns the log density of an unnormalized distribution. With the unnormalized version of the function, Stan does not define what the normalization constant will be, though usually as many terms as possible are dropped to make the calculation fast. Dropping a constant sigma term, normal_lupdf would be equivalent to:\n\\[\n\\textsf{normal\\_lupdf}(x | \\mu, \\sigma) =\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nAll functions ending in _lpdf have a corresponding _lupdf version which evaluates and returns the unnormalized density. The same is true for _lpmf and _lupmf.\n\n\n\nThe increment log density statement looks syntactically like compound addition and assignment (see the compound arithmetic/assignment section, it is treated as a primitive statement because target is not itself a variable. So, even though\ntarget += lp;\nis a legal statement, the corresponding long form is not legal.\ntarget = target + lp;  // BAD, target is not a variable\n\n\n\nThe target += ... statement accepts an argument in place of ... for any expression type, including integers, reals, vectors, row vectors, matrices, and arrays of any dimensionality, including arrays of vectors and matrices. For container arguments, their sum will be added to the total log density.\n\n\n\nTo access accumulated log density up to the current execution point, the function target() may be used.\n\n\n\n\nStan supports writing probability statements also in sampling notation, such as\ny ~ normal(mu, sigma);\nThe name “sampling statement” is meant to be suggestive, not interpreted literally. Conceptually, the variable y, which may be an unknown parameter or known, modeled data, is being declared to have the distribution indicated by the right-hand side of the sampling statement. The above statement can read as “y is distributed as normal distribution with parameters mu and sigma.”. The symbol ~ is called tilde.\nExecuting such a statement does not perform any sampling. In Stan, a sampling statement is merely a notational convenience following the typical notation used to present models in the literature. The above sampling statement could be expressed as a direct increment on the total log probability density as\ntarget += normal_lpdf(y | mu, sigma);\nIn general, a sampling statement of the form\ny ~ dist(theta1, ..., thetaN);\ninvolving subexpressions y and theta1 through thetaN (including the case where N is zero) will be well formed if and only if the corresponding assignment statement is well-formed. For densities allowing real y values, the log probability density function is used,\ntarget += dist_lpdf(y | theta1, ..., thetaN);\nFor those restricted to integer y values, the log probability mass function is used,\ntarget += dist_lpmf(y | theta1, ..., thetaN);\nThis will be well formed if and only if dist_lpdf(y | theta1,   ..., thetaN) or dist_lpmf(y | theta1, ..., thetaN) is a well-formed expression of type real.\n\n\nAlthough both lead to the same sampling behavior in Stan, there is one critical difference between using the sampling statement, as in\ny ~ normal(mu, sigma);\nand explicitly incrementing the log probability function, as in\ntarget += normal_lpdf(y | mu, sigma);\nThe sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions, y, mu, or sigma, involve only constants, data variables, and transformed data variables.\n\n\n\nThe left-hand side of a sampling statement may be a complex expression. For instance, it is legal syntactically to write\nparameters {\n  real&lt;lower=0&gt; beta;\n}\n// ...\nmodel {\n  log(beta) ~ normal(mu, sigma);\n}\nUnfortunately, this is not enough to properly model beta as having a lognormal distribution. Whenever a nonlinear transform is applied to a parameter, such as the logarithm function being applied to beta here, and then used on the left-hand side of a sampling statement or on the left of a vertical bar in a log pdf function, an adjustment must be made to account for the differential change in scale and ensure beta gets the correct distribution. The correction required is to add the log Jacobian of the transform to the target log density; see the change of variables section for full definitions. For the case above, the following adjustment will account for the log transform.3\ntarget += - log(abs(y));\n\n\n\nStan supports truncating distributions with lower bounds, upper bounds, or both.\n\n\nA probability density function \\(p(x)\\) for a continuous distribution may be truncated to an interval \\([a, b]\\) to define a new density \\(p_{[a, b]}(x)\\) with support \\([a, b]\\) by setting\n\\[\np_{[a, b]}(x)\n= \\frac{p(x)}\n       {\\int_a^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) for a discrete distribution may be truncated to the closed interval \\([a, b]\\) by\n\\[\np_{[a, b]}(x) = \\frac{p(x)}\n                  {\\sum_{u = a}^b p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x)\n= \\frac{p(x)}\n       {\\int_a^{\\infty} p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x) = \\frac{p(x)}\n                  {\\sum_{a &lt;= u} p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty, b]}(x)\n= \\frac{p(x)}\n       {\\int_{-\\infty}^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty,b]}(x) = \\frac{p(x)}\n                  {\\sum_{u &lt;= b} p(u)}.\n\\]\n\n\n\nGiven a probability function \\(p_X(x)\\) for a random variable \\(X\\), its cumulative distribution function (cdf) \\(F_X(x)\\) is defined to be the probability that \\(X \\leq x\\),\n\\[\nF_X(x) = \\Pr[X \\leq x].\n\\]\nThe upper-case variable \\(X\\) is the random variable whereas the lower-case variable \\(x\\) is just an ordinary bound variable. For continuous random variables, the definition of the cdf works out to\n\\[\nF_X(x) \\ = \\ \\int_{-\\infty}^{x} p_X(u) \\, du,\n\\]\nFor discrete variables, the cdf is defined to include the upper bound given by the argument,\n\\[\nF_X(x) = \\sum_{u \\leq x} p_X(u).\n\\]\n\n\n\nThe complementary cumulative distribution function (ccdf) in both the continuous and discrete cases is given by\n\\[\nF^C_X(x)\n\\ = \\ \\Pr[X &gt; x]\n\\ = \\ 1 - F_X(x).\n\\]\nUnlike the cdf, the ccdf is exclusive of the bound, hence the event \\(X &gt; x\\) rather than the cdf’s event \\(X \\leq x\\).\nFor continuous distributions, the ccdf works out to\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\int_{-\\infty}^x p_X(u) \\, du\n\\ = \\ \\int_x^{\\infty} p_X(u) \\, du.\n\\]\nThe lower boundary can be included in the integration bounds because it is a single point on a line and hence has no probability mass. For the discrete case, the lower bound must be excluded in the summation explicitly by summing over \\(u &gt; x\\),\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\sum_{u \\leq x} p_X(u)\n\\ = \\ \\sum_{u &gt; x} p_X(u).\n\\]\nCumulative distribution functions provide the necessary integral calculations to define truncated distributions. For truncation with lower and upper bounds, the denominator is defined by \\[\n\\int_a^b p(u) \\, du = F_X(b) - F_X(a).\n\\] This allows truncated distributions to be defined as \\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a)}.\n\\]\nFor discrete distributions, a slightly more complicated form is required to explicitly insert the lower truncation point, which is otherwise excluded from \\(F_X(b) - F_X(a)\\),\n\\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a) + p_X(a)}.\n\\]\n\n\n\nStan allows probability functions to be truncated. For example, a truncated unit normal distributions restricted to \\([-0.5, 2.1]\\) can be coded with the following sampling statement.\ny ~ normal(0, 1) T[-0.5, 2.1];\nTruncated distributions are translated as an additional term in the accumulated log density function plus error checking to make sure the variate in the sampling statement is within the bounds of the truncation.\nIn general, the truncation bounds and parameters may be parameters or local variables.\nBecause the example above involves a continuous distribution, it behaves the same way as the following more verbose form.\ny ~ normal(0, 1);\nif (y &lt; -0.5 || y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -log_diff_exp(normal_lcdf(2.1 | 0, 1),\n}\n                          normal_lcdf(-0.5 | 0, 1));\nBecause a Stan program defines a log density function, all calculations are on the log scale. The function normal_lcdf is the log of the cumulative normal distribution function and the function log_diff_exp(a, b) is a more arithmetically stable form of log(exp(a) - exp(b)).\nFor a discrete distribution, another term is necessary in the denominator to account for the excluded boundary. The truncated discrete distribution\ny ~ poisson(3.7) T[2, 10];\nbehaves in the same way as the following code.\ny ~ poisson(3.7);\nif (y &lt; 2 || y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         log_diff_exp(poisson_lcdf(10 | 3.7),\n                                      poisson_lcdf(2 | 3.7)));\n}\nRecall that log_sum_exp(a, b) is just the arithmetically stable form of log(exp(a) + exp(b)).\n\n\n\nFor truncating with only a lower bound, the upper limit is left blank.\ny ~ normal(0, 1) T[-0.5, ];\nThis truncated sampling statement has the same behavior as the following code.\ny ~ normal(0, 1);\nif (y &lt; -0.5) {\n  target += negative_infinity();\n} else {\n  target += -normal_lccdf(-0.5 | 0, 1);\n}\nThe normal_lccdf function is the normal complementary cumulative distribution function.\nAs with lower and upper truncation, the discrete case requires a more complicated denominator to add back in the probability mass for the lower bound. Thus\ny ~ poisson(3.7) T[2, ];\nbehaves the same way as\ny ~ poisson(3.7);\nif (y &lt; 2) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         poisson_lccdf(2 | 3.7));\n}\n\n\n\nTo truncate with only an upper bound, the lower bound is left blank. The upper truncated sampling statement\ny ~ normal(0, 1) T[ , 2.1];\nproduces the same result as the following code.\ntarget += normal_lpdf(y | 0, 1);\nif (y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -normal_lcdf(2.1 | 0, 1);\n}\nWith only an upper bound, the discrete case does not need a boundary adjustment. The upper-truncated sampling statement\ny ~ poisson(3.7) T[ , 10];\nbehaves the same way as the following code.\ny ~ poisson(3.7);\nif (y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -poisson_lcdf(10 | 3.7);\n}\n\n\n\nIn all cases, the truncation is only well formed if the appropriate log density or mass function and necessary log cumulative distribution functions are defined. Not every distribution built into Stan has log cdf and log ccdfs defined, nor will every user-defined distribution. The discrete probability function documentations describes the available discrete and continuous cumulative distribution functions; most univariate distributions have log cdf and log ccdf functions.\n\n\n\nFor continuous distributions, truncation points must be expressions of type int or real. For discrete distributions, truncation points must be expressions of type int.\n\n\n\nFor a truncated sampling statement, if the value sampled is not within the bounds specified by the truncation expression, the result is zero probability and the entire statement adds \\(-\\infty\\) to the total log probability, which in turn results in the sample being rejected.\n\n\n\nVectorization of distribution functions with truncation is available if the underlying distribution, lcdf, and lccdf functions meet the required signatures.\nThe equivalent code for a vectorized truncation depends on which of the variables are non-scalars (arrays, vectors, etc.):\n\nIf the variate y is the only non-scalar, the result is the same as described in the above sections, but the lcdf/lccdf calculation is multiplied by size(y).\nIf the other arguments to the distribution are non-scalars, then the vectorized version of the lcdf/lccdf is used. These functions return the sum of their terms, so no multiplication by the size is needed.\nThe exception to the above is when a non-variate is a vector and both a lower and upper bound are specified in the truncation. In this case, a for loop is generated over the elements of the non-scalar arguments. This is required since the log_diff_exp of two sums is not the same as the sum of the pairwise log_diff_exp operations.\n\nNote that while a lower-and-upper truncated distribution may generate a for-loop internally as part of translating the truncation statement, this is still preferable to manually constructing a loop, since the distribution function itself can still be evaluated in a vectorized manner.\n\n\n\n\n\nSuppose N is a variable of type int, y is a one-dimensional array of type array[] real, and mu and sigma are variables of type real. Furthermore, suppose that n has not been defined as a variable. Then the following is a well-formed for-loop statement.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nThe loop variable is n, the loop bounds are the values in the range 1:N, and the body is the statement following the loop bounds.\n\n\nThe type of the loop variable is int. Unlike in C++ and similarly to R, this variable must not be declared explicitly.\nThe bounds in a for loop must be integers. Unlike in R, the loop is always interpreted as an upward counting loop. The range L:H will cause the loop to execute the loop with the loop variable taking on all integer values greater than or equal to L and less than or equal to H. For example, the loop for (n in 2:5) will cause the body of the for loop to be executed with n equal to 2, 3, 4, and 5, in order. The variable and bound for (n in   5:2) will not execute anything because there are no integers greater than or equal to 5 and less than or equal to 2.\nThe scope of the loop variable is limited to the body of the loop.\n\n\n\nUnlike in BUGS, Stan allows variables to be reassigned. For example, the variable theta in the following program is reassigned in each iteration of the loop.\nfor (n in 1:N) {\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nSuch reassignment is not permitted in BUGS. In BUGS, for loops are declarative, defining plates in directed graphical model notation, which can be thought of as repeated substructures in the graphical model. Therefore, it is illegal in BUGS or JAGS to have a for loop that repeatedly reassigns a value to a variable.4\nIn Stan, assignments are executed in the order they are encountered. As a consequence, the following Stan program has a very different interpretation than the previous one.\nfor (n in 1:N) {\n  y[n] ~ bernoulli(theta);\n  theta = inv_logit(alpha + x[n] * beta);\n}\nIn this program, theta is assigned after it is used in the probability statement. This presupposes it was defined before the first loop iteration (otherwise behavior is undefined), and then each loop uses the assignment from the previous iteration.\nStan loops may be used to accumulate values. Thus it is possible to sum the values of an array directly using code such as the following.\ntotal = 0.0;\nfor (n in 1:N) {\n  total = total + x[n];\n}\nAfter the for loop is executed, the variable total will hold the sum of the elements in the array x. This example was purely pedagogical; it is easier and more efficient to write\ntotal = sum(x);\nA variable inside (or outside) a loop may even be reassigned multiple times, as in the following legal code.\nfor (n in 1:100) {\n  y += y * epsilon;\n  epsilon = 0.5 * epsilon;\n  y += y * epsilon;\n}\n\n\n\n\nA second form of for loops allows iteration over elements of containers. If ys is an expression denoting a container (vector, row vector, matrix, or array) with elements of type T, then the following is a well-formed foreach statement.\nfor (y in ys) {\n  // ... do something with y ...\n}\nThe order in which elements of ys are visited is defined for container types as follows.\n\nvector, row_vector: elements visited in order, y is of type double\nmatrix: elements visited in column-major order, y is of type double\narray[] T: elements visited in order, y is of type T.\n\nConsequently, if ys is a two dimensional array array[,] real, y will be a one-dimensional array of real values (type array[] real). If ’ysis a matrix, thenywill be a real value (typereal`). To loop over all values of a two-dimensional array using foreach statements would require a doubly-nested loop,\narray[2, 3] real yss;\nfor (ys in yss) {\n  for (y in ys) {\n    // ... do something with y ...\n  }\n}\nwhereas a matrix can be looped over in one foreach statement\nmatrix[2, 3] yss;\nfor (y in yss) {\n   // ... do something with y...\n}\nIn both cases, the loop variable y is of type real.  The elements of the matrix are visited in column-major order (e.g.,y[1, 1],y[2, 1],y[1, 2], ...,y[2, 3]), whereas the elements of the two-dimensional array are visited in row-major order (e.g.,y[1, 1],y[1, 2],y[1, 3],y[2, 1], ...,y[2, 3]`).\n\n\n\nStan supports full conditional statements using the same if-then-else syntax as C++. The general format is\nif (condition1)\n  statement1\nelse if (condition2)\n  statement2\n// ...\nelse if (conditionN-1)\n  statementN-1\nelse\n  statementN\nThere must be a single leading if clause, which may be followed by any number of else if clauses, all of which may be optionally followed by an else clause. Each condition must be an integer value, with non-zero values interpreted as true and the zero value as false.\nThe entire sequence of if-then-else clauses forms a single conditional statement for evaluation. The conditions are evaluated in order until one of the conditions evaluates to a non-zero value, at which point its corresponding statement is executed and the conditional statement finishes execution. If none of the conditions evaluate to a non-zero value and there is a final else clause, its statement is executed.\n\n\n\nStan supports standard while loops using the same syntax as C++. The general format is as follows.\nwhile (condition)\n  body\nThe condition must be an integer expression and the body can be any statement (or sequence of statements in curly braces).\nEvaluation of a while loop starts by evaluating the condition. If the condition evaluates to a false (zero) value, the execution of the loop terminates and control moves to the position after the loop. If the loop’s condition evaluates to a true (non-zero) value, the body statement is executed, then the whole loop is executed again. Thus the loop is continually executed as long as the condition evaluates to a true value.\nThe rest of the body of a while loop may be skipped using a continue. The loop will be exited with a break statement. See the section on continue and break statements for more details.\n\n\n\nJust as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are\nint, real, vector[K], row_vector[K], matrix[M, N].\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.\n\n\n\n\nThe one-token statements continue and break may be used within loops to alter control flow; continue causes the next iteration of the loop to run immediately, whereas break terminates the loop and causes execution to resume after the loop. Both control structures must appear in loops. Both break and continue scope to the most deeply nested loop, but pass through non-loop statements.\nAlthough these control statements may seem undesirable because of their goto-like behavior, their judicious use can greatly improve readability by reducing the level of nesting or eliminating bookkeeping inside loops.\n\n\nWhen a break statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:\nwhile (1) {\n  if (n &lt; 0) {\n    break;\n  }\n  foo(n);\n  n = n - 1;\n}\nThe while~(1) loop is a “forever” loop, because 1 is the true value, so the test always succeeds. Within the loop, if the value of n is less than 0, the loop terminates, otherwise it executes foo(n) and then decrements n. The statement above does exactly the same thing as\nwhile (n &gt;= 0) {\n  foo(n);\n  n = n - 1;\n}\nThis case is simply illustrative of the behavior; it is not a case where a break simplifies the loop.\n\n\n\nThe continue statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array x,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &lt;= 0) {\n    continue;\n  }\n  sum += x[n];\n}\nWhen the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an x[n] less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &gt; 0) {\n    sum += x[n];\n  }\n}\nWhile the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.\n\n\n\nIf there is a loop nested within a loop, a break or continue statement only breaks out of the inner loop. So\nwhile (cond1) {\n  // ...\n  while (cond2) {\n    // ...\n    if (cond3) {\n      break;\n    }\n    // ...\n  }\n  // execution continues here after break\n  // ...\n}\nIf the break is triggered by cond3 being true, execution will continue after the nested loop.\nAs with break statements, continue statements go back to the top of the most deeply nested loop in which the continue appears.\nAlthough break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.\n\n\n\n\nStan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\n\n\n\n\nThe Stan reject statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.\nLike the print statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.\nReject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable x’s value is negative.\nif (x &lt; 0) {\n  reject(\"x must not be negative; found x=\", x);\n}\n\n\nReject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the normal_lpdf function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.\nIn all cases of rejection, the interface accessing the Stan program should print the arguments to the reject statement.\n\n\nRejections in user-defined functions are just passed to the calling function or program block. Reject statements can be used in functions to validate the function arguments, allowing user-defined functions to fully emulate built-in function behavior. It is better to find out earlier rather than later when there is a problem.\n\n\n\nIn both the transformed data block and generated quantities block, rejections are fatal. This is because if initialization fails or if generating output fails, there is no way to recover values.\nReject statements placed in the transformed data block can be used to validate both the data and transformed data (if any). This allows more complicated constraints to be enforced that can be specified with Stan’s constrained variable declarations.\n\n\n\nRejections in the transformed parameters and model blocks are not in and of themselves instantly fatal. The result has the same effect as assigning a \\(-\\infty\\) log probability, which causes rejection of the current proposal in MCMC samplers and adjustment of search parameters in optimization.\nIf the log probability function results in a rejection every time it is called, the containing application (MCMC sampler or optimization) should diagnose this problem and terminate with an appropriate error message. To aid in diagnosing problems, the message for each reject statement will be printed as a result of executing it.\n\n\n\n\nRejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;\n  real&lt;lower=a, upper=b&gt; theta;\n  // ...\n}\nmodel {\n  // **wrong** needs explicit truncation\n  theta ~ normal(0, 1);\n  // ...\n}\nThis program is wrong because its truncation bounds on theta depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.\n  theta ~ normal(0, 1) T[a, b];\nThe conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over \\([a ,b]\\); in these simple univariate cases, we can overcome that with the T[ , ] notation, which essentially divides by whatever the prior integrates to over \\([a, b]\\).\nThis problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.\n  if (theta &lt; a || theta &gt; b) {\n    reject(\"theta not in (a, b)\");\n  }\n  // still **wrong**, needs T[a,b]\n  theta ~ normal(0, 1);\nIn this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.\nEven if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.",
+    "text": "The blocks of a Stan program are made up of variable declarations and statements; see the blocks chapter for details. Unlike programs in BUGS, the declarations and statements making up a Stan program are executed in the order in which they are written. Variables must be defined to have some value (as well as declared to have some type) before they are used — if they do not, the behavior is undefined.\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters. Log probability function can be constructed by using assignment statements. Statements may be grouped into sequences and into for-each loops. In addition, Stan allows local variables to be declared in blocks and also allows an empty statement consisting only of a semicolon.\n\n\nThe data and parameters blocks do not allow statements of any kind because these blocks are solely used to declare the data variables for input and the parameter variables for sampling. All other blocks allow statements. In these blocks, both variable declarations and statements are allowed. All top-level variables in a block are considered block variables. See the blocks chapter for more information about the block structure of Stan programs.\n\n\n\nAn assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,\n\na variable, or\na variable with one or more indices, or\na comma separated list of lvalues surrounded by ( and )\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y\n\n\n\n\n\n\n\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.1\ntarget += -0.5 * y * y;\nThe keyword target here is actually not a variable, and may not be accessed as such (though see below on how to access the value of target through a special function).\nIn this example, the unnormalized log probability of a unit normal variable \\(y\\) is added to the total log probability. In the general case, the argument can be any expression.2\nAn entire Stan model can be implemented this way. For instance, the following model will draw a single variable according to a unit normal probability.\nparameters {\n  real y;\n}\nmodel {\n  target += -0.5 * y * y;\n}\nThis model defines a log probability function\n\\[\n\\log p(y) = - \\, \\frac{y^2}{2} - \\log Z\n\\]\nwhere \\(Z\\) is a normalizing constant that does not depend on \\(y\\). The constant \\(Z\\) is conventionally written this way because on the linear scale, \\[\np(y) = \\frac{1}{Z} \\exp\\left(-\\frac{y^2}{2}\\right).\n\\] which is typically written without reference to \\(Z\\) as \\[\np(y) \\propto \\exp\\left(-\\frac{y^2}{2}\\right).\n\\]\nStan only requires models to be defined up to a constant that does not depend on the parameters. This is convenient because often the normalizing constant \\(Z\\) is either time-consuming to compute or intractable to evaluate.\n\n\nThe built in distribution functions in Stan are all available in normalized and unnormalized form. The normalized forms include all of the terms in the log density, and the unnormalized forms drop terms which are not directly or indirectly a function of the model parameters.\nFor instance, the normal_lpdf function returns the log density of a normal distribution:\n\\[\n\\textsf{normal\\_lpdf}(x | \\mu, \\sigma) =\n-\\log \\left( \\sigma \\sqrt{2 \\pi} \\right)\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nThe normal_lupdf function returns the log density of an unnormalized distribution. With the unnormalized version of the function, Stan does not define what the normalization constant will be, though usually as many terms as possible are dropped to make the calculation fast. Dropping a constant sigma term, normal_lupdf would be equivalent to:\n\\[\n\\textsf{normal\\_lupdf}(x | \\mu, \\sigma) =\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nAll functions ending in _lpdf have a corresponding _lupdf version which evaluates and returns the unnormalized density. The same is true for _lpmf and _lupmf.\n\n\n\nThe increment log density statement looks syntactically like compound addition and assignment (see the compound arithmetic/assignment section, it is treated as a primitive statement because target is not itself a variable. So, even though\ntarget += lp;\nis a legal statement, the corresponding long form is not legal.\ntarget = target + lp;  // BAD, target is not a variable\n\n\n\nThe target += ... statement accepts an argument in place of ... for any expression type, including integers, reals, vectors, row vectors, matrices, and arrays of any dimensionality, including arrays of vectors and matrices. For container arguments, their sum will be added to the total log density.\n\n\n\nTo access accumulated log density up to the current execution point, the function target() may be used.\n\n\n\n\nStan supports writing probability statements also in sampling notation, such as\ny ~ normal(mu, sigma);\nThe name “sampling statement” is meant to be suggestive, not interpreted literally. Conceptually, the variable y, which may be an unknown parameter or known, modeled data, is being declared to have the distribution indicated by the right-hand side of the sampling statement. The above statement can read as “y is distributed as normal distribution with parameters mu and sigma.”. The symbol ~ is called tilde.\nExecuting such a statement does not perform any sampling. In Stan, a sampling statement is merely a notational convenience following the typical notation used to present models in the literature. The above sampling statement could be expressed as a direct increment on the total log probability density as\ntarget += normal_lpdf(y | mu, sigma);\nIn general, a sampling statement of the form\ny ~ dist(theta1, ..., thetaN);\ninvolving subexpressions y and theta1 through thetaN (including the case where N is zero) will be well formed if and only if the corresponding assignment statement is well-formed. For densities allowing real y values, the log probability density function is used,\ntarget += dist_lpdf(y | theta1, ..., thetaN);\nFor those restricted to integer y values, the log probability mass function is used,\ntarget += dist_lpmf(y | theta1, ..., thetaN);\nThis will be well formed if and only if dist_lpdf(y | theta1,   ..., thetaN) or dist_lpmf(y | theta1, ..., thetaN) is a well-formed expression of type real.\n\n\nAlthough both lead to the same sampling behavior in Stan, there is one critical difference between using the sampling statement, as in\ny ~ normal(mu, sigma);\nand explicitly incrementing the log probability function, as in\ntarget += normal_lpdf(y | mu, sigma);\nThe sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions, y, mu, or sigma, involve only constants, data variables, and transformed data variables.\n\n\n\nThe left-hand side of a sampling statement may be a complex expression. For instance, it is legal syntactically to write\nparameters {\n  real&lt;lower=0&gt; beta;\n}\n// ...\nmodel {\n  log(beta) ~ normal(mu, sigma);\n}\nUnfortunately, this is not enough to properly model beta as having a lognormal distribution. Whenever a nonlinear transform is applied to a parameter, such as the logarithm function being applied to beta here, and then used on the left-hand side of a sampling statement or on the left of a vertical bar in a log pdf function, an adjustment must be made to account for the differential change in scale and ensure beta gets the correct distribution. The correction required is to add the log Jacobian of the transform to the target log density; see the change of variables section for full definitions. For the case above, the following adjustment will account for the log transform.3\ntarget += - log(abs(y));\n\n\n\nStan supports truncating distributions with lower bounds, upper bounds, or both.\n\n\nA probability density function \\(p(x)\\) for a continuous distribution may be truncated to an interval \\([a, b]\\) to define a new density \\(p_{[a, b]}(x)\\) with support \\([a, b]\\) by setting\n\\[\np_{[a, b]}(x)\n= \\frac{p(x)}\n       {\\int_a^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) for a discrete distribution may be truncated to the closed interval \\([a, b]\\) by\n\\[\np_{[a, b]}(x) = \\frac{p(x)}\n                  {\\sum_{u = a}^b p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x)\n= \\frac{p(x)}\n       {\\int_a^{\\infty} p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x) = \\frac{p(x)}\n                  {\\sum_{a &lt;= u} p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty, b]}(x)\n= \\frac{p(x)}\n       {\\int_{-\\infty}^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty,b]}(x) = \\frac{p(x)}\n                  {\\sum_{u &lt;= b} p(u)}.\n\\]\n\n\n\nGiven a probability function \\(p_X(x)\\) for a random variable \\(X\\), its cumulative distribution function (cdf) \\(F_X(x)\\) is defined to be the probability that \\(X \\leq x\\),\n\\[\nF_X(x) = \\Pr[X \\leq x].\n\\]\nThe upper-case variable \\(X\\) is the random variable whereas the lower-case variable \\(x\\) is just an ordinary bound variable. For continuous random variables, the definition of the cdf works out to\n\\[\nF_X(x) \\ = \\ \\int_{-\\infty}^{x} p_X(u) \\, du,\n\\]\nFor discrete variables, the cdf is defined to include the upper bound given by the argument,\n\\[\nF_X(x) = \\sum_{u \\leq x} p_X(u).\n\\]\n\n\n\nThe complementary cumulative distribution function (ccdf) in both the continuous and discrete cases is given by\n\\[\nF^C_X(x)\n\\ = \\ \\Pr[X &gt; x]\n\\ = \\ 1 - F_X(x).\n\\]\nUnlike the cdf, the ccdf is exclusive of the bound, hence the event \\(X &gt; x\\) rather than the cdf’s event \\(X \\leq x\\).\nFor continuous distributions, the ccdf works out to\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\int_{-\\infty}^x p_X(u) \\, du\n\\ = \\ \\int_x^{\\infty} p_X(u) \\, du.\n\\]\nThe lower boundary can be included in the integration bounds because it is a single point on a line and hence has no probability mass. For the discrete case, the lower bound must be excluded in the summation explicitly by summing over \\(u &gt; x\\),\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\sum_{u \\leq x} p_X(u)\n\\ = \\ \\sum_{u &gt; x} p_X(u).\n\\]\nCumulative distribution functions provide the necessary integral calculations to define truncated distributions. For truncation with lower and upper bounds, the denominator is defined by \\[\n\\int_a^b p(u) \\, du = F_X(b) - F_X(a).\n\\] This allows truncated distributions to be defined as \\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a)}.\n\\]\nFor discrete distributions, a slightly more complicated form is required to explicitly insert the lower truncation point, which is otherwise excluded from \\(F_X(b) - F_X(a)\\),\n\\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a) + p_X(a)}.\n\\]\n\n\n\nStan allows probability functions to be truncated. For example, a truncated unit normal distributions restricted to \\([-0.5, 2.1]\\) can be coded with the following sampling statement.\ny ~ normal(0, 1) T[-0.5, 2.1];\nTruncated distributions are translated as an additional term in the accumulated log density function plus error checking to make sure the variate in the sampling statement is within the bounds of the truncation.\nIn general, the truncation bounds and parameters may be parameters or local variables.\nBecause the example above involves a continuous distribution, it behaves the same way as the following more verbose form.\ny ~ normal(0, 1);\nif (y &lt; -0.5 || y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -log_diff_exp(normal_lcdf(2.1 | 0, 1),\n}\n                          normal_lcdf(-0.5 | 0, 1));\nBecause a Stan program defines a log density function, all calculations are on the log scale. The function normal_lcdf is the log of the cumulative normal distribution function and the function log_diff_exp(a, b) is a more arithmetically stable form of log(exp(a) - exp(b)).\nFor a discrete distribution, another term is necessary in the denominator to account for the excluded boundary. The truncated discrete distribution\ny ~ poisson(3.7) T[2, 10];\nbehaves in the same way as the following code.\ny ~ poisson(3.7);\nif (y &lt; 2 || y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         log_diff_exp(poisson_lcdf(10 | 3.7),\n                                      poisson_lcdf(2 | 3.7)));\n}\nRecall that log_sum_exp(a, b) is just the arithmetically stable form of log(exp(a) + exp(b)).\n\n\n\nFor truncating with only a lower bound, the upper limit is left blank.\ny ~ normal(0, 1) T[-0.5, ];\nThis truncated sampling statement has the same behavior as the following code.\ny ~ normal(0, 1);\nif (y &lt; -0.5) {\n  target += negative_infinity();\n} else {\n  target += -normal_lccdf(-0.5 | 0, 1);\n}\nThe normal_lccdf function is the normal complementary cumulative distribution function.\nAs with lower and upper truncation, the discrete case requires a more complicated denominator to add back in the probability mass for the lower bound. Thus\ny ~ poisson(3.7) T[2, ];\nbehaves the same way as\ny ~ poisson(3.7);\nif (y &lt; 2) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         poisson_lccdf(2 | 3.7));\n}\n\n\n\nTo truncate with only an upper bound, the lower bound is left blank. The upper truncated sampling statement\ny ~ normal(0, 1) T[ , 2.1];\nproduces the same result as the following code.\ntarget += normal_lpdf(y | 0, 1);\nif (y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -normal_lcdf(2.1 | 0, 1);\n}\nWith only an upper bound, the discrete case does not need a boundary adjustment. The upper-truncated sampling statement\ny ~ poisson(3.7) T[ , 10];\nbehaves the same way as the following code.\ny ~ poisson(3.7);\nif (y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -poisson_lcdf(10 | 3.7);\n}\n\n\n\nIn all cases, the truncation is only well formed if the appropriate log density or mass function and necessary log cumulative distribution functions are defined. Not every distribution built into Stan has log cdf and log ccdfs defined, nor will every user-defined distribution. The discrete probability function documentations describes the available discrete and continuous cumulative distribution functions; most univariate distributions have log cdf and log ccdf functions.\n\n\n\nFor continuous distributions, truncation points must be expressions of type int or real. For discrete distributions, truncation points must be expressions of type int.\n\n\n\nFor a truncated sampling statement, if the value sampled is not within the bounds specified by the truncation expression, the result is zero probability and the entire statement adds \\(-\\infty\\) to the total log probability, which in turn results in the sample being rejected.\n\n\n\nVectorization of distribution functions with truncation is available if the underlying distribution, lcdf, and lccdf functions meet the required signatures.\nThe equivalent code for a vectorized truncation depends on which of the variables are non-scalars (arrays, vectors, etc.):\n\nIf the variate y is the only non-scalar, the result is the same as described in the above sections, but the lcdf/lccdf calculation is multiplied by size(y).\nIf the other arguments to the distribution are non-scalars, then the vectorized version of the lcdf/lccdf is used. These functions return the sum of their terms, so no multiplication by the size is needed.\nThe exception to the above is when a non-variate is a vector and both a lower and upper bound are specified in the truncation. In this case, a for loop is generated over the elements of the non-scalar arguments. This is required since the log_diff_exp of two sums is not the same as the sum of the pairwise log_diff_exp operations.\n\nNote that while a lower-and-upper truncated distribution may generate a for-loop internally as part of translating the truncation statement, this is still preferable to manually constructing a loop, since the distribution function itself can still be evaluated in a vectorized manner.\n\n\n\n\n\nSuppose N is a variable of type int, y is a one-dimensional array of type array[] real, and mu and sigma are variables of type real. Furthermore, suppose that n has not been defined as a variable. Then the following is a well-formed for-loop statement.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nThe loop variable is n, the loop bounds are the values in the range 1:N, and the body is the statement following the loop bounds.\n\n\nThe type of the loop variable is int. Unlike in C++ and similarly to R, this variable must not be declared explicitly.\nThe bounds in a for loop must be integers. Unlike in R, the loop is always interpreted as an upward counting loop. The range L:H will cause the loop to execute the loop with the loop variable taking on all integer values greater than or equal to L and less than or equal to H. For example, the loop for (n in 2:5) will cause the body of the for loop to be executed with n equal to 2, 3, 4, and 5, in order. The variable and bound for (n in   5:2) will not execute anything because there are no integers greater than or equal to 5 and less than or equal to 2.\nThe scope of the loop variable is limited to the body of the loop.\n\n\n\nUnlike in BUGS, Stan allows variables to be reassigned. For example, the variable theta in the following program is reassigned in each iteration of the loop.\nfor (n in 1:N) {\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nSuch reassignment is not permitted in BUGS. In BUGS, for loops are declarative, defining plates in directed graphical model notation, which can be thought of as repeated substructures in the graphical model. Therefore, it is illegal in BUGS or JAGS to have a for loop that repeatedly reassigns a value to a variable.4\nIn Stan, assignments are executed in the order they are encountered. As a consequence, the following Stan program has a very different interpretation than the previous one.\nfor (n in 1:N) {\n  y[n] ~ bernoulli(theta);\n  theta = inv_logit(alpha + x[n] * beta);\n}\nIn this program, theta is assigned after it is used in the probability statement. This presupposes it was defined before the first loop iteration (otherwise behavior is undefined), and then each loop uses the assignment from the previous iteration.\nStan loops may be used to accumulate values. Thus it is possible to sum the values of an array directly using code such as the following.\ntotal = 0.0;\nfor (n in 1:N) {\n  total = total + x[n];\n}\nAfter the for loop is executed, the variable total will hold the sum of the elements in the array x. This example was purely pedagogical; it is easier and more efficient to write\ntotal = sum(x);\nA variable inside (or outside) a loop may even be reassigned multiple times, as in the following legal code.\nfor (n in 1:100) {\n  y += y * epsilon;\n  epsilon = 0.5 * epsilon;\n  y += y * epsilon;\n}\n\n\n\n\nA second form of for loops allows iteration over elements of containers. If ys is an expression denoting a container (vector, row vector, matrix, or array) with elements of type T, then the following is a well-formed foreach statement.\nfor (y in ys) {\n  // ... do something with y ...\n}\nThe order in which elements of ys are visited is defined for container types as follows.\n\nvector, row_vector: elements visited in order, y is of type double\nmatrix: elements visited in column-major order, y is of type double\narray[] T: elements visited in order, y is of type T.\n\nConsequently, if ys is a two dimensional array array[,] real, y will be a one-dimensional array of real values (type array[] real). If ’ysis a matrix, thenywill be a real value (typereal`). To loop over all values of a two-dimensional array using foreach statements would require a doubly-nested loop,\narray[2, 3] real yss;\nfor (ys in yss) {\n  for (y in ys) {\n    // ... do something with y ...\n  }\n}\nwhereas a matrix can be looped over in one foreach statement\nmatrix[2, 3] yss;\nfor (y in yss) {\n   // ... do something with y...\n}\nIn both cases, the loop variable y is of type real.  The elements of the matrix are visited in column-major order (e.g.,y[1, 1],y[2, 1],y[1, 2], ...,y[2, 3]), whereas the elements of the two-dimensional array are visited in row-major order (e.g.,y[1, 1],y[1, 2],y[1, 3],y[2, 1], ...,y[2, 3]`).\n\n\n\nStan supports full conditional statements using the same if-then-else syntax as C++. The general format is\nif (condition1)\n  statement1\nelse if (condition2)\n  statement2\n// ...\nelse if (conditionN-1)\n  statementN-1\nelse\n  statementN\nThere must be a single leading if clause, which may be followed by any number of else if clauses, all of which may be optionally followed by an else clause. Each condition must be an integer value, with non-zero values interpreted as true and the zero value as false.\nThe entire sequence of if-then-else clauses forms a single conditional statement for evaluation. The conditions are evaluated in order until one of the conditions evaluates to a non-zero value, at which point its corresponding statement is executed and the conditional statement finishes execution. If none of the conditions evaluate to a non-zero value and there is a final else clause, its statement is executed.\n\n\n\nStan supports standard while loops using the same syntax as C++. The general format is as follows.\nwhile (condition)\n  body\nThe condition must be an integer expression and the body can be any statement (or sequence of statements in curly braces).\nEvaluation of a while loop starts by evaluating the condition. If the condition evaluates to a false (zero) value, the execution of the loop terminates and control moves to the position after the loop. If the loop’s condition evaluates to a true (non-zero) value, the body statement is executed, then the whole loop is executed again. Thus the loop is continually executed as long as the condition evaluates to a true value.\nThe rest of the body of a while loop may be skipped using a continue. The loop will be exited with a break statement. See the section on continue and break statements for more details.\n\n\n\nJust as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are listed in the types table under “local”.\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.\n\n\n\n\nThe one-token statements continue and break may be used within loops to alter control flow; continue causes the next iteration of the loop to run immediately, whereas break terminates the loop and causes execution to resume after the loop. Both control structures must appear in loops. Both break and continue scope to the most deeply nested loop, but pass through non-loop statements.\nAlthough these control statements may seem undesirable because of their goto-like behavior, their judicious use can greatly improve readability by reducing the level of nesting or eliminating bookkeeping inside loops.\n\n\nWhen a break statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:\nwhile (1) {\n  if (n &lt; 0) {\n    break;\n  }\n  foo(n);\n  n = n - 1;\n}\nThe while~(1) loop is a “forever” loop, because 1 is the true value, so the test always succeeds. Within the loop, if the value of n is less than 0, the loop terminates, otherwise it executes foo(n) and then decrements n. The statement above does exactly the same thing as\nwhile (n &gt;= 0) {\n  foo(n);\n  n = n - 1;\n}\nThis case is simply illustrative of the behavior; it is not a case where a break simplifies the loop.\n\n\n\nThe continue statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array x,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &lt;= 0) {\n    continue;\n  }\n  sum += x[n];\n}\nWhen the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an x[n] less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &gt; 0) {\n    sum += x[n];\n  }\n}\nWhile the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.\n\n\n\nIf there is a loop nested within a loop, a break or continue statement only breaks out of the inner loop. So\nwhile (cond1) {\n  // ...\n  while (cond2) {\n    // ...\n    if (cond3) {\n      break;\n    }\n    // ...\n  }\n  // execution continues here after break\n  // ...\n}\nIf the break is triggered by cond3 being true, execution will continue after the nested loop.\nAs with break statements, continue statements go back to the top of the most deeply nested loop in which the continue appears.\nAlthough break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.\n\n\n\n\nStan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\nNote that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.\n\n\n\n\nThe Stan reject statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.\nLike the print statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.\nReject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable x’s value is negative.\nif (x &lt; 0) {\n  reject(\"x must not be negative; found x=\", x);\n}\n\n\nReject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the normal_lpdf function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.\nIn all cases of rejection, the interface accessing the Stan program should print the arguments to the reject statement.\n\n\nRejections in user-defined functions are just passed to the calling function or program block. Reject statements can be used in functions to validate the function arguments, allowing user-defined functions to fully emulate built-in function behavior. It is better to find out earlier rather than later when there is a problem.\n\n\n\nIn both the transformed data block and generated quantities block, rejections are fatal. This is because if initialization fails or if generating output fails, there is no way to recover values.\nReject statements placed in the transformed data block can be used to validate both the data and transformed data (if any). This allows more complicated constraints to be enforced that can be specified with Stan’s constrained variable declarations.\n\n\n\nRejections in the transformed parameters and model blocks are not in and of themselves instantly fatal. The result has the same effect as assigning a \\(-\\infty\\) log probability, which causes rejection of the current proposal in MCMC samplers and adjustment of search parameters in optimization.\nIf the log probability function results in a rejection every time it is called, the containing application (MCMC sampler or optimization) should diagnose this problem and terminate with an appropriate error message. To aid in diagnosing problems, the message for each reject statement will be printed as a result of executing it.\n\n\n\n\nRejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;\n  real&lt;lower=a, upper=b&gt; theta;\n  // ...\n}\nmodel {\n  // **wrong** needs explicit truncation\n  theta ~ normal(0, 1);\n  // ...\n}\nThis program is wrong because its truncation bounds on theta depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.\n  theta ~ normal(0, 1) T[a, b];\nThe conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over \\([a ,b]\\); in these simple univariate cases, we can overcome that with the T[ , ] notation, which essentially divides by whatever the prior integrates to over \\([a, b]\\).\nThis problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.\n  if (theta &lt; a || theta &gt; b) {\n    reject(\"theta not in (a, b)\");\n  }\n  // still **wrong**, needs T[a,b]\n  theta ~ normal(0, 1);\nIn this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.\nEven if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.",
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-    "text": "An assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,\n\na variable, or\na variable with one or more indices.\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y",
+    "text": "An assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,\n\na variable, or\na variable with one or more indices, or\na comma separated list of lvalues surrounded by ( and )\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y",
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-    "text": "Just as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are\nint, real, vector[K], row_vector[K], matrix[M, N].\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.",
+    "text": "Just as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are listed in the types table under “local”.\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.",
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     "href": "reference-manual/statements.html#print-statements.section",
     "title": "Statements",
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-    "text": "Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).",
+    "text": "Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\nNote that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.",
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     "title": "User-Defined Functions",
     "section": "",
-    "text": "Stan allows users to define their own functions. The basic syntax is a simplified version of that used in C and C++. This chapter specifies how functions are declared, defined, and used in Stan.\n\n\nUser-defined functions appear in a special function-definition block before all of the other program blocks.\nfunctions {\n   // ... function declarations and definitions ...\n}\ndata {\n  // ...\nFunction definitions and declarations may appear in any order. Forward declarations are allowed but not required.\n\n\n\nThe rules for function naming and function-argument naming are the same as for other variables; see the section on variables for more information on valid identifiers. For example,\nreal foo(real mu, real sigma);\ndeclares a function named foo with two argument variables of types real and real. The arguments are named mu and sigma, but that is not part of the declaration.\n\n\nMultiple user-defined functions may have the same name if they have different sequences of argument types. This is known as function overloading.\nFor example, the following two functions are both defined with the name add_up\nreal add_up(real a, real b){\n  return a + b;\n}\n\nreal add_up(real a, real b, real c){\n  return a + b + c;\n}\nThe return types of overloaded functions do not need to be the same. One could define an additional add_up function as follows\nint add_up(int a, int b){\n  return a + b;\n}\nThat being said, functions may not use the same name if their signature only differs by the return type.\nFor example, the following is not permitted\n// illegal\nreal baz(int x);\nint baz(int x);\nFunction names used in the Stan standard library may be overloaded by user-defined functions. Exceptions to this are the reduce_sum family of functions and ODE integrators, which cannot be overloaded.\n\n\n\n\nAll function arguments are mandatory—there are no default values.\n\n\nFunctions with non-void return types are called just like any other built-in function in Stan—they are applied to appropriately typed arguments to produce an expression, which has a value when executed.\n\n\n\nFunctions with void return types may be applied to arguments and used as statements. These act like sampling statements or print statements. Such uses are only appropriate for functions that act through side effects, such as incrementing the log probability accumulator, printing, or raising exceptions.\n\n\n\nOverloaded functions alongside type promotion can result in situations where there are multiple valid interpretations of a function call. Stan requires that there be a unique signature which minimizes the number of promotions required.\nConsider the following two overloaded functions\nreal foo(int a, real b);\nreal foo(real a, int b);\nThese functions do not have a unique minimum when called with two integer arguments foo(1,2), and therefore cannot be called as such.\nPromotion of integers to complex numbers is considered as two separate promotions, one from int to real and a second from real to complex. Consider the following functions with real and complex signatures\nreal bar(real x);\nreal bar(complex z);\nA call bar(5) with an integer argument will be resolved to bar(real) because it only requires a single promotion, whereas the promotion to a complex number requires two promotions.\n\n\n\nThe rules for calling functions work the same way as assignment as far as promotion goes. This means that we can promote arguments to the type expected by function arguments. For example, the following will work.\nreal foo(real x) { return ... };\n...\nint a = 5;\nreal b = foo(a); // a promoted to type real\nIn addition to promoting int to real, Stan also promotes real to complex, and by transitivity, int to complex. This also works for containers, so an array of int may be assigned to an array of real of the same shape. And we can also promote vector to complex_vector and similarly for row vectors and matrices.\n\n\n\nFunctions whose name ends in _lpdf or _lpmf (density and mass functions) may be used as probability functions and may be used in place of parameterized distributions on the right-hand-side of sampling statements.\n\n\n\nFunctions of certain types are restricted on scope of usage. Functions whose names end in _lp assume access to the log probability accumulator and are only available in the transformed parameter and model blocks.\nFunctions whose names end in _rng assume access to the random number generator and may only be used within the generated quantities block, transformed data block, and within user-defined functions ending in _rng.\nFunctions whose names end in _lpdf and _lpmf can be used anywhere. However, _lupdf and _lupmf functions can only be used in the model block or user-defined probability functions.\nSee the section on function bodies for more information on these special types of function.\n\n\n\n\nStan’s functions all have declared types for both arguments and returned value. As with built-in functions, user-defined functions are only declared for base argument type and dimensionality. This requires a different syntax than for declaring other variables. The choice of language was made so that return types and argument types could use the same declaration syntax.\nThe type void may not be used as an argument type, only a return type for a function with side effects.\n\n\nThe base variable types are integer, real, complex, vector, row_vector, and matrix. No lower-bound or upper-bound constraints are allowed (e.g., real&lt;lower=0&gt; is illegal). Specialized constrained types are also not allowed (e.g., simplex is illegal).\nTuple types of the form tuple(T1, ..., TN) are also allowed, with all of the types T1 to TN being function argument types (i.e., no constraints and no sizes).\n\n\n\nArguments and return types may be arrays, and these are indicated with optional brackets and commas as would be used for indexing. For example, int denotes a single integer argument or return, whereas array[] real indicates a one-dimensional array of reals, array[,] real a two-dimensional array and array[,,] real a three-dimensional array; whitespace is optional, as usual.\nThe dimensions for vectors and matrices are not included, so that matrix is the type of a single matrix argument or return type. Thus if a variable is declared as matrix a, then a has two indexing dimensions, so that a[1] is a row vector and a[1, 1] a real value. Matrices implicitly have two indexing dimensions. The type declaration matrix[ , ] b specifies that b is a two-dimensional array of matrices, for a total of four indexing dimensions, with b[1, 1, 1, 1] picking out a real value.\n\n\n\nFunction argument and return types are not themselves checked for dimensionality. A matrix of any size may be passed in as a matrix argument. Nevertheless, a user-defined function might call a function (such as a multivariate normal density) that itself does dimensionality checks.\nDimensions of function return values will be checked if they’re assigned to a previously declared variable. They may also be checked if they are used as the argument to a function.\nAny errors raised by calls to functions inside user functions or return type mismatches are simply passed on; this typically results in a warning message and rejection of a proposal during sampling or optimization.\n\n\n\nSome of Stan’s built-in functions, like the differential equation solvers, have arguments that must be data. Such data-only arguments must be expressions involving only data, transformed data, and generated quantity variables.\nIn user-defined functions, the qualifier data may be placed before an argument type declaration to indicate that the argument must be data only. For example,\nreal foo(data real x) {\n  return x^2;\n}\nrequires the argument x to be data only.\nDeclaring an argument data only allows type inference to proceed in the body of the function so that, for example, the variable may be used as a data-only argument to a built-in function.\n\n\n\n\nThe body of a function is between an open curly brace ({) and close curly brace (}). The body may contain local variable declarations at the top of the function body’s block and these scope the same way as local variables used in any other statement block.\nAny user-defined function may be used in the function body regardless of the order in which the function definitions appear in the file. Self-recursive and mutually recursive functions are possible without any additional declarations.\nThe only restrictions on statements in function bodies are external, and determine whether the log probability accumulator or random number generators are available; see the rest of this section for details.\n\n\nFunctions that call random number generating functions in their bodies must have a name that ends in _rng; attempts to use random-number generators in other functions lead to a compile-time error.\nLike other random number generating functions, user-defined functions with names that end in _rng may be used only in the generated quantities block and transformed data block, or within the bodies of user-defined functions ending in _rng. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions that include sampling statements or log probability increment statements must have a name that ends in _lp. Attempts to use sampling statements or increment log probability statements in other functions lead to a compile-time error.\nLike the target log density increment statement and sampling statements, user-defined functions with names that end in _lp may only be used in blocks where the log probability accumulator is accessible, namely the transformed parameters and model blocks. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions whose names end in _lpdf and _lpmf (density and mass functions) can be used as probability functions in sampling statements. As with the built-in functions, the first argument will appear on the left of the sampling statement operator (~) in the sampling statement and the other arguments follow. For example, suppose a function returning the log of the density of y given parameter theta allows the use of the sampling statement is defined as follows.\nreal foo_lpdf(real y, vector theta) { ... }\nNote that for function definitions, the comma is used rather than the vertical bar.\nFor every custom _lpdf and _lpmf defined there is a corresponding _lupdf and _lupmf defined automatically. The _lupdf and _lupmf versions of the functions cannot be defined directly (to do so will produce an error). The difference in the _lpdf and _lpmf and the corresponding _lupdf and _lupmf functions is that if any other unnormalized density functions are used inside the user-defined function, the _lpdf and _lpmf forms of the user-defined function will change these densities to be normalized. The _lupdf and _lupmf forms of the user-defined functions will instead allow other unnormalized density functions to drop additive constants.\nThe sampling shorthand\nz ~ foo(phi);\nwill have the same effect as incrementing the target with the log of the unnormalized density:\ntarget += foo_lupdf(z | phi);\nOther _lupdf and _lupmf functions used in the definition of foo_lpdf will drop additive constants when foo_lupdf is called and will not drop additive constants when foo_lpdf is called.\nIf there are _lupdf and _lupmf functions used inside the following call to foo_lpdf, they will be forced to normalize (return the equivalent of their _lpdf and _lpmf forms):\ntarget += foo_lpdf(z | phi);\nIf there are no _lupdf or _lupmf functions used in the definition of foo_lpdf, then there will be no difference between a foo_lpdf or foo_lupdf call.\nThe unnormalized _lupdf and _lupmf functions can only be used in the model block or in user-defined probability functions (those ending in _lpdf or _lpmf).\nThe same syntax and shorthand that works for _lpdf also works for log probability mass functions with suffixes _lpmf.\nA function that is going to be accessed as distributions must return the log of the density or mass function it defines.\n\n\n\n\nWithin function definition bodies, the parameters may be used like any other variable. But the parameters are constant in the sense that they can’t be assigned to (i.e., can’t appear on the left side of an assignment (=) statement). In other words, their value remains constant throughout the function body. Attempting to assign a value to a function parameter value will raise a compile-time error.1\nLocal variables may be declared at the top of the function block and scope as usual.\n\n\n\nNon-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.\n\n\n\n\n\n\nA function can be declared without a return value by using void in place of a return type. Note that the type void may only be used as a return type—arguments may not be declared to be of type void.\n\n\n\nA void function may be used as a statement.\nBecause there is no return, such a usage is only for side effects, such as incrementing the log probability function, printing, or raising an error.\n\n\n\nIn a return statement within a void function’s definition, the return keyword is followed immediately by a semicolon (;) rather than by the expression whose value is returned.\n\n\n\n\nStan supports forward declarations, which look like function definitions without bodies. For example,\nreal unit_normal_lpdf(real y);\ndeclares a function named unit_normal_lpdf that consumes a single real-valued input and produces a real-valued output. Declaring a function without a definition is only really useful when using an extension which supplies the definition in C++ rather than in the Stan code itself. How exactly this can be accomplished will differ depending on your Stan interface.\nA function definition with a body simultaneously declares and defines the named function, as in\nreal unit_normal_lpdf(real y) {\n  return -0.5 * square(y);\n}\nA function can be declared and (perhaps separately) defined at most once. However, functions with different argument types are considered distinct even if they have the same name; see the section on function overloading.",
+    "text": "Stan allows users to define their own functions. The basic syntax is a simplified version of that used in C and C++. This chapter specifies how functions are declared, defined, and used in Stan.\n\n\nUser-defined functions appear in a special function-definition block before all of the other program blocks.\nfunctions {\n   // ... function declarations and definitions ...\n}\ndata {\n  // ...\nFunction definitions and declarations may appear in any order. Forward declarations are allowed but not required.\n\n\n\nThe rules for function naming and function-argument naming are the same as for other variables; see the section on variables for more information on valid identifiers. For example,\nreal foo(real mu, real sigma);\ndeclares a function named foo with two argument variables of types real and real. The arguments are named mu and sigma, but that is not part of the declaration.\n\n\nMultiple user-defined functions may have the same name if they have different sequences of argument types. This is known as function overloading.\nFor example, the following two functions are both defined with the name add_up\nreal add_up(real a, real b){\n  return a + b;\n}\n\nreal add_up(real a, real b, real c){\n  return a + b + c;\n}\nThe return types of overloaded functions do not need to be the same. One could define an additional add_up function as follows\nint add_up(int a, int b){\n  return a + b;\n}\nThat being said, functions may not use the same name if their signature only differs by the return type.\nFor example, the following is not permitted\n// illegal\nreal baz(int x);\nint baz(int x);\nFunction names used in the Stan standard library may be overloaded by user-defined functions. Exceptions to this are the reduce_sum family of functions and ODE integrators, which cannot be overloaded.\n\n\n\n\nAll function arguments are mandatory—there are no default values.\n\n\nFunctions with non-void return types are called just like any other built-in function in Stan—they are applied to appropriately typed arguments to produce an expression, which has a value when executed.\n\n\n\nFunctions with void return types may be applied to arguments and used as statements. These act like sampling statements or print statements. Such uses are only appropriate for functions that act through side effects, such as incrementing the log probability accumulator, printing, or raising exceptions.\n\n\n\nOverloaded functions alongside type promotion can result in situations where there are multiple valid interpretations of a function call. Stan requires that there be a unique signature which minimizes the number of promotions required.\nConsider the following two overloaded functions\nreal foo(int a, real b);\nreal foo(real a, int b);\nThese functions do not have a unique minimum when called with two integer arguments foo(1,2), and therefore cannot be called as such.\nPromotion of integers to complex numbers is considered as two separate promotions, one from int to real and a second from real to complex. Consider the following functions with real and complex signatures\nreal bar(real x);\nreal bar(complex z);\nA call bar(5) with an integer argument will be resolved to bar(real) because it only requires a single promotion, whereas the promotion to a complex number requires two promotions.\n\n\n\nThe rules for calling functions work the same way as assignment as far as promotion goes. This means that we can promote arguments to the type expected by function arguments. For example, the following will work.\nreal foo(real x) { return ... };\n...\nint a = 5;\nreal b = foo(a); // a promoted to type real\nIn addition to promoting int to real, Stan also promotes real to complex, and by transitivity, int to complex. This also works for containers, so an array of int may be assigned to an array of real of the same shape. And we can also promote vector to complex_vector and similarly for row vectors and matrices.\n\n\n\nFunctions whose name ends in _lpdf or _lpmf (density and mass functions) may be used as probability functions and may be used in place of parameterized distributions on the right-hand-side of sampling statements.\n\n\n\nFunctions of certain types are restricted on scope of usage. Functions whose names end in _lp assume access to the log probability accumulator and are only available in the transformed parameter and model blocks.\nFunctions whose names end in _rng assume access to the random number generator and may only be used within the generated quantities block, transformed data block, and within user-defined functions ending in _rng.\nFunctions whose names end in _lpdf and _lpmf can be used anywhere. However, _lupdf and _lupmf functions can only be used in the model block or user-defined probability functions.\nSee the section on function bodies for more information on these special types of function.\n\n\n\n\nStan’s functions all have declared types for both arguments and returned value. As with built-in functions, user-defined functions are only declared for base argument type and dimensionality. This requires a different syntax than for declaring other variables. The choice of language was made so that return types and argument types could use the same declaration syntax.\nThe type void may not be used as an argument type, only a return type for a function with side effects.\n\n\nThe base variable types are integer, real, complex, vector, row_vector, and matrix. No lower-bound or upper-bound constraints are allowed (e.g., real&lt;lower=0&gt; is illegal). Specialized constrained types are also not allowed (e.g., simplex is illegal).\nTuple types of the form tuple(T1, ..., TN) are also allowed, with all of the types T1 to TN being function argument types (i.e., no constraints and no sizes).\n\n\n\nArguments and return types may be arrays, and these are indicated with optional brackets and commas as would be used for indexing. For example, int denotes a single integer argument or return, whereas array[] real indicates a one-dimensional array of reals, array[,] real a two-dimensional array and array[,,] real a three-dimensional array; whitespace is optional, as usual.\nThe dimensions for vectors and matrices are not included, so that matrix is the type of a single matrix argument or return type. Thus if a variable is declared as matrix a, then a has two indexing dimensions, so that a[1] is a row vector and a[1, 1] a real value. Matrices implicitly have two indexing dimensions. The type declaration matrix[ , ] b specifies that b is a two-dimensional array of matrices, for a total of four indexing dimensions, with b[1, 1, 1, 1] picking out a real value.\n\n\n\nFunction argument and return types are not themselves checked for dimensionality. A matrix of any size may be passed in as a matrix argument. Nevertheless, a user-defined function might call a function (such as a multivariate normal density) that itself does dimensionality checks.\nDimensions of function return values will be checked if they’re assigned to a previously declared variable. They may also be checked if they are used as the argument to a function.\nAny errors raised by calls to functions inside user functions or return type mismatches are simply passed on; this typically results in a warning message and rejection of a proposal during sampling or optimization.\n\n\n\nSome of Stan’s built-in functions, like the differential equation solvers, have arguments that must be data. Such data-only arguments must be expressions involving only data, transformed data, and generated quantity variables.\nIn user-defined functions, the qualifier data may be placed before an argument type declaration to indicate that the argument must be data only. For example,\nreal foo(data real x) {\n  return x^2;\n}\nrequires the argument x to be data only.\nDeclaring an argument data only allows type inference to proceed in the body of the function so that, for example, the variable may be used as a data-only argument to a built-in function.\n\n\n\n\nThe body of a function is between an open curly brace ({) and close curly brace (}). The body may contain local variable declarations at the top of the function body’s block and these scope the same way as local variables used in any other statement block.\nAny user-defined function may be used in the function body regardless of the order in which the function definitions appear in the file. Self-recursive and mutually recursive functions are possible without any additional declarations.\nThe only restrictions on statements in function bodies are external, and determine whether the log probability accumulator or random number generators are available; see the rest of this section for details.\n\n\nFunctions that call random number generating functions in their bodies must have a name that ends in _rng; attempts to use random-number generators in other functions lead to a compile-time error.\nLike other random number generating functions, user-defined functions with names that end in _rng may be used only in the generated quantities block and transformed data block, or within the bodies of user-defined functions ending in _rng. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions that include sampling statements or log probability increment statements must have a name that ends in _lp. Attempts to use sampling statements or increment log probability statements in other functions lead to a compile-time error.\nLike the target log density increment statement and sampling statements, user-defined functions with names that end in _lp may only be used in blocks where the log probability accumulator is accessible, namely the transformed parameters and model blocks. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions whose names end in _lpdf and _lpmf (density and mass functions) can be used as probability functions in sampling statements. As with the built-in functions, the first argument will appear on the left of the sampling statement operator (~) in the sampling statement and the other arguments follow. For example, suppose a function returning the log of the density of y given parameter theta allows the use of the sampling statement is defined as follows.\nreal foo_lpdf(real y, vector theta) { ... }\nNote that for function definitions, the comma is used rather than the vertical bar.\nFor every custom _lpdf and _lpmf defined there is a corresponding _lupdf and _lupmf defined automatically. The _lupdf and _lupmf versions of the functions cannot be defined directly (to do so will produce an error). The difference in the _lpdf and _lpmf and the corresponding _lupdf and _lupmf functions is that if any other unnormalized density functions are used inside the user-defined function, the _lpdf and _lpmf forms of the user-defined function will change these densities to be normalized. The _lupdf and _lupmf forms of the user-defined functions will instead allow other unnormalized density functions to drop additive constants.\nThe sampling shorthand\nz ~ foo(phi);\nwill have the same effect as incrementing the target with the log of the unnormalized density:\ntarget += foo_lupdf(z | phi);\nOther _lupdf and _lupmf functions used in the definition of foo_lpdf will drop additive constants when foo_lupdf is called and will not drop additive constants when foo_lpdf is called.\nIf there are _lupdf and _lupmf functions used inside the following call to foo_lpdf, they will be forced to normalize (return the equivalent of their _lpdf and _lpmf forms):\ntarget += foo_lpdf(z | phi);\nIf there are no _lupdf or _lupmf functions used in the definition of foo_lpdf, then there will be no difference between a foo_lpdf or foo_lupdf call.\nThe unnormalized _lupdf and _lupmf functions can only be used in the model block or in user-defined probability functions (those ending in _lpdf or _lpmf).\nThe same syntax and shorthand that works for _lpdf also works for log probability mass functions with suffixes _lpmf.\nA function that is going to be accessed as distributions must return the log of the density or mass function it defines.\n\n\n\n\nWithin function definition bodies, the parameters may be used like any other variable. But the parameters are constant in the sense that they can’t be assigned to (i.e., can’t appear on the left side of an assignment (=) statement). In other words, their value remains constant throughout the function body. Attempting to assign a value to a function parameter value will raise a compile-time error.1\nLocal variables may be declared at the top of the function block and scope as usual.\n\n\n\nNon-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nAn exception is made for “obviously infinite” loops like while (1), which contain a return statement and no break statements. The only way to exit such a loop is to return, so they are considered as returning statements.\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.\n\n\n\n\n\n\nA function can be declared without a return value by using void in place of a return type. Note that the type void may only be used as a return type—arguments may not be declared to be of type void.\n\n\n\nA void function may be used as a statement.\nBecause there is no return, such a usage is only for side effects, such as incrementing the log probability function, printing, or raising an error.\n\n\n\nIn a return statement within a void function’s definition, the return keyword is followed immediately by a semicolon (;) rather than by the expression whose value is returned.\n\n\n\n\nStan supports forward declarations, which look like function definitions without bodies. For example,\nreal unit_normal_lpdf(real y);\ndeclares a function named unit_normal_lpdf that consumes a single real-valued input and produces a real-valued output. Declaring a function without a definition is only really useful when using an extension which supplies the definition in C++ rather than in the Stan code itself. How exactly this can be accomplished will differ depending on your Stan interface.\nA function definition with a body simultaneously declares and defines the named function, as in\nreal unit_normal_lpdf(real y) {\n  return -0.5 * square(y);\n}\nA function can be declared and (perhaps separately) defined at most once. However, functions with different argument types are considered distinct even if they have the same name; see the section on function overloading.",
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     "title": "User-Defined Functions",
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-    "text": "Non-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.",
+    "text": "Non-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nAn exception is made for “obviously infinite” loops like while (1), which contain a return statement and no break statements. The only way to exit such a loop is to return, so they are considered as returning statements.\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.",
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     "title": "Latent Discrete Parameters",
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-    "text": "Stan does not support sampling discrete parameters. So it is not possible to directly translate BUGS or JAGS models with discrete parameters (i.e., discrete stochastic nodes). Nevertheless, it is possible to code many models that involve bounded discrete parameters by marginalizing out the discrete parameters.1\nThis chapter shows how to code several widely-used models involving latent discrete parameters. The next chapter, the clustering chapter, on clustering models, considers further models involving latent discrete parameters.\n\n\nAlthough it requires some algebra on the joint probability function, a pleasant byproduct of the required calculations is the posterior expectation of the marginalized variable, which is often the quantity of interest for a model. This allows far greater exploration of the tails of the distribution as well as more efficient sampling on an iteration-by-iteration basis because the expectation at all possible values is being used rather than itself being estimated through sampling a discrete parameter.\nStandard optimization algorithms, including expectation maximization (EM), are often provided in applied statistics papers to describe maximum likelihood estimation algorithms. Such derivations provide exactly the marginalization needed for coding the model in Stan.\n\n\n\nThe first example is a model of coal mining disasters in the U.K. for the years 1851–1962.2\n\n\nFonnesbeck et al. (2013, sec. 3.1) provides a Poisson model of disaster \\(D_t\\) in year \\(t\\) with two rate parameters, an early rate (\\(e\\)) and late rate (\\(l\\)), that change at a given point in time \\(s\\). The full model expressed using a latent discrete parameter \\(s\\) is \\[\\begin{align*}\ne   &\\sim  \\textsf{exponential}(r_e) \\\\\nl   &\\sim  \\textsf{exponential}(r_l) \\\\\ns   &\\sim  \\textsf{uniform}(1, T) \\\\\nD_t &\\sim  \\textsf{Poisson}(t &lt; s \\; ? \\; e \\: : \\: l)\n\\end{align*}\\]\nThe last line uses the conditional operator (also known as the ternary operator), which is borrowed from C and related languages. The conditional operator has the same behavior as its counterpart in C++.3\nIt uses a compact notation involving separating its three arguments by a question mark (?) and a colon (:). The conditional operator is defined by \\[\nc \\; ? \\; x_1 \\: : \\: x_2\n=\n\\begin{cases}\n\\ x_1 & \\quad\\text{if } c \\text{ is true (i.e., non-zero), and} \\\\\n\\ x_2 & \\quad\\text{if } c \\text{ is false (i.e., zero).}\n\\end{cases}\n\\]\n\n\n\nTo code this model in Stan, the discrete parameter \\(s\\) must be marginalized out to produce a model defining the log of the probability function \\(p(e,l,D_t)\\). The full joint probability factors as \\[\\begin{align*}\np(e,l,s,D) &=  p(e) \\, p(l) \\, p(s) \\, p(D \\mid s, e, l) \\\\\n&= \\textsf{exponential}(e \\mid r_e) \\ \\textsf{exponential}(l \\mid r_l) \\,\n    \\textsf{uniform}(s \\mid 1, T) \\\\\n& \\qquad \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nTo marginalize, an alternative factorization into prior and likelihood is used, \\[\np(e,l,D) = p(e,l) \\, p(D \\mid e,l),\n\\]\nwhere the likelihood is defined by marginalizing \\(s\\) as \\[\\begin{align*}\np(D \\mid e,l) &= \\sum_{s=1}^T p(s, D \\mid e,l) \\\\\n&= \\sum_{s=1}^T p(s) \\, p(D \\mid s,e,l) \\\\\n&= \\sum_{s=1}^T \\textsf{uniform}(s \\mid 1,T) \\,\n    \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nStan operates on the log scale and thus requires the log likelihood, \\[\\begin{align*}\n\\log p(D \\mid e,l)\n&= \\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{s=1}^T\n    \\left( \\log \\textsf{uniform}(s \\mid 1, T) \\vphantom{\\sum_{t=1}^T}\\right. \\\\\n&\\qquad \\left.\n    + \\sum_{t=1}^T \\log \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l)\n\\right),\n\\end{align*}\\] where the log sum of exponents function is defined by \\[\n\\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{n=1}^N \\, \\alpha_n =\n\\log \\sum_{n=1}^N \\exp(\\alpha_n).\n\\]\nThe log sum of exponents function allows the model to be coded directly in Stan using the built-in function log_sum_exp, which provides both arithmetic stability and efficiency for mixture model calculations.\n\n\n\nThe Stan program for the change point model is shown in the figure below. The transformed parameter lp[s] stores the quantity \\(\\log p(s, D \\mid e, l)\\).\ndata {\n  real&lt;lower=0&gt; r_e;\n  real&lt;lower=0&gt; r_l;\n\n  int&lt;lower=1&gt; T;\n  array[T] int&lt;lower=0&gt; D;\n}\ntransformed data {\n  real log_unif;\n  log_unif = -log(T);\n}\nparameters {\n  real&lt;lower=0&gt; e;\n  real&lt;lower=0&gt; l;\n}\ntransformed parameters {\n  vector[T] lp;\n  lp = rep_vector(log_unif, T);\n  for (s in 1:T) {\n    for (t in 1:T) {\n      lp[s] = lp[s] + poisson_lpmf(D[t] | t &lt; s ? e : l);\n    }\n  }\n}\nmodel {\n  e ~ exponential(r_e);\n  l ~ exponential(r_l);\n  target += log_sum_exp(lp);\n}\nA change point model in which disaster rates D[t] have one rate, e, before the change point and a different rate, l, after the change point. The change point itself, s, is marginalized out as described in the text.\nAlthough the change-point model is coded directly, the doubly nested loop used for s and t is quadratic in T. Luke Wiklendt pointed out that a linear alternative can be achieved by the use of dynamic programming similar to the forward-backward algorithm for Hidden Markov models; he submitted a slight variant of the following code to replace the transformed parameters block of the above Stan program.\ntransformed parameters {\n    vector[T] lp;\n    {\n      vector[T + 1] lp_e;\n      vector[T + 1] lp_l;\n      lp_e[1] = 0;\n      lp_l[1] = 0;\n      for (t in 1:T) {\n        lp_e[t + 1] = lp_e[t] + poisson_lpmf(D[t] | e);\n        lp_l[t + 1] = lp_l[t] + poisson_lpmf(D[t] | l);\n      }\n      lp = rep_vector(log_unif + lp_l[T + 1], T)\n           + head(lp_e, T) - head(lp_l, T);\n    }\n  }\nAs should be obvious from looking at it, it has linear complexity in T rather than quadratic. The result for the mining-disaster data is about 20 times faster; the improvement will be greater for larger T.\nThe key to understanding Wiklendt’s dynamic programming version is to see that head(lp_e) holds the forward values, whereas lp_l[T + 1] - head(lp_l, T) holds the backward values; the clever use of subtraction allows lp_l to be accumulated naturally in the forward direction.\n\n\n\nThis model is easy to fit using MCMC with NUTS in its default configuration. Convergence is fast and sampling produces roughly one effective sample every two iterations. Because it is a relatively small model (the inner double loop over time is roughly 20,000 steps), it is fast.\nThe value of lp for each iteration for each change point is available because it is declared as a transformed parameter. If the value of lp were not of interest, it could be coded as a local variable in the model block and thus avoid the I/O overhead of saving values every iteration.\n\n\n\nThe value of lp[s] in a given iteration is given by \\(\\log\np(s,D \\mid e,l)\\) for the values of the early and late rates, \\(e\\) and \\(l\\), in the iteration. In each iteration after convergence, the early and late disaster rates, \\(e\\) and \\(l\\), are drawn from the posterior \\(p(e,l \\mid D)\\) by MCMC sampling and the associated lp calculated. The value of lp may be normalized to calculate \\(p(s \\mid e,l,D)\\) in each iteration, based on on the current values of \\(e\\) and \\(l\\). Averaging over iterations provides an unnormalized probability estimate of the change point being \\(s\\) (see below for the normalizing constant), \\[\\begin{align*}\np(s \\mid D) &\\propto q(s \\mid D) \\\\\n&= \\frac{1}{M} \\sum_{m=1}^{M} \\exp(\\texttt{lp}[m,s]).\n\\end{align*}\\] where \\(\\texttt{lp}[m,s]\\) represents the value of lp in posterior draw \\(m\\) for change point \\(s\\). By averaging over draws, \\(e\\) and \\(l\\) are themselves marginalized out, and the result has no dependence on a given iteration’s value for \\(e\\) and \\(l\\). A final normalization then produces the quantity of interest, the posterior probability of the change point being \\(s\\) conditioned on the data \\(D\\), \\[\np(s \\mid D) = \\frac{q(s \\mid D)}{\\sum_{s'=1}^T q(s' \\mid D)}.\n\\]\nA plot of the values of \\(\\log p(s \\mid D)\\) computed using Stan 2.4’s default MCMC implementation is shown in the posterior plot.\nLog probability of change point being in year, calculated analytically.\n\n\n\nAnalytical change-point posterior\n\n\nThe frequency of change points generated by sampling the discrete change points.\n\n\n\nSampled change-point posterior\n\n\nIn order their range of estimates be visible, the first plot is on the log scale and the second plot on the linear scale; note the narrower range of years in the second plot resulting from sampling. The posterior mean of \\(s\\) is roughly 1891.\n\n\n\nThe generated quantities block may be used to draw discrete parameter values using the built-in pseudo-random number generators. For example, with lp defined as above, the following program draws a random value for s at every iteration.\ngenerated quantities {\n  int&lt;lower=1, upper=T&gt; s;\n  s = categorical_logit_rng(lp);\n}\nA posterior histogram of draws for \\(s\\) is shown on the second change point posterior figure above.\nCompared to working in terms of expectations, discrete sampling is highly inefficient, especially for tails of distributions, so this approach should only be used if draws from a distribution are explicitly required. Otherwise, expectations should be computed in the generated quantities block based on the posterior distribution for s given by softmax(lp).\n\n\n\nThe discrete sample generated for \\(s\\) can be used to calculate covariance with other parameters. Although the sampling approach is straightforward, it is more statistically efficient (in the sense of requiring far fewer iterations for the same degree of accuracy) to calculate these covariances in expectation using lp.\n\n\n\nThere is no obstacle in principle to allowing multiple change points. The only issue is that computation increases from linear to quadratic in marginalizing out two change points, cubic for three change points, and so on. There are three parameters, e, m, and l, and two loops for the change point and then one over time, with log densities being stored in a matrix.\nmatrix[T, T] lp;\nlp = rep_matrix(log_unif, T);\nfor (s1 in 1:T) {\n  for (s2 in 1:T) {\n    for (t in 1:T) {\n      lp[s1,s2] = lp[s1,s2]\n        + poisson_lpmf(D[t] | t &lt; s1 ? e : (t &lt; s2 ? m : l));\n    }\n  }\n}\nThe matrix can then be converted back to a vector using to_vector before being passed to log_sum_exp.\n\n\n\n\nA widely applied field method in ecology is to capture (or sight) animals, mark them (e.g., by tagging), then release them. This process is then repeated one or more times, and is often done for populations on an ongoing basis. The resulting data may be used to estimate population size.\nThe first subsection describes a simple mark-recapture model that does not involve any latent discrete parameters. The following subsections describes the Cormack-Jolly-Seber model, which involves latent discrete parameters for animal death.\n\n\nIn the simplest case, a one-stage mark-recapture study produces the following data\n\n\\(M\\) : number of animals marked in first capture,\n\\(C\\) : number animals in second capture, and\n\\(R\\) : number of marked animals in second capture.\n\nThe estimand of interest is\n\n\\(N\\) : number of animals in the population.\n\nDespite the notation, the model will take \\(N\\) to be a continuous parameter; just because the population must be finite doesn’t mean the parameter representing it must be. The parameter will be used to produce a real-valued estimate of the population size.\nThe Lincoln-Petersen (Lincoln 1930; Petersen 1896) method for estimating population size is \\[\n\\hat{N} = \\frac{M C}{R}.\n\\]\nThis population estimate would arise from a probabilistic model in which the number of recaptured animals is distributed binomially, \\[\nR \\sim \\textsf{binomial}(C, M / N)\n\\] given the total number of animals captured in the second round (\\(C\\)) with a recapture probability of \\(M/N\\), the fraction of the total population \\(N\\) marked in the first round.\ndata {\n  int&lt;lower=0&gt; M;\n  int&lt;lower=0&gt; C;\n  int&lt;lower=0, upper=min(M, C)&gt; R;\n}\nparameters {\n  real&lt;lower=(C - R + M)&gt; N;\n}\nmodel {\n  R ~ binomial(C, M / N);\n}\nA probabilistic formulation of the Lincoln-Petersen estimator for population size based on data from a one-step mark-recapture study. The lower bound on \\(N\\) is necessary to efficiently eliminate impossible values.\nThe probabilistic variant of the Lincoln-Petersen estimator can be directly coded in Stan as shown in the Lincon-Petersen model figure. The Lincoln-Petersen estimate is the maximum likelihood estimate (MLE) for this model.\nTo ensure the MLE is the Lincoln-Petersen estimate, an improper uniform prior for \\(N\\) is used; this could (and should) be replaced with a more informative prior if possible, based on knowledge of the population under study.\nThe one tricky part of the model is the lower bound \\(C - R + M\\) placed on the population size \\(N\\). Values below this bound are impossible because it is otherwise not possible to draw \\(R\\) samples out of the \\(C\\) animals recaptured. Implementing this lower bound is necessary to ensure sampling and optimization can be carried out in an unconstrained manner with unbounded support for parameters on the transformed (unconstrained) space. The lower bound in the declaration for \\(C\\) implies a variable transform \\(f : (C-R+M,\\infty) \\rightarrow (-\\infty,+\\infty)\\) defined by \\(f(N) = \\log(N - (C - R + M))\\); the reference manual contains full details of all constrained parameter transforms.\n\n\n\nThe Cormack-Jolly-Seber (CJS) model (Cormack 1964; Jolly 1965; Seber 1965) is an open-population model in which the population may change over time due to death; the presentation here draws heavily on Schofield (2007).\nThe basic data are\n\n\\(I\\): number of individuals,\n\\(T\\): number of capture periods, and\n\\(y_{i,t}\\): Boolean indicating if individual \\(i\\) was captured at time \\(t\\).\n\nEach individual is assumed to have been captured at least once because an individual only contributes information conditionally after they have been captured the first time.\nThere are two Bernoulli parameters in the model,\n\n\\(\\phi_t\\) : probability that animal alive at time \\(t\\) survives until \\(t + 1\\) and\n\\(p_t\\) : probability that animal alive at time \\(t\\) is captured at time \\(t\\).\n\nThese parameters will both be given uniform priors, but information should be used to tighten these priors in practice.\nThe CJS model also employs a latent discrete parameter \\(z_{i,t}\\) indicating for each individual \\(i\\) whether it is alive at time \\(t\\), distributed as \\[\nz_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t-1} \\; ? \\; 0 \\: : \\: \\phi_{t-1}).\n\\]\nThe conditional prevents the model positing zombies; once an animal is dead, it stays dead. The data distribution is then simple to express conditional on \\(z\\) as \\[\ny_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t} \\; ? \\; 0 \\: : \\: p_t).\n\\]\nThe conditional enforces the constraint that dead animals cannot be captured.\n\n\n\nThis subsection presents an implementation of the model in terms of counts for different history profiles for individuals over three capture times. It assumes exchangeability of the animals in that each is assigned the same capture and survival probabilities.\nIn order to ease the marginalization of the latent discrete parameter \\(z_{i,t}\\), the Stan models rely on a derived quantity \\(\\chi_t\\) for the probability that an individual is never captured again if it is alive at time \\(t\\) (if it is dead, the recapture probability is zero). this quantity is defined recursively by \\[\n\\chi_t\n=\n\\begin{cases}\n1 & \\quad\\text{if } t = T \\\\\n(1 - \\phi_t) + \\phi_t (1 - p_{t+1}) \\chi_{t+1}\n  & \\quad\\text{if } t &lt; T\n\\end{cases}\n\\]\nThe base case arises because if an animal was captured in the last time period, the probability it is never captured again is 1 because there are no more capture periods. The recursive case defining \\(\\chi_{t}\\) in terms of \\(\\chi_{t+1}\\) involves two possibilities: (1) not surviving to the next time period, with probability \\((1 - \\phi_t)\\), or (2) surviving to the next time period with probability \\(\\phi_t\\), not being captured in the next time period with probability \\((1 - p_{t+1})\\), and not being captured again after being alive in period \\(t+1\\) with probability \\(\\chi_{t+1}\\).\nWith three capture times, there are eight captured/not-captured profiles an individual may have. These may be naturally coded as binary numbers as follows.\n\\[\n\\begin{array}{crclc}\n\\hline\n& \\qquad\\qquad & captures & \\qquad\\qquad &  \\\\\n\\mathrm{profile} & 1 & 2 & 3 & \\mathrm{probability} \\\\\n\\hline\n0 & - & - & - & n/a \\\\\n1 & - & - & + & n/a \\\\\n2 & - & + & - & \\chi_2 \\\\\n3 & - & + & + & \\phi_2 \\, p_3 \\\\\n4 & + & - & - & \\chi_1 \\\\\n5 & + & - & + & \\phi_1 \\, (1 - p_2) \\, \\phi_2 \\, p_3 \\\\\n6 & + & + & - & \\phi_1 \\, p_2 \\, \\chi_2 \\\\\n7 & + & + & + & \\phi_1 \\, p_2 \\, \\phi_2 \\, p_3 \\\\\n\\hline\n\\end{array}\n\\]\nHistory 0, for animals that are never captured, is unobservable because only animals that are captured are observed. History 1, for animals that are only captured in the last round, provides no information for the CJS model, because capture/non-capture status is only informative when conditioned on earlier captures. For the remaining cases, the contribution to the likelihood is provided in the final column.\nBy defining these probabilities in terms of \\(\\chi\\) directly, there is no need for a latent binary parameter indicating whether an animal is alive at time \\(t\\) or not. The definition of \\(\\chi\\) is typically used to define the likelihood (i.e., marginalize out the latent discrete parameter) for the CJS model (Schofield 2007).\nThe Stan model defines \\(\\chi\\) as a transformed parameter based on parameters \\(\\phi\\) and \\(p\\). In the model block, the log probability is incremented for each history based on its count. This second step is similar to collecting Bernoulli observations into a binomial or categorical observations into a multinomial, only it is coded directly in the Stan program using target += rather than being part of a built-in probability function.\nThe following is the Stan program for the Cormack-Jolly-Seber mark-recapture model that considers counts of individuals with observation histories of being observed or not in three capture periods\ndata {\n  array[7] int&lt;lower=0&gt; history;\n}\nparameters {\n  array[2] real&lt;lower=0, upper=1&gt; phi;\n  array[3] real&lt;lower=0, upper=1&gt; p;\n}\ntransformed parameters {\n  array[2] real&lt;lower=0, upper=1&gt; chi;\n  chi[2] = (1 - phi[2]) + phi[2] * (1 - p[3]);\n  chi[1] = (1 - phi[1]) + phi[1] * (1 - p[2]) * chi[2];\n}\nmodel {\n  target += history[2] * log(chi[2]);\n  target += history[3] * (log(phi[2]) + log(p[3]));\n  target += history[4] * (log(chi[1]));\n  target += history[5] * (log(phi[1]) + log1m(p[2])\n                            + log(phi[2]) + log(p[3]));\n  target += history[6] * (log(phi[1]) + log(p[2])\n                            + log(chi[2]));\n  target += history[7] * (log(phi[1]) + log(p[2])\n                            + log(phi[2]) + log(p[3]));\n}\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; beta3;\n  beta3 = phi[2] * p[3];\n}\n\n\nThe parameters \\(\\phi_2\\) and \\(p_3\\), the probability of death at time 2 and probability of capture at time 3 are not identifiable, because both may be used to account for lack of capture at time 3. Their product, \\(\\beta_3 = \\phi_2 \\, p_3\\), is identified. The Stan model defines beta3 as a generated quantity. Unidentified parameters pose a problem for Stan’s samplers’ adaptation. Although the problem posed for adaptation is mild here because the parameters are bounded and thus have proper uniform priors, it would be better to formulate an identified parameterization. One way to do this would be to formulate a hierarchical model for the \\(p\\) and \\(\\phi\\) parameters.\n\n\n\n\nThis section presents a version of the Cormack-Jolly-Seber (CJS) model cast at the individual level rather than collectively as in the previous subsection. It also extends the model to allow an arbitrary number of time periods. The data will consist of the number \\(T\\) of capture events, the number \\(I\\) of individuals, and a boolean flag \\(y_{i,t}\\) indicating if individual \\(i\\) was observed at time \\(t\\). In Stan,\ndata {\n  int&lt;lower=2&gt; T;\n  int&lt;lower=0&gt; I;\n  array[I, T] int&lt;lower=0, upper=1&gt; y;\n}\nThe advantages to the individual-level model is that it becomes possible to add individual “random effects” that affect survival or capture probability, as well as to avoid the combinatorics involved in unfolding \\(2^T\\) observation histories for \\(T\\) capture times.\n\n\nThe individual CJS model is written involves several function definitions. The first two are used in the transformed data block to compute the first and last time period in which an animal was captured.4\nfunctions {\n  int first_capture(array[] int y_i) {\n    for (k in 1:size(y_i)) {\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  int last_capture(array[] int y_i) {\n    for (k_rev in 0:(size(y_i) - 1)) {\n      int k;\n      k = size(y_i) - k_rev;\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  // ...\n}\nThese two functions are used to define the first and last capture time for each individual in the transformed data block.5\ntransformed data {\n  array[I] int&lt;lower=0, upper=T&gt; first;\n  array[I] int&lt;lower=0, upper=T&gt; last;\n  vector&lt;lower=0, upper=I&gt;[T] n_captured;\n  for (i in 1:I) {\n    first[i] = first_capture(y[i]);\n  }\n  for (i in 1:I) {\n    last[i] = last_capture(y[i]);\n  }\n  n_captured = rep_vector(0, T);\n  for (t in 1:T) {\n    for (i in 1:I) {\n      if (y[i, t]) {\n        n_captured[t] = n_captured[t] + 1;\n      }\n    }\n  }\n}\nThe transformed data block also defines n_captured[t], which is the total number of captures at time t. The variable n_captured is defined as a vector instead of an integer array so that it can be used in an elementwise vector operation in the generated quantities block to model the population estimates at each time point.\nThe parameters and transformed parameters are as before, but now there is a function definition for computing the entire vector chi, the probability that if an individual is alive at t that it will never be captured again.\nparameters {\n  vector&lt;lower=0, upper=1&gt;[T - 1] phi;\n  vector&lt;lower=0, upper=1&gt;[T] p;\n}\ntransformed parameters {\n  vector&lt;lower=0, upper=1&gt;[T] chi;\n  chi = prob_uncaptured(T, p, phi);\n}\nThe definition of prob_uncaptured, from the functions block, is\nfunctions {\n  // ...\n  vector prob_uncaptured(int T, vector p, vector phi) {\n    vector[T] chi;\n    chi[T] = 1.0;\n    for (t in 1:(T - 1)) {\n      int t_curr;\n      int t_next;\n      t_curr = T - t;\n      t_next = t_curr + 1;\n      chi[t_curr] = (1 - phi[t_curr])\n                     + phi[t_curr]\n                       * (1 - p[t_next])\n                       * chi[t_next];\n    }\n    return chi;\n  }\n}\nThe function definition directly follows the mathematical definition of \\(\\chi_t\\), unrolling the recursion into an iteration and defining the elements of chi from T down to 1.\n\n\n\nGiven the precomputed quantities, the model block directly encodes the CJS model’s log likelihood function. All parameters are left with their default uniform priors and the model simply encodes the log probability of the observations q given the parameters p and phi as well as the transformed parameter chi defined in terms of p and phi.\nmodel {\n  for (i in 1:I) {\n    if (first[i] &gt; 0) {\n      for (t in (first[i]+1):last[i]) {\n        1 ~ bernoulli(phi[t - 1]);\n        y[i, t] ~ bernoulli(p[t]);\n      }\n      1 ~ bernoulli(chi[last[i]]);\n    }\n  }\n}\nThe outer loop is over individuals, conditional skipping individuals i which are never captured. The never-captured check depends on the convention of the first-capture and last-capture functions returning 0 for first if an individual is never captured.\nThe inner loop for individual i first increments the log probability based on the survival of the individual with probability phi[t - 1]. The outcome of 1 is fixed because the individual must survive between the first and last capture (i.e., no zombies). The loop starts after the first capture, because all information in the CJS model is conditional on the first capture.\nIn the inner loop, the observed capture status y[i, t] for individual i at time t has a Bernoulli distribution based on the capture probability p[t] at time t.\nAfter the inner loop, the probability of an animal never being seen again after being observed at time last[i] is included, because last[i] was defined to be the last time period in which animal i was observed.\n\n\n\nAs with the collective model described in the previous subsection, this model does not identify phi[T - 1] and p[T], but does identify their product, beta. Thus beta is defined as a generated quantity to monitor convergence and report.\ngenerated quantities {\n  real beta;\n  // ...\n\n  beta = phi[T - 1] * p[T];\n  // ...\n}\nThe parameter p[1] is also not modeled and will just be uniform between 0 and 1. A more finely articulated model might have a hierarchical or time-series component, in which case p[1] would be an unknown initial condition and both phi[T - 1] and p[T] could be identified.\n\n\n\nThe generated quantities also calculates an estimate of the population mean at each time t in the same way as in the simple mark-recapture model as the number of individuals captured at time t divided by the probability of capture at time t. This is done with the elementwise division operation for vectors (./) in the generated quantities block.\ngenerated quantities {\n  // ...\n  vector&lt;lower=0&gt;[T] pop;\n  // ...\n  pop = n_captured ./ p;\n  pop[1] = -1;\n}\n\n\n\nAll individuals are modeled as having the same capture probability, but this model could be easily generalized to use a logistic regression here based on individual-level inputs to be used as predictors.\n\n\n\n\n\nAlthough seemingly disparate tasks, the rating/coding/annotation of items with categories and diagnostic testing for disease or other conditions, share several characteristics which allow their statistical properties to be modeled similarly.\n\n\nSuppose you have diagnostic tests for a condition of varying sensitivity and specificity. Sensitivity is the probability a test returns positive when the patient has the condition and specificity is the probability that a test returns negative when the patient does not have the condition. For example, mammograms and puncture biopsy tests both test for the presence of breast cancer. Mammograms have high sensitivity and low specificity, meaning lots of false positives, whereas puncture biopsies are the opposite, with low sensitivity and high specificity, meaning lots of false negatives.\nThere are several estimands of interest in such studies. An epidemiological study may be interested in the prevalence of a kind of infection, such as malaria, in a population. A test development study might be interested in the diagnostic accuracy of a new test. A health care worker performing tests might be interested in the disease status of a particular patient.\n\n\n\nHumans are often given the task of coding (equivalently rating or annotating) data. For example, journal or grant reviewers rate submissions, a political study may code campaign commercials as to whether they are attack ads or not, a natural language processing study might annotate Tweets as to whether they are positive or negative in overall sentiment, or a dentist looking at an X-ray classifies a patient as having a cavity or not. In all of these cases, the data coders play the role of the diagnostic tests and all of the same estimands are in play — data coder accuracy and bias, true categories of items being coded, or the prevalence of various categories of items in the data.\n\n\n\nIn this section, only categorical ratings are considered, and the challenge in the modeling for Stan is to marginalize out the discrete parameters.\nDawid and Skene (1979) introduce a noisy-measurement model for coding and apply it in the epidemiological setting of coding what doctors say about patient histories; the same model can be used for diagnostic procedures.\n\n\nThe data for the model consists of \\(J\\) raters (diagnostic tests), \\(I\\) items (patients), and \\(K\\) categories (condition statuses) to annotate, with \\(y_{i, j} \\in \\{1, \\dotsc, K\\}\\) being the rating provided by rater \\(j\\) for item \\(i\\). In a diagnostic test setting for a particular condition, the raters are diagnostic procedures and often \\(K=2\\), with values signaling the presence or absence of the condition.6\nIt is relatively straightforward to extend Dawid and Skene’s model to deal with the situation where not every rater rates each item exactly once.\n\n\n\n\nThe model is based on three parameters, the first of which is discrete:\n\n\\(z_i\\) : a value in \\(\\{1, \\dotsc, K\\}\\) indicating the true category of item \\(i\\),\n\\(\\pi\\) : a \\(K\\)-simplex for the prevalence of the \\(K\\) categories in the population, and\n\\(\\theta_{j,k}\\) : a \\(K\\)-simplex for the response of annotator \\(j\\) to an item of true category \\(k\\).\n\n\n\n\nThe true category of an item is assumed to be generated by a simple categorical distribution based on item prevalence, \\[\nz_i \\sim \\textsf{categorical}(\\pi).\n\\]\nThe rating \\(y_{i, j}\\) provided for item \\(i\\) by rater \\(j\\) is modeled as a categorical response of rater \\(i\\) to an item of category \\(z_i\\),7 \\[\ny_{i, j} \\sim \\textsf{categorical}(\\theta_{j,\\pi_{z[i]}}).\n\\]\n\n\nDawid and Skene provided maximum likelihood estimates for \\(\\theta\\) and \\(\\pi\\), which allows them to generate probability estimates for each \\(z_i\\).\nTo mimic Dawid and Skene’s maximum likelihood model, the parameters \\(\\theta_{j,k}\\) and \\(\\pi\\) can be given uniform priors over \\(K\\)-simplexes. It is straightforward to generalize to Dirichlet priors, \\[\n\\pi \\sim \\textsf{Dirichlet}(\\alpha)\n\\] and \\[\n\\theta_{j,k} \\sim \\textsf{Dirichlet}(\\beta_k)\n\\] with fixed hyperparameters \\(\\alpha\\) (a vector) and \\(\\beta\\) (a matrix or array of vectors). The prior for \\(\\theta_{j,k}\\) must be allowed to vary in \\(k\\), so that, for instance, \\(\\beta_{k,k}\\) is large enough to allow the prior to favor better-than-chance annotators over random or adversarial ones.\nBecause there are \\(J\\) coders, it would be natural to extend the model to include a hierarchical prior for \\(\\beta\\) and to partially pool the estimates of coder accuracy and bias.\n\n\n\nBecause the true category parameter \\(z\\) is discrete, it must be marginalized out of the joint posterior in order to carry out sampling or maximum likelihood estimation in Stan. The joint posterior factors as \\[\np(y, \\theta, \\pi) = p(y \\mid \\theta,\\pi) \\, p(\\pi) \\, p(\\theta),\n\\] where \\(p(y \\mid \\theta,\\pi)\\) is derived by marginalizing \\(z\\) out of \\[\np(z, y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\left( \\textsf{categorical}(z_i \\mid \\pi)\n                     \\prod_{j=1}^J\n                     \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, z[i]})\n              \\right).\n\\]\nThis can be done item by item, with \\[\np(y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\sum_{k=1}^K\n  \\left( \\textsf{categorical}(k \\mid \\pi)\n         \\prod_{j=1}^J\n         \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n  \\right).\n\\]\nIn the missing data model, only the observed labels would be used in the inner product.\nDawid and Skene (1979) derive exactly the same equation in their Equation (2.7), required for the E-step in their expectation maximization (EM) algorithm. Stan requires the marginalized probability function on the log scale, \\[\\begin{align*}\n\\log p(y \\mid \\theta, \\pi)\n&= \\sum_{i=1}^I \\log \\left( \\sum_{k=1}^K \\exp\n    \\left(\\log \\textsf{categorical}(k \\mid \\pi) \\vphantom{\\sum_{j=1}^J}\\right.\\right.\n    \\left.\\left. + \\ \\sum_{j=1}^J\n           \\log \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n    \\right) \\right),\n\\end{align*}\\] which can be directly coded using Stan’s built-in log_sum_exp function.\n\n\n\n\nThe Stan program for the Dawid and Skene model is provided below (Dawid and Skene 1979).\ndata {\n  int&lt;lower=2&gt; K;\n  int&lt;lower=1&gt; I;\n  int&lt;lower=1&gt; J;\n\n  array[I, J] int&lt;lower=1, upper=K&gt; y;\n\n  vector&lt;lower=0&gt;[K] alpha;\n  vector&lt;lower=0&gt;[K] beta[K];\n}\nparameters {\n  simplex[K] pi;\n  array[J, K] simplex[K] theta;\n}\ntransformed parameters {\n  array[I] vector[K] log_q_z;\n  for (i in 1:I) {\n    log_q_z[i] = log(pi);\n    for (j in 1:J) {\n      for (k in 1:K) {\n        log_q_z[i, k] = log_q_z[i, k]\n                         + log(theta[j, k, y[i, j]]);\n      }\n    }\n  }\n}\nmodel {\n  pi ~ dirichlet(alpha);\n  for (j in 1:J) {\n    for (k in 1:K) {\n      theta[j, k] ~ dirichlet(beta[k]);\n    }\n  }\n\n  for (i in 1:I) {\n    target += log_sum_exp(log_q_z[i]);\n  }\n}\nThe model marginalizes out the discrete parameter \\(z\\), storing the unnormalized conditional probability \\(\\log q(z_i=k|\\theta,\\pi)\\) in log_q_z[i, k].\nThe Stan model converges quickly and mixes well using NUTS starting at diffuse initial points, unlike the equivalent model implemented with Gibbs sampling over the discrete parameter. Reasonable weakly informative priors are \\(\\alpha_k = 3\\) and \\(\\beta_{k,k} = 2.5 K\\) and \\(\\beta_{k,k'} = 1\\) if \\(k \\neq k'\\). Taking \\(\\alpha\\) and \\(\\beta_k\\) to be unit vectors and applying optimization will produce the same answer as the expectation maximization (EM) algorithm of Dawid and Skene (1979).\n\n\nThe quantity log_q_z[i] is defined as a transformed parameter. It encodes the (unnormalized) log of \\(p(z_i \\mid \\theta,\n\\pi)\\). Each iteration provides a value conditioned on that iteration’s values for \\(\\theta\\) and \\(\\pi\\). Applying the softmax function to log_q_z[i] provides a simplex corresponding to the probability mass function of \\(z_i\\) in the posterior. These may be averaged across the iterations to provide the posterior probability distribution over each \\(z_i\\).\n\n\n\n\n\n\n\nThis section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z] \\mid \\Theta)] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
+    "text": "Stan does not support sampling discrete parameters. So it is not possible to directly translate BUGS or JAGS models with discrete parameters (i.e., discrete stochastic nodes). Nevertheless, it is possible to code many models that involve bounded discrete parameters by marginalizing out the discrete parameters.1\nThis chapter shows how to code several widely-used models involving latent discrete parameters. The next chapter, the clustering chapter, on clustering models, considers further models involving latent discrete parameters.\n\n\nAlthough it requires some algebra on the joint probability function, a pleasant byproduct of the required calculations is the posterior expectation of the marginalized variable, which is often the quantity of interest for a model. This allows far greater exploration of the tails of the distribution as well as more efficient sampling on an iteration-by-iteration basis because the expectation at all possible values is being used rather than itself being estimated through sampling a discrete parameter.\nStandard optimization algorithms, including expectation maximization (EM), are often provided in applied statistics papers to describe maximum likelihood estimation algorithms. Such derivations provide exactly the marginalization needed for coding the model in Stan.\n\n\n\nThe first example is a model of coal mining disasters in the U.K. for the years 1851–1962.2\n\n\nFonnesbeck et al. (2013, sec. 3.1) provides a Poisson model of disaster \\(D_t\\) in year \\(t\\) with two rate parameters, an early rate (\\(e\\)) and late rate (\\(l\\)), that change at a given point in time \\(s\\). The full model expressed using a latent discrete parameter \\(s\\) is \\[\\begin{align*}\ne   &\\sim  \\textsf{exponential}(r_e) \\\\\nl   &\\sim  \\textsf{exponential}(r_l) \\\\\ns   &\\sim  \\textsf{uniform}(1, T) \\\\\nD_t &\\sim  \\textsf{Poisson}(t &lt; s \\; ? \\; e \\: : \\: l)\n\\end{align*}\\]\nThe last line uses the conditional operator (also known as the ternary operator), which is borrowed from C and related languages. The conditional operator has the same behavior as its counterpart in C++.3\nIt uses a compact notation involving separating its three arguments by a question mark (?) and a colon (:). The conditional operator is defined by \\[\nc \\; ? \\; x_1 \\: : \\: x_2\n=\n\\begin{cases}\n\\ x_1 & \\quad\\text{if } c \\text{ is true (i.e., non-zero), and} \\\\\n\\ x_2 & \\quad\\text{if } c \\text{ is false (i.e., zero).}\n\\end{cases}\n\\]\n\n\n\nTo code this model in Stan, the discrete parameter \\(s\\) must be marginalized out to produce a model defining the log of the probability function \\(p(e,l,D_t)\\). The full joint probability factors as \\[\\begin{align*}\np(e,l,s,D) &=  p(e) \\, p(l) \\, p(s) \\, p(D \\mid s, e, l) \\\\\n&= \\textsf{exponential}(e \\mid r_e) \\ \\textsf{exponential}(l \\mid r_l) \\,\n    \\textsf{uniform}(s \\mid 1, T) \\\\\n& \\qquad \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nTo marginalize, an alternative factorization into prior and likelihood is used, \\[\np(e,l,D) = p(e,l) \\, p(D \\mid e,l),\n\\]\nwhere the likelihood is defined by marginalizing \\(s\\) as \\[\\begin{align*}\np(D \\mid e,l) &= \\sum_{s=1}^T p(s, D \\mid e,l) \\\\\n&= \\sum_{s=1}^T p(s) \\, p(D \\mid s,e,l) \\\\\n&= \\sum_{s=1}^T \\textsf{uniform}(s \\mid 1,T) \\,\n    \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nStan operates on the log scale and thus requires the log likelihood, \\[\\begin{align*}\n\\log p(D \\mid e,l)\n&= \\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{s=1}^T\n    \\left( \\log \\textsf{uniform}(s \\mid 1, T) \\vphantom{\\sum_{t=1}^T}\\right. \\\\\n&\\qquad \\left.\n    + \\sum_{t=1}^T \\log \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l)\n\\right),\n\\end{align*}\\] where the log sum of exponents function is defined by \\[\n\\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{n=1}^N \\, \\alpha_n =\n\\log \\sum_{n=1}^N \\exp(\\alpha_n).\n\\]\nThe log sum of exponents function allows the model to be coded directly in Stan using the built-in function log_sum_exp, which provides both arithmetic stability and efficiency for mixture model calculations.\n\n\n\nThe Stan program for the change point model is shown in the figure below. The transformed parameter lp[s] stores the quantity \\(\\log p(s, D \\mid e, l)\\).\ndata {\n  real&lt;lower=0&gt; r_e;\n  real&lt;lower=0&gt; r_l;\n\n  int&lt;lower=1&gt; T;\n  array[T] int&lt;lower=0&gt; D;\n}\ntransformed data {\n  real log_unif;\n  log_unif = -log(T);\n}\nparameters {\n  real&lt;lower=0&gt; e;\n  real&lt;lower=0&gt; l;\n}\ntransformed parameters {\n  vector[T] lp;\n  lp = rep_vector(log_unif, T);\n  for (s in 1:T) {\n    for (t in 1:T) {\n      lp[s] = lp[s] + poisson_lpmf(D[t] | t &lt; s ? e : l);\n    }\n  }\n}\nmodel {\n  e ~ exponential(r_e);\n  l ~ exponential(r_l);\n  target += log_sum_exp(lp);\n}\nA change point model in which disaster rates D[t] have one rate, e, before the change point and a different rate, l, after the change point. The change point itself, s, is marginalized out as described in the text.\nAlthough the change-point model is coded directly, the doubly nested loop used for s and t is quadratic in T. Luke Wiklendt pointed out that a linear alternative can be achieved by the use of dynamic programming similar to the forward-backward algorithm for Hidden Markov models; he submitted a slight variant of the following code to replace the transformed parameters block of the above Stan program.\ntransformed parameters {\n    vector[T] lp;\n    {\n      vector[T + 1] lp_e;\n      vector[T + 1] lp_l;\n      lp_e[1] = 0;\n      lp_l[1] = 0;\n      for (t in 1:T) {\n        lp_e[t + 1] = lp_e[t] + poisson_lpmf(D[t] | e);\n        lp_l[t + 1] = lp_l[t] + poisson_lpmf(D[t] | l);\n      }\n      lp = rep_vector(log_unif + lp_l[T + 1], T)\n           + head(lp_e, T) - head(lp_l, T);\n    }\n  }\nAs should be obvious from looking at it, it has linear complexity in T rather than quadratic. The result for the mining-disaster data is about 20 times faster; the improvement will be greater for larger T.\nThe key to understanding Wiklendt’s dynamic programming version is to see that head(lp_e) holds the forward values, whereas lp_l[T + 1] - head(lp_l, T) holds the backward values; the clever use of subtraction allows lp_l to be accumulated naturally in the forward direction.\n\n\n\nThis model is easy to fit using MCMC with NUTS in its default configuration. Convergence is fast and sampling produces roughly one effective sample every two iterations. Because it is a relatively small model (the inner double loop over time is roughly 20,000 steps), it is fast.\nThe value of lp for each iteration for each change point is available because it is declared as a transformed parameter. If the value of lp were not of interest, it could be coded as a local variable in the model block and thus avoid the I/O overhead of saving values every iteration.\n\n\n\nThe value of lp[s] in a given iteration is given by \\(\\log\np(s,D \\mid e,l)\\) for the values of the early and late rates, \\(e\\) and \\(l\\), in the iteration. In each iteration after convergence, the early and late disaster rates, \\(e\\) and \\(l\\), are drawn from the posterior \\(p(e,l \\mid D)\\) by MCMC sampling and the associated lp calculated. The value of lp may be normalized to calculate \\(p(s \\mid e,l,D)\\) in each iteration, based on on the current values of \\(e\\) and \\(l\\). Averaging over iterations provides an unnormalized probability estimate of the change point being \\(s\\) (see below for the normalizing constant), \\[\\begin{align*}\np(s \\mid D) &\\propto q(s \\mid D) \\\\\n&= \\frac{1}{M} \\sum_{m=1}^{M} \\exp(\\texttt{lp}[m,s]).\n\\end{align*}\\] where \\(\\texttt{lp}[m,s]\\) represents the value of lp in posterior draw \\(m\\) for change point \\(s\\). By averaging over draws, \\(e\\) and \\(l\\) are themselves marginalized out, and the result has no dependence on a given iteration’s value for \\(e\\) and \\(l\\). A final normalization then produces the quantity of interest, the posterior probability of the change point being \\(s\\) conditioned on the data \\(D\\), \\[\np(s \\mid D) = \\frac{q(s \\mid D)}{\\sum_{s'=1}^T q(s' \\mid D)}.\n\\]\nA plot of the values of \\(\\log p(s \\mid D)\\) computed using Stan 2.4’s default MCMC implementation is shown in the posterior plot.\nLog probability of change point being in year, calculated analytically.\n\n\n\nAnalytical change-point posterior\n\n\nThe frequency of change points generated by sampling the discrete change points.\n\n\n\nSampled change-point posterior\n\n\nIn order their range of estimates be visible, the first plot is on the log scale and the second plot on the linear scale; note the narrower range of years in the second plot resulting from sampling. The posterior mean of \\(s\\) is roughly 1891.\n\n\n\nThe generated quantities block may be used to draw discrete parameter values using the built-in pseudo-random number generators. For example, with lp defined as above, the following program draws a random value for s at every iteration.\ngenerated quantities {\n  int&lt;lower=1, upper=T&gt; s;\n  s = categorical_logit_rng(lp);\n}\nA posterior histogram of draws for \\(s\\) is shown on the second change point posterior figure above.\nCompared to working in terms of expectations, discrete sampling is highly inefficient, especially for tails of distributions, so this approach should only be used if draws from a distribution are explicitly required. Otherwise, expectations should be computed in the generated quantities block based on the posterior distribution for s given by softmax(lp).\n\n\n\nThe discrete sample generated for \\(s\\) can be used to calculate covariance with other parameters. Although the sampling approach is straightforward, it is more statistically efficient (in the sense of requiring far fewer iterations for the same degree of accuracy) to calculate these covariances in expectation using lp.\n\n\n\nThere is no obstacle in principle to allowing multiple change points. The only issue is that computation increases from linear to quadratic in marginalizing out two change points, cubic for three change points, and so on. There are three parameters, e, m, and l, and two loops for the change point and then one over time, with log densities being stored in a matrix.\nmatrix[T, T] lp;\nlp = rep_matrix(log_unif, T);\nfor (s1 in 1:T) {\n  for (s2 in 1:T) {\n    for (t in 1:T) {\n      lp[s1,s2] = lp[s1,s2]\n        + poisson_lpmf(D[t] | t &lt; s1 ? e : (t &lt; s2 ? m : l));\n    }\n  }\n}\nThe matrix can then be converted back to a vector using to_vector before being passed to log_sum_exp.\n\n\n\n\nA widely applied field method in ecology is to capture (or sight) animals, mark them (e.g., by tagging), then release them. This process is then repeated one or more times, and is often done for populations on an ongoing basis. The resulting data may be used to estimate population size.\nThe first subsection describes a simple mark-recapture model that does not involve any latent discrete parameters. The following subsections describes the Cormack-Jolly-Seber model, which involves latent discrete parameters for animal death.\n\n\nIn the simplest case, a one-stage mark-recapture study produces the following data\n\n\\(M\\) : number of animals marked in first capture,\n\\(C\\) : number animals in second capture, and\n\\(R\\) : number of marked animals in second capture.\n\nThe estimand of interest is\n\n\\(N\\) : number of animals in the population.\n\nDespite the notation, the model will take \\(N\\) to be a continuous parameter; just because the population must be finite doesn’t mean the parameter representing it must be. The parameter will be used to produce a real-valued estimate of the population size.\nThe Lincoln-Petersen (Lincoln 1930; Petersen 1896) method for estimating population size is \\[\n\\hat{N} = \\frac{M C}{R}.\n\\]\nThis population estimate would arise from a probabilistic model in which the number of recaptured animals is distributed binomially, \\[\nR \\sim \\textsf{binomial}(C, M / N)\n\\] given the total number of animals captured in the second round (\\(C\\)) with a recapture probability of \\(M/N\\), the fraction of the total population \\(N\\) marked in the first round.\ndata {\n  int&lt;lower=0&gt; M;\n  int&lt;lower=0&gt; C;\n  int&lt;lower=0, upper=min(M, C)&gt; R;\n}\nparameters {\n  real&lt;lower=(C - R + M)&gt; N;\n}\nmodel {\n  R ~ binomial(C, M / N);\n}\nA probabilistic formulation of the Lincoln-Petersen estimator for population size based on data from a one-step mark-recapture study. The lower bound on \\(N\\) is necessary to efficiently eliminate impossible values.\nThe probabilistic variant of the Lincoln-Petersen estimator can be directly coded in Stan as shown in the Lincon-Petersen model figure. The Lincoln-Petersen estimate is the maximum likelihood estimate (MLE) for this model.\nTo ensure the MLE is the Lincoln-Petersen estimate, an improper uniform prior for \\(N\\) is used; this could (and should) be replaced with a more informative prior if possible, based on knowledge of the population under study.\nThe one tricky part of the model is the lower bound \\(C - R + M\\) placed on the population size \\(N\\). Values below this bound are impossible because it is otherwise not possible to draw \\(R\\) samples out of the \\(C\\) animals recaptured. Implementing this lower bound is necessary to ensure sampling and optimization can be carried out in an unconstrained manner with unbounded support for parameters on the transformed (unconstrained) space. The lower bound in the declaration for \\(C\\) implies a variable transform \\(f : (C-R+M,\\infty) \\rightarrow (-\\infty,+\\infty)\\) defined by \\(f(N) = \\log(N - (C - R + M))\\); the reference manual contains full details of all constrained parameter transforms.\n\n\n\nThe Cormack-Jolly-Seber (CJS) model (Cormack 1964; Jolly 1965; Seber 1965) is an open-population model in which the population may change over time due to death; the presentation here draws heavily on Schofield (2007).\nThe basic data are\n\n\\(I\\): number of individuals,\n\\(T\\): number of capture periods, and\n\\(y_{i,t}\\): Boolean indicating if individual \\(i\\) was captured at time \\(t\\).\n\nEach individual is assumed to have been captured at least once because an individual only contributes information conditionally after they have been captured the first time.\nThere are two Bernoulli parameters in the model,\n\n\\(\\phi_t\\) : probability that animal alive at time \\(t\\) survives until \\(t + 1\\) and\n\\(p_t\\) : probability that animal alive at time \\(t\\) is captured at time \\(t\\).\n\nThese parameters will both be given uniform priors, but information should be used to tighten these priors in practice.\nThe CJS model also employs a latent discrete parameter \\(z_{i,t}\\) indicating for each individual \\(i\\) whether it is alive at time \\(t\\), distributed as \\[\nz_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t-1} \\; ? \\; 0 \\: : \\: \\phi_{t-1}).\n\\]\nThe conditional prevents the model positing zombies; once an animal is dead, it stays dead. The data distribution is then simple to express conditional on \\(z\\) as \\[\ny_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t} \\; ? \\; 0 \\: : \\: p_t).\n\\]\nThe conditional enforces the constraint that dead animals cannot be captured.\n\n\n\nThis subsection presents an implementation of the model in terms of counts for different history profiles for individuals over three capture times. It assumes exchangeability of the animals in that each is assigned the same capture and survival probabilities.\nIn order to ease the marginalization of the latent discrete parameter \\(z_{i,t}\\), the Stan models rely on a derived quantity \\(\\chi_t\\) for the probability that an individual is never captured again if it is alive at time \\(t\\) (if it is dead, the recapture probability is zero). this quantity is defined recursively by \\[\n\\chi_t\n=\n\\begin{cases}\n1 & \\quad\\text{if } t = T \\\\\n(1 - \\phi_t) + \\phi_t (1 - p_{t+1}) \\chi_{t+1}\n  & \\quad\\text{if } t &lt; T\n\\end{cases}\n\\]\nThe base case arises because if an animal was captured in the last time period, the probability it is never captured again is 1 because there are no more capture periods. The recursive case defining \\(\\chi_{t}\\) in terms of \\(\\chi_{t+1}\\) involves two possibilities: (1) not surviving to the next time period, with probability \\((1 - \\phi_t)\\), or (2) surviving to the next time period with probability \\(\\phi_t\\), not being captured in the next time period with probability \\((1 - p_{t+1})\\), and not being captured again after being alive in period \\(t+1\\) with probability \\(\\chi_{t+1}\\).\nWith three capture times, there are eight captured/not-captured profiles an individual may have. These may be naturally coded as binary numbers as follows.\n\\[\n\\begin{array}{crclc}\n\\hline\n& \\qquad\\qquad & captures & \\qquad\\qquad &  \\\\\n\\mathrm{profile} & 1 & 2 & 3 & \\mathrm{probability} \\\\\n\\hline\n0 & - & - & - & n/a \\\\\n1 & - & - & + & n/a \\\\\n2 & - & + & - & \\chi_2 \\\\\n3 & - & + & + & \\phi_2 \\, p_3 \\\\\n4 & + & - & - & \\chi_1 \\\\\n5 & + & - & + & \\phi_1 \\, (1 - p_2) \\, \\phi_2 \\, p_3 \\\\\n6 & + & + & - & \\phi_1 \\, p_2 \\, \\chi_2 \\\\\n7 & + & + & + & \\phi_1 \\, p_2 \\, \\phi_2 \\, p_3 \\\\\n\\hline\n\\end{array}\n\\]\nHistory 0, for animals that are never captured, is unobservable because only animals that are captured are observed. History 1, for animals that are only captured in the last round, provides no information for the CJS model, because capture/non-capture status is only informative when conditioned on earlier captures. For the remaining cases, the contribution to the likelihood is provided in the final column.\nBy defining these probabilities in terms of \\(\\chi\\) directly, there is no need for a latent binary parameter indicating whether an animal is alive at time \\(t\\) or not. The definition of \\(\\chi\\) is typically used to define the likelihood (i.e., marginalize out the latent discrete parameter) for the CJS model (Schofield 2007).\nThe Stan model defines \\(\\chi\\) as a transformed parameter based on parameters \\(\\phi\\) and \\(p\\). In the model block, the log probability is incremented for each history based on its count. This second step is similar to collecting Bernoulli observations into a binomial or categorical observations into a multinomial, only it is coded directly in the Stan program using target += rather than being part of a built-in probability function.\nThe following is the Stan program for the Cormack-Jolly-Seber mark-recapture model that considers counts of individuals with observation histories of being observed or not in three capture periods\ndata {\n  array[7] int&lt;lower=0&gt; history;\n}\nparameters {\n  array[2] real&lt;lower=0, upper=1&gt; phi;\n  array[3] real&lt;lower=0, upper=1&gt; p;\n}\ntransformed parameters {\n  array[2] real&lt;lower=0, upper=1&gt; chi;\n  chi[2] = (1 - phi[2]) + phi[2] * (1 - p[3]);\n  chi[1] = (1 - phi[1]) + phi[1] * (1 - p[2]) * chi[2];\n}\nmodel {\n  target += history[2] * log(chi[2]);\n  target += history[3] * (log(phi[2]) + log(p[3]));\n  target += history[4] * (log(chi[1]));\n  target += history[5] * (log(phi[1]) + log1m(p[2])\n                            + log(phi[2]) + log(p[3]));\n  target += history[6] * (log(phi[1]) + log(p[2])\n                            + log(chi[2]));\n  target += history[7] * (log(phi[1]) + log(p[2])\n                            + log(phi[2]) + log(p[3]));\n}\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; beta3;\n  beta3 = phi[2] * p[3];\n}\n\n\nThe parameters \\(\\phi_2\\) and \\(p_3\\), the probability of death at time 2 and probability of capture at time 3 are not identifiable, because both may be used to account for lack of capture at time 3. Their product, \\(\\beta_3 = \\phi_2 \\, p_3\\), is identified. The Stan model defines beta3 as a generated quantity. Unidentified parameters pose a problem for Stan’s samplers’ adaptation. Although the problem posed for adaptation is mild here because the parameters are bounded and thus have proper uniform priors, it would be better to formulate an identified parameterization. One way to do this would be to formulate a hierarchical model for the \\(p\\) and \\(\\phi\\) parameters.\n\n\n\n\nThis section presents a version of the Cormack-Jolly-Seber (CJS) model cast at the individual level rather than collectively as in the previous subsection. It also extends the model to allow an arbitrary number of time periods. The data will consist of the number \\(T\\) of capture events, the number \\(I\\) of individuals, and a boolean flag \\(y_{i,t}\\) indicating if individual \\(i\\) was observed at time \\(t\\). In Stan,\ndata {\n  int&lt;lower=2&gt; T;\n  int&lt;lower=0&gt; I;\n  array[I, T] int&lt;lower=0, upper=1&gt; y;\n}\nThe advantages to the individual-level model is that it becomes possible to add individual “random effects” that affect survival or capture probability, as well as to avoid the combinatorics involved in unfolding \\(2^T\\) observation histories for \\(T\\) capture times.\n\n\nThe individual CJS model is written involves several function definitions. The first two are used in the transformed data block to compute the first and last time period in which an animal was captured.4\nfunctions {\n  int first_capture(array[] int y_i) {\n    for (k in 1:size(y_i)) {\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  int last_capture(array[] int y_i) {\n    for (k_rev in 0:(size(y_i) - 1)) {\n      int k;\n      k = size(y_i) - k_rev;\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  // ...\n}\nThese two functions are used to define the first and last capture time for each individual in the transformed data block.5\ntransformed data {\n  array[I] int&lt;lower=0, upper=T&gt; first;\n  array[I] int&lt;lower=0, upper=T&gt; last;\n  vector&lt;lower=0, upper=I&gt;[T] n_captured;\n  for (i in 1:I) {\n    first[i] = first_capture(y[i]);\n  }\n  for (i in 1:I) {\n    last[i] = last_capture(y[i]);\n  }\n  n_captured = rep_vector(0, T);\n  for (t in 1:T) {\n    for (i in 1:I) {\n      if (y[i, t]) {\n        n_captured[t] = n_captured[t] + 1;\n      }\n    }\n  }\n}\nThe transformed data block also defines n_captured[t], which is the total number of captures at time t. The variable n_captured is defined as a vector instead of an integer array so that it can be used in an elementwise vector operation in the generated quantities block to model the population estimates at each time point.\nThe parameters and transformed parameters are as before, but now there is a function definition for computing the entire vector chi, the probability that if an individual is alive at t that it will never be captured again.\nparameters {\n  vector&lt;lower=0, upper=1&gt;[T - 1] phi;\n  vector&lt;lower=0, upper=1&gt;[T] p;\n}\ntransformed parameters {\n  vector&lt;lower=0, upper=1&gt;[T] chi;\n  chi = prob_uncaptured(T, p, phi);\n}\nThe definition of prob_uncaptured, from the functions block, is\nfunctions {\n  // ...\n  vector prob_uncaptured(int T, vector p, vector phi) {\n    vector[T] chi;\n    chi[T] = 1.0;\n    for (t in 1:(T - 1)) {\n      int t_curr;\n      int t_next;\n      t_curr = T - t;\n      t_next = t_curr + 1;\n      chi[t_curr] = (1 - phi[t_curr])\n                     + phi[t_curr]\n                       * (1 - p[t_next])\n                       * chi[t_next];\n    }\n    return chi;\n  }\n}\nThe function definition directly follows the mathematical definition of \\(\\chi_t\\), unrolling the recursion into an iteration and defining the elements of chi from T down to 1.\n\n\n\nGiven the precomputed quantities, the model block directly encodes the CJS model’s log likelihood function. All parameters are left with their default uniform priors and the model simply encodes the log probability of the observations q given the parameters p and phi as well as the transformed parameter chi defined in terms of p and phi.\nmodel {\n  for (i in 1:I) {\n    if (first[i] &gt; 0) {\n      for (t in (first[i]+1):last[i]) {\n        1 ~ bernoulli(phi[t - 1]);\n        y[i, t] ~ bernoulli(p[t]);\n      }\n      1 ~ bernoulli(chi[last[i]]);\n    }\n  }\n}\nThe outer loop is over individuals, conditional skipping individuals i which are never captured. The never-captured check depends on the convention of the first-capture and last-capture functions returning 0 for first if an individual is never captured.\nThe inner loop for individual i first increments the log probability based on the survival of the individual with probability phi[t - 1]. The outcome of 1 is fixed because the individual must survive between the first and last capture (i.e., no zombies). The loop starts after the first capture, because all information in the CJS model is conditional on the first capture.\nIn the inner loop, the observed capture status y[i, t] for individual i at time t has a Bernoulli distribution based on the capture probability p[t] at time t.\nAfter the inner loop, the probability of an animal never being seen again after being observed at time last[i] is included, because last[i] was defined to be the last time period in which animal i was observed.\n\n\n\nAs with the collective model described in the previous subsection, this model does not identify phi[T - 1] and p[T], but does identify their product, beta. Thus beta is defined as a generated quantity to monitor convergence and report.\ngenerated quantities {\n  real beta;\n  // ...\n\n  beta = phi[T - 1] * p[T];\n  // ...\n}\nThe parameter p[1] is also not modeled and will just be uniform between 0 and 1. A more finely articulated model might have a hierarchical or time-series component, in which case p[1] would be an unknown initial condition and both phi[T - 1] and p[T] could be identified.\n\n\n\nThe generated quantities also calculates an estimate of the population mean at each time t in the same way as in the simple mark-recapture model as the number of individuals captured at time t divided by the probability of capture at time t. This is done with the elementwise division operation for vectors (./) in the generated quantities block.\ngenerated quantities {\n  // ...\n  vector&lt;lower=0&gt;[T] pop;\n  // ...\n  pop = n_captured ./ p;\n  pop[1] = -1;\n}\n\n\n\nAll individuals are modeled as having the same capture probability, but this model could be easily generalized to use a logistic regression here based on individual-level inputs to be used as predictors.\n\n\n\n\n\nAlthough seemingly disparate tasks, the rating/coding/annotation of items with categories and diagnostic testing for disease or other conditions, share several characteristics which allow their statistical properties to be modeled similarly.\n\n\nSuppose you have diagnostic tests for a condition of varying sensitivity and specificity. Sensitivity is the probability a test returns positive when the patient has the condition and specificity is the probability that a test returns negative when the patient does not have the condition. For example, mammograms and puncture biopsy tests both test for the presence of breast cancer. Mammograms have high sensitivity and low specificity, meaning lots of false positives, whereas puncture biopsies are the opposite, with low sensitivity and high specificity, meaning lots of false negatives.\nThere are several estimands of interest in such studies. An epidemiological study may be interested in the prevalence of a kind of infection, such as malaria, in a population. A test development study might be interested in the diagnostic accuracy of a new test. A health care worker performing tests might be interested in the disease status of a particular patient.\n\n\n\nHumans are often given the task of coding (equivalently rating or annotating) data. For example, journal or grant reviewers rate submissions, a political study may code campaign commercials as to whether they are attack ads or not, a natural language processing study might annotate Tweets as to whether they are positive or negative in overall sentiment, or a dentist looking at an X-ray classifies a patient as having a cavity or not. In all of these cases, the data coders play the role of the diagnostic tests and all of the same estimands are in play — data coder accuracy and bias, true categories of items being coded, or the prevalence of various categories of items in the data.\n\n\n\nIn this section, only categorical ratings are considered, and the challenge in the modeling for Stan is to marginalize out the discrete parameters.\nDawid and Skene (1979) introduce a noisy-measurement model for coding and apply it in the epidemiological setting of coding what doctors say about patient histories; the same model can be used for diagnostic procedures.\n\n\nThe data for the model consists of \\(J\\) raters (diagnostic tests), \\(I\\) items (patients), and \\(K\\) categories (condition statuses) to annotate, with \\(y_{i, j} \\in \\{1, \\dotsc, K\\}\\) being the rating provided by rater \\(j\\) for item \\(i\\). In a diagnostic test setting for a particular condition, the raters are diagnostic procedures and often \\(K=2\\), with values signaling the presence or absence of the condition.6\nIt is relatively straightforward to extend Dawid and Skene’s model to deal with the situation where not every rater rates each item exactly once.\n\n\n\n\nThe model is based on three parameters, the first of which is discrete:\n\n\\(z_i\\) : a value in \\(\\{1, \\dotsc, K\\}\\) indicating the true category of item \\(i\\),\n\\(\\pi\\) : a \\(K\\)-simplex for the prevalence of the \\(K\\) categories in the population, and\n\\(\\theta_{j,k}\\) : a \\(K\\)-simplex for the response of annotator \\(j\\) to an item of true category \\(k\\).\n\n\n\n\nThe true category of an item is assumed to be generated by a simple categorical distribution based on item prevalence, \\[\nz_i \\sim \\textsf{categorical}(\\pi).\n\\]\nThe rating \\(y_{i, j}\\) provided for item \\(i\\) by rater \\(j\\) is modeled as a categorical response of rater \\(i\\) to an item of category \\(z_i\\),7 \\[\ny_{i, j} \\sim \\textsf{categorical}(\\theta_{j,\\pi_{z[i]}}).\n\\]\n\n\nDawid and Skene provided maximum likelihood estimates for \\(\\theta\\) and \\(\\pi\\), which allows them to generate probability estimates for each \\(z_i\\).\nTo mimic Dawid and Skene’s maximum likelihood model, the parameters \\(\\theta_{j,k}\\) and \\(\\pi\\) can be given uniform priors over \\(K\\)-simplexes. It is straightforward to generalize to Dirichlet priors, \\[\n\\pi \\sim \\textsf{Dirichlet}(\\alpha)\n\\] and \\[\n\\theta_{j,k} \\sim \\textsf{Dirichlet}(\\beta_k)\n\\] with fixed hyperparameters \\(\\alpha\\) (a vector) and \\(\\beta\\) (a matrix or array of vectors). The prior for \\(\\theta_{j,k}\\) must be allowed to vary in \\(k\\), so that, for instance, \\(\\beta_{k,k}\\) is large enough to allow the prior to favor better-than-chance annotators over random or adversarial ones.\nBecause there are \\(J\\) coders, it would be natural to extend the model to include a hierarchical prior for \\(\\beta\\) and to partially pool the estimates of coder accuracy and bias.\n\n\n\nBecause the true category parameter \\(z\\) is discrete, it must be marginalized out of the joint posterior in order to carry out sampling or maximum likelihood estimation in Stan. The joint posterior factors as \\[\np(y, \\theta, \\pi) = p(y \\mid \\theta,\\pi) \\, p(\\pi) \\, p(\\theta),\n\\] where \\(p(y \\mid \\theta,\\pi)\\) is derived by marginalizing \\(z\\) out of \\[\np(z, y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\left( \\textsf{categorical}(z_i \\mid \\pi)\n                     \\prod_{j=1}^J\n                     \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, z[i]})\n              \\right).\n\\]\nThis can be done item by item, with \\[\np(y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\sum_{k=1}^K\n  \\left( \\textsf{categorical}(k \\mid \\pi)\n         \\prod_{j=1}^J\n         \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n  \\right).\n\\]\nIn the missing data model, only the observed labels would be used in the inner product.\nDawid and Skene (1979) derive exactly the same equation in their Equation (2.7), required for the E-step in their expectation maximization (EM) algorithm. Stan requires the marginalized probability function on the log scale, \\[\\begin{align*}\n\\log p(y \\mid \\theta, \\pi)\n&= \\sum_{i=1}^I \\log \\left( \\sum_{k=1}^K \\exp\n    \\left(\\log \\textsf{categorical}(k \\mid \\pi) \\vphantom{\\sum_{j=1}^J}\\right.\\right.\n    \\left.\\left. + \\ \\sum_{j=1}^J\n           \\log \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n    \\right) \\right),\n\\end{align*}\\] which can be directly coded using Stan’s built-in log_sum_exp function.\n\n\n\n\nThe Stan program for the Dawid and Skene model is provided below (Dawid and Skene 1979).\ndata {\n  int&lt;lower=2&gt; K;\n  int&lt;lower=1&gt; I;\n  int&lt;lower=1&gt; J;\n\n  array[I, J] int&lt;lower=1, upper=K&gt; y;\n\n  vector&lt;lower=0&gt;[K] alpha;\n  vector&lt;lower=0&gt;[K] beta[K];\n}\nparameters {\n  simplex[K] pi;\n  array[J, K] simplex[K] theta;\n}\ntransformed parameters {\n  array[I] vector[K] log_q_z;\n  for (i in 1:I) {\n    log_q_z[i] = log(pi);\n    for (j in 1:J) {\n      for (k in 1:K) {\n        log_q_z[i, k] = log_q_z[i, k]\n                         + log(theta[j, k, y[i, j]]);\n      }\n    }\n  }\n}\nmodel {\n  pi ~ dirichlet(alpha);\n  for (j in 1:J) {\n    for (k in 1:K) {\n      theta[j, k] ~ dirichlet(beta[k]);\n    }\n  }\n\n  for (i in 1:I) {\n    target += log_sum_exp(log_q_z[i]);\n  }\n}\nThe model marginalizes out the discrete parameter \\(z\\), storing the unnormalized conditional probability \\(\\log q(z_i=k|\\theta,\\pi)\\) in log_q_z[i, k].\nThe Stan model converges quickly and mixes well using NUTS starting at diffuse initial points, unlike the equivalent model implemented with Gibbs sampling over the discrete parameter. Reasonable weakly informative priors are \\(\\alpha_k = 3\\) and \\(\\beta_{k,k} = 2.5 K\\) and \\(\\beta_{k,k'} = 1\\) if \\(k \\neq k'\\). Taking \\(\\alpha\\) and \\(\\beta_k\\) to be unit vectors and applying optimization will produce the same answer as the expectation maximization (EM) algorithm of Dawid and Skene (1979).\n\n\nThe quantity log_q_z[i] is defined as a transformed parameter. It encodes the (unnormalized) log of \\(p(z_i \\mid \\theta,\n\\pi)\\). Each iteration provides a value conditioned on that iteration’s values for \\(\\theta\\) and \\(\\pi\\). Applying the softmax function to log_q_z[i] provides a simplex corresponding to the probability mass function of \\(z_i\\) in the posterior. These may be averaged across the iterations to provide the posterior probability distribution over each \\(z_i\\).\n\n\n\n\n\n\n\nThis section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z)] \\mid \\Theta] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
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-    "text": "This section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z] \\mid \\Theta)] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
+    "text": "This section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z)] \\mid \\Theta] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
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diff --git a/docs/2_34/site_libs/bootstrap/bootstrap-dark.min.css b/docs/2_34/site_libs/bootstrap/bootstrap-dark.min.css
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+@import"https://fonts.googleapis.com/css2?family=Source+Sans+Pro:wght@300;400;700&display=swap";:root{--stan-bg: #242424;--stan-highlight: #053A42;--stan-secondary: #AED1E4}/*!
+ * Bootstrap  v5.3.1 (https://getbootstrap.com/)
+ * Copyright 2011-2023 The Bootstrap Authors
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none}.btn-outline-primary{--bs-btn-color: #2780e3;--bs-btn-border-color: #2780e3;--bs-btn-hover-color: #fff;--bs-btn-hover-bg: #2780e3;--bs-btn-hover-border-color: #2780e3;--bs-btn-focus-shadow-rgb: 39, 128, 227;--bs-btn-active-color: #fff;--bs-btn-active-bg: #2780e3;--bs-btn-active-border-color: #2780e3;--bs-btn-active-shadow: inset 0 3px 5px rgba(0, 0, 0, 0.125);--bs-btn-disabled-color: #2780e3;--bs-btn-disabled-bg: transparent;--bs-btn-disabled-border-color: #2780e3;--bs-btn-bg: transparent;--bs-gradient: none}.btn-outline-secondary{--bs-btn-color: #343a40;--bs-btn-border-color: #343a40;--bs-btn-hover-color: #fff;--bs-btn-hover-bg: #343a40;--bs-btn-hover-border-color: #343a40;--bs-btn-focus-shadow-rgb: 52, 58, 64;--bs-btn-active-color: #fff;--bs-btn-active-bg: #343a40;--bs-btn-active-border-color: #343a40;--bs-btn-active-shadow: inset 0 3px 5px rgba(0, 0, 0, 0.125);--bs-btn-disabled-color: #343a40;--bs-btn-disabled-bg: transparent;--bs-btn-disabled-border-color: #343a40;--bs-btn-bg: 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var(--bs-dropdown-item-padding-x);clear:both;font-weight:400;color:var(--bs-dropdown-link-color);text-align:inherit;text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;white-space:nowrap;background-color:rgba(0,0,0,0);border:0}.dropdown-item:hover,.dropdown-item:focus{color:var(--bs-dropdown-link-hover-color);background-color:var(--bs-dropdown-link-hover-bg)}.dropdown-item.active,.dropdown-item:active{color:var(--bs-dropdown-link-active-color);text-decoration:none;background-color:var(--bs-dropdown-link-active-bg)}.dropdown-item.disabled,.dropdown-item:disabled{color:var(--bs-dropdown-link-disabled-color);pointer-events:none;background-color:rgba(0,0,0,0)}.dropdown-menu.show{display:block}.dropdown-header{display:block;padding:var(--bs-dropdown-header-padding-y) 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auto}.btn-group>.btn-check:checked+.btn,.btn-group>.btn-check:focus+.btn,.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active,.btn-group-vertical>.btn-check:checked+.btn,.btn-group-vertical>.btn-check:focus+.btn,.btn-group-vertical>.btn:hover,.btn-group-vertical>.btn:focus,.btn-group-vertical>.btn:active,.btn-group-vertical>.btn.active{z-index:1}.btn-toolbar{display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;justify-content:flex-start;-webkit-justify-content:flex-start}.btn-toolbar .input-group{width:auto}.btn-group>:not(.btn-check:first-child)+.btn,.btn-group>.btn-group:not(:first-child){margin-left:calc(1px*-1)}.dropdown-toggle-split{padding-right:.5625rem;padding-left:.5625rem}.dropdown-toggle-split::after,.dropup .dropdown-toggle-split::after,.dropend .dropdown-toggle-split::after{margin-left:0}.dropstart .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:.375rem;padding-left:.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:.75rem;padding-left:.75rem}.btn-group-vertical{flex-direction:column;-webkit-flex-direction:column;align-items:flex-start;-webkit-align-items:flex-start;justify-content:center;-webkit-justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:calc(1px*-1)}.nav{--bs-nav-link-padding-x: 1rem;--bs-nav-link-padding-y: 0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color: #AED1E4;--bs-nav-link-hover-color: #8ba7b6;--bs-nav-link-disabled-color: rgba(250, 241, 228, 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#2780e3}.nav-pills .nav-link.active,.nav-pills .show>.nav-link{color:var(--bs-nav-pills-link-active-color);background-color:var(--bs-nav-pills-link-active-bg)}.nav-underline{--bs-nav-underline-gap: 1rem;--bs-nav-underline-border-width: 0.125rem;--bs-nav-underline-link-active-color: #000;gap:var(--bs-nav-underline-gap)}.nav-underline .nav-link{padding-right:0;padding-left:0;border-bottom:var(--bs-nav-underline-border-width) solid rgba(0,0,0,0)}.nav-underline .nav-link:hover,.nav-underline .nav-link:focus{border-bottom-color:currentcolor}.nav-underline .nav-link.active,.nav-underline .show>.nav-link{font-weight:700;color:var(--bs-nav-underline-link-active-color);border-bottom-color:currentcolor}.nav-fill>.nav-link,.nav-fill .nav-item{flex:1 1 auto;-webkit-flex:1 1 auto;text-align:center}.nav-justified>.nav-link,.nav-justified .nav-item{flex-basis:0;-webkit-flex-basis:0;flex-grow:1;-webkit-flex-grow:1;text-align:center}.nav-fill .nav-item .nav-link,.nav-justified .nav-item 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var(--bs-navbar-hover-color);--bs-nav-link-disabled-color: var(--bs-navbar-disabled-color);display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;list-style:none}.navbar-nav .nav-link.active,.navbar-nav .nav-link.show{color:var(--bs-navbar-active-color)}.navbar-nav .dropdown-menu{position:static}.navbar-text{padding-top:.5rem;padding-bottom:.5rem;color:var(--bs-navbar-color)}.navbar-text a,.navbar-text a:hover,.navbar-text a:focus{color:var(--bs-navbar-active-color)}.navbar-collapse{flex-basis:100%;-webkit-flex-basis:100%;flex-grow:1;-webkit-flex-grow:1;align-items:center;-webkit-align-items:center}.navbar-toggler{padding:var(--bs-navbar-toggler-padding-y) var(--bs-navbar-toggler-padding-x);font-size:var(--bs-navbar-toggler-font-size);line-height:1;color:var(--bs-navbar-color);background-color:rgba(0,0,0,0);border:var(--bs-border-width) solid 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992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #bdbdbd;--bs-navbar-hover-color: rgba(174, 209, 228, 0.8);--bs-navbar-disabled-color: rgba(189, 189, 189, 0.75);--bs-navbar-active-color: #aed1e4;--bs-navbar-brand-color: #bdbdbd;--bs-navbar-brand-hover-color: #aed1e4;--bs-navbar-toggler-border-color: rgba(189, 189, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 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0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #FAF1E4;--bs-accordion-bg: #181818;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #FAF1E4;--bs-accordion-btn-bg: #181818;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23FAF1E4'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(250, 241, 228, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(250, 241, 228, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #AED1E4;--bs-pagination-bg: #181818;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #8ba7b6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #8ba7b6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(250, 241, 228, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #FAF1E4;--bs-list-group-bg: #181818;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(250, 241, 228, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #FAF1E4;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(250, 241, 228, 0.75);--bs-list-group-disabled-bg: #181818;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(24, 24, 24, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(250, 241, 228, 0.75);--bs-toast-header-bg: rgba(24, 24, 24, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #181818;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;flex-wrap:wrap;-webkit-flex-wrap:wrap;align-items:center;-webkit-align-items:center;justify-content:flex-end;-webkit-justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid var(--bs-modal-footer-border-color)}.modal-footer>*{margin:calc(var(--bs-modal-footer-gap)*.5)}@media(min-width: 576px){.modal{--bs-modal-margin: 1.75rem;--bs-modal-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15)}.modal-dialog{max-width:var(--bs-modal-width);margin-right:auto;margin-left:auto}.modal-sm{--bs-modal-width: 300px}}@media(min-width: 992px){.modal-lg,.modal-xl{--bs-modal-width: 800px}}@media(min-width: 1200px){.modal-xl{--bs-modal-width: 1140px}}.modal-fullscreen{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen .modal-content{height:100%;border:0}.modal-fullscreen .modal-body{overflow-y:auto}@media(max-width: 575.98px){.modal-fullscreen-sm-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-sm-down .modal-content{height:100%;border:0}.modal-fullscreen-sm-down .modal-body{overflow-y:auto}}@media(max-width: 767.98px){.modal-fullscreen-md-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-md-down .modal-content{height:100%;border:0}.modal-fullscreen-md-down .modal-body{overflow-y:auto}}@media(max-width: 991.98px){.modal-fullscreen-lg-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-lg-down .modal-content{height:100%;border:0}.modal-fullscreen-lg-down .modal-body{overflow-y:auto}}@media(max-width: 1199.98px){.modal-fullscreen-xl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xl-down .modal-content{height:100%;border:0}.modal-fullscreen-xl-down .modal-body{overflow-y:auto}}@media(max-width: 1399.98px){.modal-fullscreen-xxl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xxl-down .modal-content{height:100%;border:0}.modal-fullscreen-xxl-down .modal-body{overflow-y:auto}}.tooltip{--bs-tooltip-zindex: 1080;--bs-tooltip-max-width: 200px;--bs-tooltip-padding-x: 0.5rem;--bs-tooltip-padding-y: 0.25rem;--bs-tooltip-margin: ;--bs-tooltip-font-size:0.875rem;--bs-tooltip-color: #181818;--bs-tooltip-bg: #000;--bs-tooltip-border-radius: 0.25rem;--bs-tooltip-opacity: 0.9;--bs-tooltip-arrow-width: 0.8rem;--bs-tooltip-arrow-height: 0.4rem;z-index:var(--bs-tooltip-zindex);display:block;margin:var(--bs-tooltip-margin);font-family:"Source Sans Pro",-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,"Helvetica Neue",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI 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var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout 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.nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-sm .navbar-nav-scroll{overflow:visible}.navbar-expand-sm .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-sm .navbar-toggler{display:none}.navbar-expand-sm .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-sm .offcanvas .offcanvas-header{display:none}.navbar-expand-sm .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 768px){.navbar-expand-md{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-md .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-md .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-md .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-md .navbar-nav-scroll{overflow:visible}.navbar-expand-md .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-md .navbar-toggler{display:none}.navbar-expand-md .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-md .offcanvas .offcanvas-header{display:none}.navbar-expand-md .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1200px){.navbar-expand-xl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xl .navbar-nav-scroll{overflow:visible}.navbar-expand-xl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xl .navbar-toggler{display:none}.navbar-expand-xl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xl .offcanvas .offcanvas-header{display:none}.navbar-expand-xl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1400px){.navbar-expand-xxl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xxl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xxl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xxl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xxl .navbar-nav-scroll{overflow:visible}.navbar-expand-xxl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xxl .navbar-toggler{display:none}.navbar-expand-xxl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #F2E5BD;--bs-navbar-hover-color: rgba(147, 176, 241, 0.8);--bs-navbar-disabled-color: rgba(242, 229, 189, 0.75);--bs-navbar-active-color: #93b0f1;--bs-navbar-brand-color: #F2E5BD;--bs-navbar-brand-hover-color: #93b0f1;--bs-navbar-toggler-border-color: rgba(242, 229, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #fff;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #fff;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #343a40;--bs-accordion-btn-bg: #fff;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23343a40'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #fff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 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140deg), #20c997 var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #3c7fbb;color:#fff}.bg-gradient-teal-purple{--bslib-color-fg: #fff;--bslib-color-bg: #3a918c;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #3a918c;color:#fff}.bg-gradient-teal-pink{--bslib-color-fg: #fff;--bslib-color-bg: #709193;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #709193;color:#fff}.bg-gradient-teal-red{--bslib-color-fg: #fff;--bslib-color-bg: #797971;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #797971;color:#fff}.bg-gradient-teal-orange{--bslib-color-fg: #000;--bslib-color-bg: #73be7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #73be7a;color:#000}.bg-gradient-teal-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #79a764;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #79a764;color:#fff}.bg-gradient-teal-green{--bslib-color-fg: #000;--bslib-color-bg: #2cc164;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #2cc164;color:#000}.bg-gradient-teal-cyan{--bslib-color-fg: #fff;--bslib-color-bg: #509aa5;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #9954bb var(--bg-gradient-end, 180%)) #509aa5;color:#fff}.bg-gradient-cyan-blue{--bslib-color-fg: #fff;--bslib-color-bg: #6b66cb;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #2780e3 var(--bg-gradient-end, 180%)) #6b66cb;color:#fff}.bg-gradient-cyan-indigo{--bslib-color-fg: #fff;--bslib-color-bg: #8539d1;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #8539d1;color:#fff}.bg-gradient-cyan-purple{--bslib-color-fg: #fff;--bslib-color-bg: #834ba2;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #834ba2;color:#fff}.bg-gradient-cyan-pink{--bslib-color-fg: #fff;--bslib-color-bg: #b94ba8;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #b94ba8;color:#fff}.bg-gradient-cyan-red{--bslib-color-fg: #fff;--bslib-color-bg: #c23287;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #c23287;color:#fff}.bg-gradient-cyan-orange{--bslib-color-fg: #fff;--bslib-color-bg: #bc788f;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #bc788f;color:#fff}.bg-gradient-cyan-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #c2617a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #c2617a;color:#fff}.bg-gradient-cyan-green{--bslib-color-fg: #fff;--bslib-color-bg: #757b7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #757b7a;color:#fff}.bg-gradient-cyan-teal{--bslib-color-fg: #fff;--bslib-color-bg: #6983ad;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #20c997 var(--bg-gradient-end, 180%)) #6983ad;color:#fff}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--bslib-sidebar-fg: var(--bs-emphasis-color, black);--bslib-sidebar-main-fg: var(--bs-card-color, var(--bs-body-color));--bslib-sidebar-main-bg: var(--bs-card-bg, var(--bs-body-bg));--bslib-sidebar-toggle-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.1);--bslib-sidebar-padding: calc(var(--bslib-spacer) * 1.5);--bslib-sidebar-icon-size: var(--bslib-spacer, 1rem);--bslib-sidebar-icon-button-size: calc(var(--bslib-sidebar-icon-size, 1rem) * 2);--bslib-sidebar-padding-icon: calc(var(--bslib-sidebar-icon-button-size, 2rem) * 1.5);--bslib-collapse-toggle-border-radius: var(--bs-border-radius, 0.25rem);--bslib-collapse-toggle-transform: 0deg;--bslib-sidebar-toggle-transition-easing: cubic-bezier(1, 0, 0, 1);--bslib-collapse-toggle-right-transform: 180deg;--bslib-sidebar-column-main: minmax(0, 1fr);display:grid !important;grid-template-columns:min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px)) var(--bslib-sidebar-column-main);position:relative;transition:grid-template-columns ease-in-out var(--bslib-sidebar-transition-duration);border:var(--bslib-sidebar-border);border-radius:var(--bslib-sidebar-border-radius)}@media(prefers-reduced-motion: reduce){.bslib-sidebar-layout{transition:none}}.bslib-sidebar-layout[data-bslib-sidebar-border=false]{border:none}.bslib-sidebar-layout[data-bslib-sidebar-border-radius=false]{border-radius:initial}.bslib-sidebar-layout>.main,.bslib-sidebar-layout>.sidebar{grid-row:1/2;border-radius:inherit;overflow:auto}.bslib-sidebar-layout>.main{grid-column:2/3;border-top-left-radius:0;border-bottom-left-radius:0;padding:var(--bslib-sidebar-padding);transition:padding var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout .sidebar-title{font-size:1.25rem;line-height:1.25;margin-top:0;margin-bottom:1rem;padding-bottom:1rem;border-bottom:var(--bslib-sidebar-border)}.bslib-sidebar-layout.sidebar-right{grid-template-columns:var(--bslib-sidebar-column-main) min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px))}.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/2;border-top-right-radius:0;border-bottom-right-radius:0;border-top-left-radius:inherit;border-bottom-left-radius:inherit}.bslib-sidebar-layout.sidebar-right>.sidebar{grid-column:2/3;border-right:none;border-left:var(--bslib-sidebar-vert-border);border-top-left-radius:0;border-bottom-left-radius:0}.bslib-sidebar-layout.sidebar-right>.collapse-toggle{grid-column:2/3;left:var(--bslib-sidebar-icon-size);right:unset;border:var(--bslib-collapse-toggle-border)}.bslib-sidebar-layout.sidebar-right>.collapse-toggle>.collapse-icon{transform:rotateY(var(--bslib-collapse-toggle-right-transform))}.bslib-sidebar-layout.sidebar-collapsed{--bslib-collapse-toggle-transform: 180deg;--bslib-collapse-toggle-right-transform: 0deg;--bslib-sidebar-vert-border: none;grid-template-columns:0 minmax(0, 1fr)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right{grid-template-columns:minmax(0, 1fr) 0}.bslib-sidebar-layout.sidebar-collapsed:not(.transitioning)>.sidebar>*{display:none}.bslib-sidebar-layout.sidebar-collapsed>.main{border-radius:inherit}.bslib-sidebar-layout.sidebar-collapsed:not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed>.collapse-toggle{color:var(--bslib-sidebar-main-fg);top:calc(var(--bslib-sidebar-overlap-counter, 0)*(var(--bslib-sidebar-icon-size) + var(--bslib-sidebar-padding)) + var(--bslib-sidebar-icon-size, 1rem)/2);right:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px))}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.collapse-toggle{left:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px));right:unset}@media(min-width: 576px){.bslib-sidebar-layout.transitioning>.sidebar>.sidebar-content{display:none}}@media(max-width: 575.98px){.bslib-sidebar-layout[data-bslib-sidebar-open=desktop]{--bslib-sidebar-js-init-collapsed: true}.bslib-sidebar-layout>.sidebar,.bslib-sidebar-layout.sidebar-right>.sidebar{border:none}.bslib-sidebar-layout>.main,.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/3}.bslib-sidebar-layout[data-bslib-sidebar-open=always]{display:block !important}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar{max-height:var(--bslib-sidebar-max-height-mobile);overflow-y:auto;border-top:var(--bslib-sidebar-vert-border)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]){grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.sidebar{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.collapse-toggle{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed.sidebar-right{grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always])>.main{opacity:0;transition:opacity var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed>.main{opacity:1}}.accordion .accordion-header{font-size:calc(1.29rem + 0.48vw);margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2;color:var(--bs-heading-color);margin-bottom:0}@media(min-width: 1200px){.accordion .accordion-header{font-size:1.65rem}}.accordion .accordion-icon:not(:empty){margin-right:.75rem;display:flex}.accordion .accordion-button:not(.collapsed){box-shadow:none}.accordion .accordion-button:not(.collapsed):focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow)}:root{--bslib-page-sidebar-title-bg: #052744;--bslib-page-sidebar-title-color: #fff}.bslib-page-title{background-color:var(--bslib-page-sidebar-title-bg);color:var(--bslib-page-sidebar-title-color);font-size:1.25rem;font-weight:300;padding:var(--bslib-spacer, 1rem);padding-left:1.5rem;margin-bottom:0;border-bottom:1px solid #dee2e6}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.form-inline{margin-top:auto;margin-bottom:auto}.nav:not(.nav-hidden).nav-stacked{flex-direction:column;-webkit-flex-direction:column;height:100%}.nav:not(.nav-hidden).nav-stacked>.bslib-nav-spacer{margin-top:auto !important}}.bslib-card{overflow:auto}.bslib-card .card-body+.card-body{padding-top:0}.bslib-card .card-body{overflow:auto}.bslib-card .card-body p{margin-top:0}.bslib-card .card-body p:last-child{margin-bottom:0}.bslib-card .card-body{max-height:var(--bslib-card-body-max-height, none)}.bslib-card[data-full-screen=true]>.card-body{max-height:var(--bslib-card-body-max-height-full-screen, none)}.bslib-card .card-header .form-group{margin-bottom:0}.bslib-card .card-header .selectize-control{margin-bottom:0}.bslib-card .card-header .selectize-control .item{margin-right:1.15rem}.bslib-card .card-footer{margin-top:auto}.bslib-card .bslib-navs-card-title{display:flex;flex-wrap:wrap;justify-content:space-between;align-items:center}.bslib-card .bslib-navs-card-title .nav{margin-left:auto}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border=true]){border:none}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border-radius=true]){border-top-left-radius:0;border-top-right-radius:0}[data-full-screen=true]{position:fixed;inset:3.5rem 1rem 1rem;height:auto !important;max-height:none !important;width:auto !important;z-index:1070}.bslib-full-screen-enter{display:none;position:absolute;bottom:var(--bslib-full-screen-enter-bottom, 0.2rem);right:var(--bslib-full-screen-enter-right, 0);top:var(--bslib-full-screen-enter-top);left:var(--bslib-full-screen-enter-left);color:var(--bslib-color-fg, var(--bs-card-color));background-color:var(--bslib-color-bg, var(--bs-card-bg, var(--bs-body-bg)));border:var(--bs-card-border-width) solid var(--bslib-color-fg, var(--bs-card-border-color));box-shadow:0 2px 4px rgba(0,0,0,.15);margin:.2rem .4rem;padding:.55rem !important;font-size:.8rem;cursor:pointer;opacity:.7;z-index:1070}.bslib-full-screen-enter:hover{opacity:1}.card[data-full-screen=false]:hover>*>.bslib-full-screen-enter{display:block}.bslib-has-full-screen .card:hover>*>.bslib-full-screen-enter{display:none}@media(max-width: 575.98px){.bslib-full-screen-enter{display:none !important}}.bslib-full-screen-exit{position:relative;top:1.35rem;font-size:.9rem;cursor:pointer;text-decoration:none;display:flex;float:right;margin-right:2.15rem;align-items:center;color:rgba(var(--bs-body-bg-rgb), 0.8)}.bslib-full-screen-exit:hover{color:rgba(var(--bs-body-bg-rgb), 1)}.bslib-full-screen-exit svg{margin-left:.5rem;font-size:1.5rem}#bslib-full-screen-overlay{position:fixed;inset:0;background-color:rgba(var(--bs-body-color-rgb), 0.6);backdrop-filter:blur(2px);-webkit-backdrop-filter:blur(2px);z-index:1069;animation:bslib-full-screen-overlay-enter 400ms cubic-bezier(0.6, 0.02, 0.65, 1) 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.nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-sm .navbar-nav-scroll{overflow:visible}.navbar-expand-sm .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-sm .navbar-toggler{display:none}.navbar-expand-sm .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-sm .offcanvas .offcanvas-header{display:none}.navbar-expand-sm .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 768px){.navbar-expand-md{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-md .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-md .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-md .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-md .navbar-nav-scroll{overflow:visible}.navbar-expand-md .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-md .navbar-toggler{display:none}.navbar-expand-md .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-md .offcanvas .offcanvas-header{display:none}.navbar-expand-md .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1200px){.navbar-expand-xl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xl .navbar-nav-scroll{overflow:visible}.navbar-expand-xl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xl .navbar-toggler{display:none}.navbar-expand-xl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xl .offcanvas .offcanvas-header{display:none}.navbar-expand-xl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1400px){.navbar-expand-xxl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xxl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xxl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xxl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xxl .navbar-nav-scroll{overflow:visible}.navbar-expand-xxl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xxl .navbar-toggler{display:none}.navbar-expand-xxl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #F2E5BD;--bs-navbar-hover-color: rgba(147, 176, 241, 0.8);--bs-navbar-disabled-color: rgba(242, 229, 189, 0.75);--bs-navbar-active-color: #93b0f1;--bs-navbar-brand-color: #F2E5BD;--bs-navbar-brand-hover-color: #93b0f1;--bs-navbar-toggler-border-color: rgba(242, 229, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #FEFBF2;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #FEFBF2;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #343a40;--bs-accordion-btn-bg: #FEFBF2;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23343a40'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #FEFBF2;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #FEFBF2;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #FEFBF2;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(254, 251, 242, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(254, 251, 242, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #FEFBF2;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 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140deg), #20c997 var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #3c7fbb;color:#fff}.bg-gradient-teal-purple{--bslib-color-fg: #fff;--bslib-color-bg: #3a918c;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #3a918c;color:#fff}.bg-gradient-teal-pink{--bslib-color-fg: #fff;--bslib-color-bg: #709193;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #709193;color:#fff}.bg-gradient-teal-red{--bslib-color-fg: #fff;--bslib-color-bg: #797971;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #797971;color:#fff}.bg-gradient-teal-orange{--bslib-color-fg: #000;--bslib-color-bg: #73be7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #73be7a;color:#000}.bg-gradient-teal-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #79a764;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #79a764;color:#fff}.bg-gradient-teal-green{--bslib-color-fg: #000;--bslib-color-bg: #2cc164;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #2cc164;color:#000}.bg-gradient-teal-cyan{--bslib-color-fg: #fff;--bslib-color-bg: #509aa5;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #9954bb var(--bg-gradient-end, 180%)) #509aa5;color:#fff}.bg-gradient-cyan-blue{--bslib-color-fg: #fff;--bslib-color-bg: #6b66cb;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #2780e3 var(--bg-gradient-end, 180%)) #6b66cb;color:#fff}.bg-gradient-cyan-indigo{--bslib-color-fg: #fff;--bslib-color-bg: #8539d1;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #8539d1;color:#fff}.bg-gradient-cyan-purple{--bslib-color-fg: #fff;--bslib-color-bg: #834ba2;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #834ba2;color:#fff}.bg-gradient-cyan-pink{--bslib-color-fg: #fff;--bslib-color-bg: #b94ba8;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #b94ba8;color:#fff}.bg-gradient-cyan-red{--bslib-color-fg: #fff;--bslib-color-bg: #c23287;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #c23287;color:#fff}.bg-gradient-cyan-orange{--bslib-color-fg: #fff;--bslib-color-bg: #bc788f;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #bc788f;color:#fff}.bg-gradient-cyan-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #c2617a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #c2617a;color:#fff}.bg-gradient-cyan-green{--bslib-color-fg: #fff;--bslib-color-bg: #757b7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #757b7a;color:#fff}.bg-gradient-cyan-teal{--bslib-color-fg: #fff;--bslib-color-bg: #6983ad;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #20c997 var(--bg-gradient-end, 180%)) #6983ad;color:#fff}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--bslib-sidebar-fg: var(--bs-emphasis-color, black);--bslib-sidebar-main-fg: var(--bs-card-color, var(--bs-body-color));--bslib-sidebar-main-bg: var(--bs-card-bg, var(--bs-body-bg));--bslib-sidebar-toggle-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.1);--bslib-sidebar-padding: calc(var(--bslib-spacer) * 1.5);--bslib-sidebar-icon-size: var(--bslib-spacer, 1rem);--bslib-sidebar-icon-button-size: calc(var(--bslib-sidebar-icon-size, 1rem) * 2);--bslib-sidebar-padding-icon: calc(var(--bslib-sidebar-icon-button-size, 2rem) * 1.5);--bslib-collapse-toggle-border-radius: var(--bs-border-radius, 0.25rem);--bslib-collapse-toggle-transform: 0deg;--bslib-sidebar-toggle-transition-easing: cubic-bezier(1, 0, 0, 1);--bslib-collapse-toggle-right-transform: 180deg;--bslib-sidebar-column-main: minmax(0, 1fr);display:grid !important;grid-template-columns:min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px)) var(--bslib-sidebar-column-main);position:relative;transition:grid-template-columns ease-in-out var(--bslib-sidebar-transition-duration);border:var(--bslib-sidebar-border);border-radius:var(--bslib-sidebar-border-radius)}@media(prefers-reduced-motion: reduce){.bslib-sidebar-layout{transition:none}}.bslib-sidebar-layout[data-bslib-sidebar-border=false]{border:none}.bslib-sidebar-layout[data-bslib-sidebar-border-radius=false]{border-radius:initial}.bslib-sidebar-layout>.main,.bslib-sidebar-layout>.sidebar{grid-row:1/2;border-radius:inherit;overflow:auto}.bslib-sidebar-layout>.main{grid-column:2/3;border-top-left-radius:0;border-bottom-left-radius:0;padding:var(--bslib-sidebar-padding);transition:padding var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout .sidebar-title{font-size:1.25rem;line-height:1.25;margin-top:0;margin-bottom:1rem;padding-bottom:1rem;border-bottom:var(--bslib-sidebar-border)}.bslib-sidebar-layout.sidebar-right{grid-template-columns:var(--bslib-sidebar-column-main) min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px))}.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/2;border-top-right-radius:0;border-bottom-right-radius:0;border-top-left-radius:inherit;border-bottom-left-radius:inherit}.bslib-sidebar-layout.sidebar-right>.sidebar{grid-column:2/3;border-right:none;border-left:var(--bslib-sidebar-vert-border);border-top-left-radius:0;border-bottom-left-radius:0}.bslib-sidebar-layout.sidebar-right>.collapse-toggle{grid-column:2/3;left:var(--bslib-sidebar-icon-size);right:unset;border:var(--bslib-collapse-toggle-border)}.bslib-sidebar-layout.sidebar-right>.collapse-toggle>.collapse-icon{transform:rotateY(var(--bslib-collapse-toggle-right-transform))}.bslib-sidebar-layout.sidebar-collapsed{--bslib-collapse-toggle-transform: 180deg;--bslib-collapse-toggle-right-transform: 0deg;--bslib-sidebar-vert-border: none;grid-template-columns:0 minmax(0, 1fr)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right{grid-template-columns:minmax(0, 1fr) 0}.bslib-sidebar-layout.sidebar-collapsed:not(.transitioning)>.sidebar>*{display:none}.bslib-sidebar-layout.sidebar-collapsed>.main{border-radius:inherit}.bslib-sidebar-layout.sidebar-collapsed:not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed>.collapse-toggle{color:var(--bslib-sidebar-main-fg);top:calc(var(--bslib-sidebar-overlap-counter, 0)*(var(--bslib-sidebar-icon-size) + var(--bslib-sidebar-padding)) + var(--bslib-sidebar-icon-size, 1rem)/2);right:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px))}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.collapse-toggle{left:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px));right:unset}@media(min-width: 576px){.bslib-sidebar-layout.transitioning>.sidebar>.sidebar-content{display:none}}@media(max-width: 575.98px){.bslib-sidebar-layout[data-bslib-sidebar-open=desktop]{--bslib-sidebar-js-init-collapsed: true}.bslib-sidebar-layout>.sidebar,.bslib-sidebar-layout.sidebar-right>.sidebar{border:none}.bslib-sidebar-layout>.main,.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/3}.bslib-sidebar-layout[data-bslib-sidebar-open=always]{display:block !important}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar{max-height:var(--bslib-sidebar-max-height-mobile);overflow-y:auto;border-top:var(--bslib-sidebar-vert-border)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]){grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.sidebar{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.collapse-toggle{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed.sidebar-right{grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always])>.main{opacity:0;transition:opacity var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed>.main{opacity:1}}.accordion .accordion-header{font-size:calc(1.29rem + 0.48vw);margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2;color:var(--bs-heading-color);margin-bottom:0}@media(min-width: 1200px){.accordion .accordion-header{font-size:1.65rem}}.accordion .accordion-icon:not(:empty){margin-right:.75rem;display:flex}.accordion .accordion-button:not(.collapsed){box-shadow:none}.accordion .accordion-button:not(.collapsed):focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow)}:root{--bslib-page-sidebar-title-bg: #052744;--bslib-page-sidebar-title-color: #fff}.bslib-page-title{background-color:var(--bslib-page-sidebar-title-bg);color:var(--bslib-page-sidebar-title-color);font-size:1.25rem;font-weight:300;padding:var(--bslib-spacer, 1rem);padding-left:1.5rem;margin-bottom:0;border-bottom:1px solid #dee2e6}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.form-inline{margin-top:auto;margin-bottom:auto}.nav:not(.nav-hidden).nav-stacked{flex-direction:column;-webkit-flex-direction:column;height:100%}.nav:not(.nav-hidden).nav-stacked>.bslib-nav-spacer{margin-top:auto !important}}.bslib-card{overflow:auto}.bslib-card .card-body+.card-body{padding-top:0}.bslib-card .card-body{overflow:auto}.bslib-card .card-body p{margin-top:0}.bslib-card .card-body p:last-child{margin-bottom:0}.bslib-card .card-body{max-height:var(--bslib-card-body-max-height, none)}.bslib-card[data-full-screen=true]>.card-body{max-height:var(--bslib-card-body-max-height-full-screen, none)}.bslib-card .card-header .form-group{margin-bottom:0}.bslib-card .card-header .selectize-control{margin-bottom:0}.bslib-card .card-header .selectize-control .item{margin-right:1.15rem}.bslib-card .card-footer{margin-top:auto}.bslib-card .bslib-navs-card-title{display:flex;flex-wrap:wrap;justify-content:space-between;align-items:center}.bslib-card .bslib-navs-card-title .nav{margin-left:auto}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border=true]){border:none}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border-radius=true]){border-top-left-radius:0;border-top-right-radius:0}[data-full-screen=true]{position:fixed;inset:3.5rem 1rem 1rem;height:auto !important;max-height:none !important;width:auto !important;z-index:1070}.bslib-full-screen-enter{display:none;position:absolute;bottom:var(--bslib-full-screen-enter-bottom, 0.2rem);right:var(--bslib-full-screen-enter-right, 0);top:var(--bslib-full-screen-enter-top);left:var(--bslib-full-screen-enter-left);color:var(--bslib-color-fg, var(--bs-card-color));background-color:var(--bslib-color-bg, var(--bs-card-bg, var(--bs-body-bg)));border:var(--bs-card-border-width) solid var(--bslib-color-fg, var(--bs-card-border-color));box-shadow:0 2px 4px rgba(0,0,0,.15);margin:.2rem .4rem;padding:.55rem !important;font-size:.8rem;cursor:pointer;opacity:.7;z-index:1070}.bslib-full-screen-enter:hover{opacity:1}.card[data-full-screen=false]:hover>*>.bslib-full-screen-enter{display:block}.bslib-has-full-screen .card:hover>*>.bslib-full-screen-enter{display:none}@media(max-width: 575.98px){.bslib-full-screen-enter{display:none !important}}.bslib-full-screen-exit{position:relative;top:1.35rem;font-size:.9rem;cursor:pointer;text-decoration:none;display:flex;float:right;margin-right:2.15rem;align-items:center;color:rgba(var(--bs-body-bg-rgb), 0.8)}.bslib-full-screen-exit:hover{color:rgba(var(--bs-body-bg-rgb), 1)}.bslib-full-screen-exit svg{margin-left:.5rem;font-size:1.5rem}#bslib-full-screen-overlay{position:fixed;inset:0;background-color:rgba(var(--bs-body-color-rgb), 0.6);backdrop-filter:blur(2px);-webkit-backdrop-filter:blur(2px);z-index:1069;animation:bslib-full-screen-overlay-enter 400ms cubic-bezier(0.6, 0.02, 0.65, 1) 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new file mode 100644
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--- /dev/null
+++ b/docs/2_34/site_libs/quarto-html/quarto-syntax-highlighting-dark.css
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+  --quarto-hl-dt-color: #81a1c1;
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 Powell, Michael J. D. 1970. <span>“A Hybrid Method for Nonlinear Equations.”</span> In <em>Numerical Methods for Nonlinear Algebraic Equations</em>, edited by P. Rabinowitz. Gordon; Breach.
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
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+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/bootstrap.html b/docs/2_34/stan-users-guide/bootstrap.html
index 3ba7e8f61..ef510d5a9 100644
--- a/docs/2_34/stan-users-guide/bootstrap.html
+++ b/docs/2_34/stan-users-guide/bootstrap.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -729,25 +732,6 @@ <h2 class="anchored" data-anchor-id="bayesian-bootstrap-and-bagging">Bayesian bo
 Huggins, Jonathan H, and Jeffrey W Miller. 2019. <span>“Using Bagged Posteriors for Robust Inference and Model Criticism.”</span> <em>arXiv</em>, no. 1912.07104.
 </div>
 </div></section></div></main> <!-- /main -->
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-   kbds.forEach(function(kbd) {
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -768,6 +752,150 @@ <h2 class="anchored" data-anchor-id="bayesian-bootstrap-and-bagging">Bayesian bo
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/clustering.html b/docs/2_34/stan-users-guide/clustering.html
index 7cd709a81..9285c7f0c 100644
--- a/docs/2_34/stan-users-guide/clustering.html
+++ b/docs/2_34/stan-users-guide/clustering.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1021,25 +1024,6 @@ <h4 class="unnumbered anchored" data-anchor-id="full-bayes-correlated-topic-mode
 <li id="fn1"><p>For clustering, the non-identifiability problems for all mixture models present a problem, whereas there is no such problem for classification. Despite the difficulties with full Bayesian inference for clustering, researchers continue to use it, often in an exploratory data analysis setting rather than for predictive modeling.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1060,6 +1044,150 @@ <h4 class="unnumbered anchored" data-anchor-id="full-bayes-correlated-topic-mode
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/complex-numbers.html b/docs/2_34/stan-users-guide/complex-numbers.html
index 618cb9722..f45729806 100644
--- a/docs/2_34/stan-users-guide/complex-numbers.html
+++ b/docs/2_34/stan-users-guide/complex-numbers.html
@@ -72,10 +72,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -716,25 +719,6 @@ <h3 class="unnumbered anchored" data-anchor-id="dependent-complex-error">Depende
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -755,6 +739,150 @@ <h3 class="unnumbered anchored" data-anchor-id="dependent-complex-error">Depende
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/cross-validation.html b/docs/2_34/stan-users-guide/cross-validation.html
index 3a716d95c..62837d868 100644
--- a/docs/2_34/stan-users-guide/cross-validation.html
+++ b/docs/2_34/stan-users-guide/cross-validation.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -886,25 +889,6 @@ <h3 class="unnumbered anchored" data-anchor-id="approximate-cross-validation">Ap
 <li id="fn1"><p>The traditional approach is to walk through a vector and replace each item with a random element from the remaining elements, which is guaranteed to only move each item once. This was not done here as it’d require new categorical <code>theta</code> because Stan does not have a uniform discrete RNG built in.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -925,6 +909,150 @@ <h3 class="unnumbered anchored" data-anchor-id="approximate-cross-validation">Ap
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/custom-probability.html b/docs/2_34/stan-users-guide/custom-probability.html
index 588ecfba8..b0c533ca7 100644
--- a/docs/2_34/stan-users-guide/custom-probability.html
+++ b/docs/2_34/stan-users-guide/custom-probability.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -665,25 +668,6 @@ <h3 class="unnumbered anchored" data-anchor-id="bivariate-normal-cumulative-dist
 <li id="fn1"><p>The problem is the (extremely!) light tails of the triangle distribution. The standard HMC and NUTS samplers can’t get into the corners of the triangle properly. Because the Stan code declares <code>y</code> to be of type <code>real&lt;lower=-1, upper=1&gt;</code>, the inverse logit transform is applied to the unconstrained variable and its log absolute derivative added to the log probability. The resulting distribution on the logit-transformed <code>y</code> is well behaved.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -704,6 +688,150 @@ <h3 class="unnumbered anchored" data-anchor-id="bivariate-normal-cumulative-dist
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
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+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
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+    // Read the current dark / light value 
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+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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diff --git a/docs/2_34/stan-users-guide/dae.html b/docs/2_34/stan-users-guide/dae.html
index a700283b5..337d1e808 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -725,25 +728,6 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
 Serban, Radu, Cosmin Petra, Alan C. Hindmarsh, Cody J. Balos, David J. Gardner, Daniel R. Reynolds, and Carol S. Woodward. 2021. <span>“User <span>Documentation</span> for <span>IDAS</span> V5.0.0.”</span> Lawrence Livermore National Laboratory.
 </div>
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@@ -764,6 +748,150 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
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+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+        el.classList.remove('notransition');
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+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
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+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
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+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
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+    } else {
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+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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diff --git a/docs/2_34/stan-users-guide/decision-analysis.html b/docs/2_34/stan-users-guide/decision-analysis.html
index 47de69987..2235c8c24 100644
--- a/docs/2_34/stan-users-guide/decision-analysis.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -738,25 +741,6 @@ <h2 class="anchored" data-anchor-id="continuous-choices">Continuous choices</h2>
 Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. <em>Bayesian Data Analysis</em>. Third Edition. London: Chapman &amp; Hall / CRC Press.
 </div>
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -777,6 +761,150 @@ <h2 class="anchored" data-anchor-id="continuous-choices">Continuous choices</h2>
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   }
   toggleBodyColorPrimary();  
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+  const enableStylesheet = (stylesheets) => {
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+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
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+    }
+    const changeGiscusTheme = () => {
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+      const sendMessage = (message) => {
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
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+  const isFileUrl = () => { 
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+  const hasAlternateSentinel = () => {  
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+    if (styleSentinel !== null) {
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+    }
+  }
+  const setStyleSentinel = (alternate) => {
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+    i.classList.add('bi');
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+  } else {
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diff --git a/docs/2_34/stan-users-guide/efficiency-tuning.html b/docs/2_34/stan-users-guide/efficiency-tuning.html
index f3867eea3..6c00116ac 100644
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1446,25 +1449,6 @@ <h2 class="anchored" data-anchor-id="using-map-reduce">Using map-reduce</h2>
 <li id="fn4"><p>Future versions of Stan may remove this inefficiency by more fully exploiting expression templates inside the Eigen C++ matrix library. This will require enhancing Eigen to deal with mixed-type arguments, such as the type <code>double</code> used for constants and the algorithmic differentiation type <code>stan::math::var</code> used for variables.<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1485,6 +1469,150 @@ <h2 class="anchored" data-anchor-id="using-map-reduce">Using map-reduce</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
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+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
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diff --git a/docs/2_34/stan-users-guide/finite-mixtures.html b/docs/2_34/stan-users-guide/finite-mixtures.html
index ae4993415..96e76b77e 100644
--- a/docs/2_34/stan-users-guide/finite-mixtures.html
+++ b/docs/2_34/stan-users-guide/finite-mixtures.html
@@ -92,10 +92,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1031,25 +1034,6 @@ <h3 class="unnumbered anchored" data-anchor-id="comparison-to-model-averaging">C
 <li id="fn1"><p>Imposing a constraint such as <span class="math inline">\(\theta &lt; 0.5\)</span> will resolve the symmetry, but fundamentally changes the model and its posterior inferences.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1070,6 +1054,150 @@ <h3 class="unnumbered anchored" data-anchor-id="comparison-to-model-averaging">C
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
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+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+    } else {
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+  const darkModeDefault = false;
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+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
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+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
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+  }
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diff --git a/docs/2_34/stan-users-guide/floating-point.html b/docs/2_34/stan-users-guide/floating-point.html
index b494f90a7..b00745985 100644
--- a/docs/2_34/stan-users-guide/floating-point.html
+++ b/docs/2_34/stan-users-guide/floating-point.html
@@ -38,10 +38,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -753,25 +756,6 @@ <h2 class="anchored" data-anchor-id="comparing-floating-point-numbers">Comparing
 <li id="fn1"><p>The notable exception is Intel’s optimizing compilers under certain optimization settings.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -792,6 +776,150 @@ <h2 class="anchored" data-anchor-id="comparing-floating-point-numbers">Comparing
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/for-bugs-users.html b/docs/2_34/stan-users-guide/for-bugs-users.html
index 5813d2430..90d8a3d1b 100644
--- a/docs/2_34/stan-users-guide/for-bugs-users.html
+++ b/docs/2_34/stan-users-guide/for-bugs-users.html
@@ -91,10 +91,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -152,7 +154,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -267,6 +269,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -819,25 +822,6 @@ <h2 class="anchored" data-anchor-id="the-stan-community">The Stan community</h2>
 <li id="fn2"><p>Most distributions have been vectorized, but currently the truncated versions may not exist.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -858,6 +842,150 @@ <h2 class="anchored" data-anchor-id="the-stan-community">The Stan community</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/gaussian-processes.html b/docs/2_34/stan-users-guide/gaussian-processes.html
index 444823659..d01b83588 100644
--- a/docs/2_34/stan-users-guide/gaussian-processes.html
+++ b/docs/2_34/stan-users-guide/gaussian-processes.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1326,25 +1329,6 @@ <h3 class="unnumbered anchored" data-anchor-id="multiple-output-gaussian-process
 <li id="fn2"><p>A boundary-avoiding prior is just one where the limit of the density is zero at the boundary, the result of which is estimates that are pushed away from the boundary.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1365,6 +1349,150 @@ <h3 class="unnumbered anchored" data-anchor-id="multiple-output-gaussian-process
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/hyperspherical-models.html b/docs/2_34/stan-users-guide/hyperspherical-models.html
index 35f97f0c6..f0e233347 100644
--- a/docs/2_34/stan-users-guide/hyperspherical-models.html
+++ b/docs/2_34/stan-users-guide/hyperspherical-models.html
@@ -92,10 +92,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -675,25 +678,6 @@ <h2 class="anchored" data-anchor-id="circular-representations-of-days-and-years"
 Muller, Mervin E. 1959. <span>“A Note on a Method for Generating Points Uniformly on n-Dimensional Spheres.”</span> <em>Commun. ACM</em> 2 (4): 19–20. <a href="https://doi.org/10.1145/377939.377946">https://doi.org/10.1145/377939.377946</a>.
 </div>
 </div></section></div></main> <!-- /main -->
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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diff --git a/docs/2_34/stan-users-guide/index.html b/docs/2_34/stan-users-guide/index.html
index 61e6882d0..b45503c2e 100644
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+++ b/docs/2_34/stan-users-guide/index.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -516,7 +519,7 @@ <h1 class="title">Stan User’s Guide</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official user’s guide for Stan. It provides example models and programming techniques for coding statistical models in Stan.</p>
+<p>This is the official user’s guide for <a href="https://mc-stan.org/">Stan</a>. It provides example models and programming techniques for coding statistical models in Stan.</p>
 <ul>
 <li><p>Part 1 gives Stan code and discussions for several important classes of models.</p></li>
 <li><p>Part 2 discusses various general Stan programming techniques that are not tied to any particular model.</p></li>
@@ -544,25 +547,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -583,6 +567,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
   toggleBodyColorPrimary();  
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+        if (!iframe) return;
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+    // properly recolor the scrollbar when toggling (#1455)
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diff --git a/docs/2_34/stan-users-guide/latent-discrete.html b/docs/2_34/stan-users-guide/latent-discrete.html
index 188bbc4d4..f64b04dab 100644
--- a/docs/2_34/stan-users-guide/latent-discrete.html
+++ b/docs/2_34/stan-users-guide/latent-discrete.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1207,7 +1210,7 @@ <h3 class="unnumbered anchored" data-anchor-id="estimating-expectations">Estimat
 \]</span> and then estimate <span class="math inline">\(\mathbb{E}[g(\Theta, Z)]\)</span> with <span class="math display">\[
 \hat{\mu} = \frac{1}{M} \sum_{i = 1}^M h(\theta^{(i)}).
 \]</span> This estimator is justified by the law of iterated expectation, the fact that <span class="math display">\[
-\mathbb{E}[h(\Theta)] = \mathbb{E}[\mathbb{E}[g(\Theta, Z] \mid \Theta)] = \mathbb{E}[g(\Theta, Z)] = \mu.
+\mathbb{E}[h(\Theta)] = \mathbb{E}[\mathbb{E}[g(\Theta, Z)] \mid \Theta] = \mathbb{E}[g(\Theta, Z)] = \mu.
 \]</span> Using this marginalized estimator provides a way to estimate the expectation of any function <span class="math inline">\(g(\cdot)\)</span> for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation <span class="math inline">\(\mathbb{E}[g(\Theta, Z) \mid \Theta]\)</span>.</p>
 </section>
 <section id="evaluating-the-conditional-expectation" class="level3 unnumbered">
@@ -1299,25 +1302,6 @@ <h3 class="unnumbered anchored" data-anchor-id="evaluating-the-conditional-expec
 <li id="fn7"><p>In the subscript, <span class="math inline">\(z_i\)</span> is written as <span class="math inline">\(z[i]\)</span> to improve legibility.<a href="#fnref7" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-
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-
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1338,6 +1322,150 @@ <h3 class="unnumbered anchored" data-anchor-id="evaluating-the-conditional-expec
     }
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   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
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+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/stan-users-guide/matrices-arrays.html b/docs/2_34/stan-users-guide/matrices-arrays.html
index 3f55070be..7967a6659 100644
--- a/docs/2_34/stan-users-guide/matrices-arrays.html
+++ b/docs/2_34/stan-users-guide/matrices-arrays.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -802,25 +805,6 @@ <h2 class="anchored" data-anchor-id="aliasing-in-stan-containers">Aliasing in St
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -841,6 +825,150 @@ <h2 class="anchored" data-anchor-id="aliasing-in-stan-containers">Aliasing in St
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
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diff --git a/docs/2_34/stan-users-guide/measurement-error.html b/docs/2_34/stan-users-guide/measurement-error.html
index 4cdc3c903..7c528cb0b 100644
--- a/docs/2_34/stan-users-guide/measurement-error.html
+++ b/docs/2_34/stan-users-guide/measurement-error.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -842,25 +845,6 @@ <h4 class="unnumbered anchored" data-anchor-id="extensions-and-alternatives">Ext
 <li id="fn2"><p>When dividing two integers, the result type is an integer and rounding will ensue if the result is not exact. See the discussion of primitive arithmetic types in the reference manual for more information.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -881,6 +865,150 @@ <h4 class="unnumbered anchored" data-anchor-id="extensions-and-alternatives">Ext
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/missing-data.html b/docs/2_34/stan-users-guide/missing-data.html
index efd0f8bf8..acb397fe4 100644
--- a/docs/2_34/stan-users-guide/missing-data.html
+++ b/docs/2_34/stan-users-guide/missing-data.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -781,25 +784,6 @@ <h2 class="anchored" data-anchor-id="missing-multivariate-data">Missing multivar
 <li id="fn2"><p>This is not the same as missing components of a multivariate predictor in a regression problem; in that case, you will need to represent the missing data as a parameter and impute missing values in order to feed them into the regression.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -820,6 +804,150 @@ <h2 class="anchored" data-anchor-id="missing-multivariate-data">Missing multivar
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
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+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
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+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/stan-users-guide/multi-indexing.html b/docs/2_34/stan-users-guide/multi-indexing.html
index 5e275a2a4..6fcbfdcf2 100644
--- a/docs/2_34/stan-users-guide/multi-indexing.html
+++ b/docs/2_34/stan-users-guide/multi-indexing.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -826,25 +829,6 @@ <h2 class="anchored" data-anchor-id="matrices-with-parameters-and-constants">Mat
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -865,6 +849,150 @@ <h2 class="anchored" data-anchor-id="matrices-with-parameters-and-constants">Mat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/odes.html b/docs/2_34/stan-users-guide/odes.html
index 5647b9efc..d5d16fa43 100644
--- a/docs/2_34/stan-users-guide/odes.html
+++ b/docs/2_34/stan-users-guide/odes.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -909,25 +912,6 @@ <h2 class="anchored" data-anchor-id="solving-a-system-of-linear-odes-using-a-mat
 <li id="fn1"><p>This example is drawn from the documentation for the Boost Numeric Odeint library <span class="citation" data-cites="AhnertMulansky:2011">(<a href="#ref-AhnertMulansky:2011" role="doc-biblioref">Ahnert and Mulansky 2011</a>)</span>, which Stan uses to implement the <code>rk45</code> and <code>ckrk</code> solver.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -948,6 +932,150 @@ <h2 class="anchored" data-anchor-id="solving-a-system-of-linear-odes-using-a-mat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
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+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
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+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      for (const sheetNode of alternateStylesheets) {
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+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+    if (!isFileUrl()) {
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+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
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+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
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+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  } else {
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diff --git a/docs/2_34/stan-users-guide/one-dimensional-integrals.html b/docs/2_34/stan-users-guide/one-dimensional-integrals.html
index cb4b9f203..4863bce40 100644
--- a/docs/2_34/stan-users-guide/one-dimensional-integrals.html
+++ b/docs/2_34/stan-users-guide/one-dimensional-integrals.html
@@ -92,10 +92,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -780,25 +783,6 @@ <h3 class="unnumbered anchored" data-anchor-id="integral-precision">Avoiding pre
 Agrawal, Nikhar, Anton Bikineev, Paul Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, et al. 2017. <span>“Double-Exponential Quadrature.”</span> https://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/double_exponential.html.
 </div>
 </div></section></div></main> <!-- /main -->
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -819,6 +803,150 @@ <h3 class="unnumbered anchored" data-anchor-id="integral-precision">Avoiding pre
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/2_34/stan-users-guide/parallelization.html b/docs/2_34/stan-users-guide/parallelization.html
index f38b772e1..1271b2ae7 100644
--- a/docs/2_34/stan-users-guide/parallelization.html
+++ b/docs/2_34/stan-users-guide/parallelization.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1002,25 +1005,6 @@ <h2 class="anchored" data-anchor-id="opencl">OpenCL</h2>
 <li id="fn3"><p>This makes the strong assumption that each group has the same number of observations!<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
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-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1041,6 +1025,150 @@ <h2 class="anchored" data-anchor-id="opencl">OpenCL</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/2_34/stan-users-guide/posterior-prediction.html b/docs/2_34/stan-users-guide/posterior-prediction.html
index 9230403f4..ae4a032c5 100644
--- a/docs/2_34/stan-users-guide/posterior-prediction.html
+++ b/docs/2_34/stan-users-guide/posterior-prediction.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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@@ -842,25 +845,6 @@ <h2 class="anchored" data-anchor-id="stand-alone-generated-quantities-and-ongoin
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diff --git a/docs/2_34/stan-users-guide/posterior-predictive-checks.html b/docs/2_34/stan-users-guide/posterior-predictive-checks.html
index 80bc40cac..01b646378 100644
--- a/docs/2_34/stan-users-guide/posterior-predictive-checks.html
+++ b/docs/2_34/stan-users-guide/posterior-predictive-checks.html
@@ -92,10 +92,12 @@
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -271,6 +273,7 @@
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     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -586,7 +589,7 @@ <h2 id="toc-title">On this page</h2>
   <ul class="collapse">
   <li><a href="#simulating-from-the-posterior-predictive-distribution" id="toc-simulating-from-the-posterior-predictive-distribution" class="nav-link" data-scroll-target="#simulating-from-the-posterior-predictive-distribution">Simulating from the posterior predictive distribution</a></li>
   <li><a href="#plotting-multiples" id="toc-plotting-multiples" class="nav-link" data-scroll-target="#plotting-multiples">Plotting multiples</a></li>
-  <li><a href="#posterior-p-values" id="toc-posterior-p-values" class="nav-link" data-scroll-target="#posterior-p-values">Posterior ``p-values’’</a>
+  <li><a href="#posterior-p-values" id="toc-posterior-p-values" class="nav-link" data-scroll-target="#posterior-p-values">Posterior ‘’p-values’’</a>
   <ul class="collapse">
   <li><a href="#which-statistics-to-test" id="toc-which-statistics-to-test" class="nav-link" data-scroll-target="#which-statistics-to-test">Which statistics to test?</a></li>
   </ul></li>
@@ -679,7 +682,7 @@ <h2 class="anchored" data-anchor-id="plotting-multiples">Plotting multiples</h2>
 <p>Now consider generating data <span class="math inline">\(y \sim \textrm{Poisson}(5)\)</span>. The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pois-pois.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-1" data-gallery="quarto-lightbox-gallery-1" title="Posterior predictive checks for Poisson data generating process and Poisson model."><img src="./img/ppc-pois-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for Poisson data generating process and Poisson model."></a></p>
+<p><a href="./img/ppc-pois-pois.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-1" data-glightbox="description: .lightbox-desc-1" title="Posterior predictive checks for Poisson data generating process and Poisson model."><img src="./img/ppc-pois-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for Poisson data generating process and Poisson model."></a></p>
 <figcaption>Posterior predictive checks for Poisson data generating process and Poisson model.</figcaption>
 </figure>
 </div>
@@ -687,14 +690,14 @@ <h2 class="anchored" data-anchor-id="plotting-multiples">Plotting multiples</h2>
 <p>Now consider generating over-dispersed data <span class="math inline">\(y \sim \textrm{negative-binomial2}(5, 1).\)</span> This has the same mean as <span class="math inline">\(\textrm{Poisson}(5)\)</span>, namely <span class="math inline">\(5\)</span>, but a standard deviation of <span class="math inline">\(\sqrt{5 + 5^2 /1} \approx 5.5.\)</span> There is no way to fit this data with the Poisson model, because a variable distributed as <span class="math inline">\(\textrm{Poisson}(\lambda)\)</span> has mean <span class="math inline">\(\lambda\)</span> and standard deviation <span class="math inline">\(\sqrt{\lambda},\)</span> which is <span class="math inline">\(\sqrt{5}\)</span> for <span class="math inline">\(\textrm{Poisson}(5).\)</span> Here’s the resulting small multiples plot, again with original data in the upper left.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-nb-pois.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-2" data-gallery="quarto-lightbox-gallery-2" title="Posterior predictive checks for negative binomial data generating process and Poisson model."><img src="./img/ppc-nb-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for negative binomial data generating process and Poisson model."></a></p>
+<p><a href="./img/ppc-nb-pois.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-2" data-glightbox="description: .lightbox-desc-2" title="Posterior predictive checks for negative binomial data generating process and Poisson model."><img src="./img/ppc-nb-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for negative binomial data generating process and Poisson model."></a></p>
 <figcaption>Posterior predictive checks for negative binomial data generating process and Poisson model.</figcaption>
 </figure>
 </div>
 <p>This time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.</p>
 </section>
 <section id="posterior-p-values" class="level2">
-<h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’’</h2>
+<h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ‘’p-values’’</h2>
 <p>If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic <span class="math inline">\(s(\cdot)\)</span> in a replicated data set exceeds that in the original data, <span class="math display">\[
 \Pr\!\left[ s(y^{\textrm{rep}}) \geq s(y) \mid y \right]
 =
@@ -702,7 +705,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 \textrm{I}\left( s(y^{\textrm{rep}}) \geq s(y) \mid y \right)
 \cdot p\left( y^{\textrm{rep}} \mid y \right)
 \, \textrm{d}{y^{\textrm{rep}}}.
-\]</span> It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified <span class="citation" data-cites="BayarriBerger:2000">(<a href="#ref-BayarriBerger:2000" role="doc-biblioref">Bayarri and Berger 2000</a>)</span>.</p>
+\]</span> It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified <span class="citation" data-cites="BayarriBerger:2000">(<a href="#ref-BayarriBerger:2000" role="doc-biblioref">Bayarri and Berger 2000</a>)</span>.</p>
 <p>Nevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.</p>
 <p>To calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.</p>
 <div class="sourceCode" id="cb4"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
@@ -718,7 +721,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 <p>For the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pvalue-nb-pois-mean.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-3" data-gallery="quarto-lightbox-gallery-3" title="Histogram of means of replicated data sets; vertical red line at mean of original data."><img src="./img/ppc-pvalue-nb-pois-mean.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Histogram of means of replicated data sets; vertical red line at mean of original data."></a></p>
+<p><a href="./img/ppc-pvalue-nb-pois-mean.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-3" data-glightbox="description: .lightbox-desc-3" title="Histogram of means of replicated data sets; vertical red line at mean of original data."><img src="./img/ppc-pvalue-nb-pois-mean.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Histogram of means of replicated data sets; vertical red line at mean of original data."></a></p>
 <figcaption>Histogram of means of replicated data sets; vertical red line at mean of original data.</figcaption>
 </figure>
 </div>
@@ -726,7 +729,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 <p>The standard deviation statistic tells a different story.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pvalue-nb-pois-sd.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-4" data-gallery="quarto-lightbox-gallery-4" title="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."><img src="./img/ppc-pvalue-nb-pois-sd.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."></a></p>
+<p><a href="./img/ppc-pvalue-nb-pois-sd.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-4" data-glightbox="description: .lightbox-desc-4" title="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."><img src="./img/ppc-pvalue-nb-pois-sd.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."></a></p>
 <figcaption>Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.</figcaption>
 </figure>
 </div>
@@ -933,25 +936,6 @@ <h3 class="unnumbered anchored" data-anchor-id="mixed-replication-for-hierarchic
 Rubin, Donald B. 1984. <span>“Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician.”</span> <em>The Annals of Statistics</em>, 1151–72.
 </div>
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@@ -1352,7 +1480,7 @@ <h3 class="unnumbered anchored" data-anchor-id="mixed-replication-for-hierarchic
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diff --git a/docs/2_34/stan-users-guide/poststratification.html b/docs/2_34/stan-users-guide/poststratification.html
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@@ -877,25 +880,6 @@ <h2 class="anchored" data-anchor-id="adding-group-level-predictors">Adding group
 Paustian, Keith, Elissa Levine, Wilfred M Post, and Irene M Ryzhova. 1997. <span>“The Use of Models to Integrate Information and Understanding of Soil <span>C</span> at the Regional Scale.”</span> <em>Geoderma</em> 79 (1-4): 227–60.
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diff --git a/docs/2_34/stan-users-guide/problematic-posteriors.html b/docs/2_34/stan-users-guide/problematic-posteriors.html
index 9760a01e8..4cc39bda9 100644
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
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@@ -985,25 +988,6 @@ <h3 class="unnumbered anchored" data-anchor-id="examples-fits-in-stan">Examples:
 <li id="fn4"><p>Tempering methods may be viewed as automated ways to carry out such a search for modes, though most MCMC tempering methods continue to search for modes on an ongoing basis; see <span class="citation" data-cites="SwendsenWang:1986 Neal:1996b">(<a href="#ref-SwendsenWang:1986" role="doc-biblioref">Swendsen and Wang 1986</a>; <a href="#ref-Neal:1996b" role="doc-biblioref">Neal 1996</a>)</span>.<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
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@@ -1024,6 +1008,150 @@ <h3 class="unnumbered anchored" data-anchor-id="examples-fits-in-stan">Examples:
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diff --git a/docs/2_34/stan-users-guide/proportionality-constants.html b/docs/2_34/stan-users-guide/proportionality-constants.html
index 712c364aa..4eb783670 100644
--- a/docs/2_34/stan-users-guide/proportionality-constants.html
+++ b/docs/2_34/stan-users-guide/proportionality-constants.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -665,25 +668,6 @@ <h2 class="anchored" data-anchor-id="limitations-on-using-_lupdf-and-_lupmf-func
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -704,6 +688,150 @@ <h2 class="anchored" data-anchor-id="limitations-on-using-_lupdf-and-_lupmf-func
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/references.html b/docs/2_34/stan-users-guide/references.html
index df2ece9e2..29e726062 100644
--- a/docs/2_34/stan-users-guide/references.html
+++ b/docs/2_34/stan-users-guide/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/regression.html b/docs/2_34/stan-users-guide/regression.html
index a2f766840..66d39c4e8 100644
--- a/docs/2_34/stan-users-guide/regression.html
+++ b/docs/2_34/stan-users-guide/regression.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1723,25 +1726,6 @@ <h3 class="unnumbered anchored" data-anchor-id="posterior-predictive-checks">Pos
 <li id="fn6"><p>Thanks to Mike Lawrence for pointing this out in the GitHub issue for the manual.<a href="#fnref6" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1762,6 +1746,150 @@ <h3 class="unnumbered anchored" data-anchor-id="posterior-predictive-checks">Pos
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
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+    i.classList.add('bi');
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diff --git a/docs/2_34/stan-users-guide/reparameterization.html b/docs/2_34/stan-users-guide/reparameterization.html
index 2759e6399..a5ce918ad 100644
--- a/docs/2_34/stan-users-guide/reparameterization.html
+++ b/docs/2_34/stan-users-guide/reparameterization.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -995,25 +998,6 @@ <h3 class="unnumbered anchored" data-anchor-id="varying-upper-and-lower-bounds">
 <li id="fn2"><p>This example is for illustrative purposes only; the recommended way to implement the lognormal distribution in Stan is with the built-in <code>lognormal</code> probability function; see the functions reference manual for details.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -1034,6 +1018,150 @@ <h3 class="unnumbered anchored" data-anchor-id="varying-upper-and-lower-bounds">
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+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
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+  const toggleColorMode = (alternate) => {
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+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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diff --git a/docs/2_34/stan-users-guide/simulation-based-calibration.html b/docs/2_34/stan-users-guide/simulation-based-calibration.html
index 1cf6fd15c..e8747779d 100644
--- a/docs/2_34/stan-users-guide/simulation-based-calibration.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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   <i class="bi"></i>
@@ -979,25 +982,6 @@ <h3 class="unnumbered anchored" data-anchor-id="when-stans-sampler-goes-wrong">W
 Talts, Sean, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. 2018. <span>“Validating <span>B</span>ayesian Inference Algorithms with Simulation-Based Calibration.”</span> <em>arXiv</em>, no. 1804.06788.
 </div>
 </div></section></div></main> <!-- /main -->
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1018,6 +1002,150 @@ <h3 class="unnumbered anchored" data-anchor-id="when-stans-sampler-goes-wrong">W
     }
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+      changeGiscusTheme();
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+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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+    i.classList.add('bi');
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index 0f7417b64..3b7860ff8 100644
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -737,25 +740,6 @@ <h2 class="anchored" data-anchor-id="ragged-data-structs.section">Ragged data st
 Gelman, Andrew, and Jennifer Hill. 2007. <em>Data Analysis Using Regression and Multilevel-Hierarchical Models</em>. Cambridge, United Kingdom: Cambridge University Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -776,6 +760,150 @@ <h2 class="anchored" data-anchor-id="ragged-data-structs.section">Ragged data st
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/style-guide.html b/docs/2_34/stan-users-guide/style-guide.html
index a6f6bd5d5..a85683fc9 100644
--- a/docs/2_34/stan-users-guide/style-guide.html
+++ b/docs/2_34/stan-users-guide/style-guide.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -833,25 +836,6 @@ <h4 class="unnumbered anchored" data-anchor-id="platform-specificity-of-newlines
 <li id="fn2"><p>This is the usual convention in both typesetting and other programming languages. Neither R nor BUGS allows breaks before an operator because they allow newlines to signal the end of an expression or statement.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -872,6 +856,150 @@ <h4 class="unnumbered anchored" data-anchor-id="platform-specificity-of-newlines
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/survival.html b/docs/2_34/stan-users-guide/survival.html
index 275201183..0c0f7f590 100644
--- a/docs/2_34/stan-users-guide/survival.html
+++ b/docs/2_34/stan-users-guide/survival.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1030,25 +1033,6 @@ <h3 class="unnumbered anchored" data-anchor-id="stan-model-for-tied-survival-tim
 Plackett, Robin L. 1975. <span>“The Analysis of Permutations.”</span> <em>Journal of the Royal Statistical Society Series C: Applied Statistics</em> 24 (2): 193–202.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1069,6 +1053,150 @@ <h3 class="unnumbered anchored" data-anchor-id="stan-model-for-tied-survival-tim
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/time-series.html b/docs/2_34/stan-users-guide/time-series.html
index 5727d3037..4c1e1bf65 100644
--- a/docs/2_34/stan-users-guide/time-series.html
+++ b/docs/2_34/stan-users-guide/time-series.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1181,25 +1184,6 @@ <h3 class="unnumbered anchored" data-anchor-id="predictive-inference">Predictive
 <li id="fn3"><p>This subsection is a lightly edited comment of Jonathan Gilligan’s on GitHub; see <a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1220,6 +1204,150 @@ <h3 class="unnumbered anchored" data-anchor-id="predictive-inference">Predictive
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/truncation-censoring.html b/docs/2_34/stan-users-guide/truncation-censoring.html
index e5ccafcf3..f692658e5 100644
--- a/docs/2_34/stan-users-guide/truncation-censoring.html
+++ b/docs/2_34/stan-users-guide/truncation-censoring.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -720,25 +723,6 @@ <h3 class="unnumbered anchored" data-anchor-id="integrating-out-censored-values"
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -759,6 +743,150 @@ <h3 class="unnumbered anchored" data-anchor-id="integrating-out-censored-values"
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/user-functions.html b/docs/2_34/stan-users-guide/user-functions.html
index c79e897c1..61d8b06bd 100644
--- a/docs/2_34/stan-users-guide/user-functions.html
+++ b/docs/2_34/stan-users-guide/user-functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -981,25 +984,6 @@ <h4 class="unnumbered anchored" data-anchor-id="truncating-above-and-below">Trun
 <li id="fn5"><p>The original code and impetus for including this in the manual came from the <a href="http://discourse.mc-stan.org/t/rng-for-truncated-distributions/3122/7">Stan forums post</a>; by user <code>lcomm</code>, who also explained truncation above and below.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1020,6 +1004,150 @@ <h4 class="unnumbered anchored" data-anchor-id="truncating-above-and-below">Trun
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/stan-users-guide/using-stanc.html b/docs/2_34/stan-users-guide/using-stanc.html
index ca3393940..0bc241490 100644
--- a/docs/2_34/stan-users-guide/using-stanc.html
+++ b/docs/2_34/stan-users-guide/using-stanc.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -943,7 +946,7 @@ <h3 class="unnumbered anchored" data-anchor-id="strict-or-nonsensical-parameter-
 </section>
 <section id="nonlinear-transformations" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="nonlinear-transformations">Nonlinear transformations</h3>
-<p>When a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.</p>
+<p>When a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the <a href="../stan-users-guide/reparameterization.html">Changes of variables</a> section.</p>
 <p>This portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.</p>
 <p>For example, consider the following program.</p>
 <div class="sourceCode" id="cb29"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
@@ -979,7 +982,7 @@ <h2 class="anchored" data-anchor-id="stanc-pretty-printing">Automatic updating a
 <p>These flags work for both <code>.stan</code> model files and <code>.stanfunctions</code> function files. They can be combined with <code>--o</code> to redirect the formatted output to a new file.</p>
 <section id="automatic-formatting" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="automatic-formatting">Automatic formatting</h3>
-<p>Invoking <code>stanc --auto-format &lt;model_file&gt;</code> will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. <em>Some</em> deprecated features, like the use of <code>#</code> for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.</p>
+<p>Invoking <code>stanc --auto-format &lt;model_file&gt;</code> will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the <a href="../stan-users-guide/style-guide.html">Stan Program Style Guide</a>. This means spacing, indentation, and line length are all regularized. <em>Some</em> deprecated features, like the use of <code>#</code> for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.</p>
 <p>By default, this will try to split lines at or before column 78. This number can be changed using <code>--max-line-length</code>.</p>
 </section>
 <section id="canonicalizing" class="level3 unnumbered">
@@ -1386,25 +1389,6 @@ <h4 class="unnumbered anchored" data-anchor-id="static-loop-unrolling">Static lo
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1425,6 +1409,150 @@ <h4 class="unnumbered anchored" data-anchor-id="static-loop-unrolling">Static lo
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/2_34/quarto_styles.css b/docs/2_34/theming/quarto_styles.css
similarity index 79%
rename from docs/2_34/quarto_styles.css
rename to docs/2_34/theming/quarto_styles.css
index 2e8ab8f40..29078a7fc 100644
--- a/docs/2_34/quarto_styles.css
+++ b/docs/2_34/theming/quarto_styles.css
@@ -1,18 +1,3 @@
-/* css styles */
-:root{
-    --old-quarto-body-bg: #FEFBF2;
-    --quarto-body-bg: #FFFDF9;
-    --quarto-body-color: #052744;
-    --quarto-text-muted: #053A42;
-    --bs-body-bg: #FEFBF2;
-    --bs-body-color: #052744;
-    --bs-dark: #052744;
-    --bs-dark-warm: #053A42;
-    --warm-beige: #F2E5BD;
-    --palest-beige: #FEFCF9;
-    --aquatic-blue: #3E8EBC;
-}
-
 /* quarto website styling */
 
 @media (min-width: 1020px) {
@@ -55,7 +40,7 @@
 
 .document-example {
   opacity: 0.9;
-  padding: 6px; 
+  padding: 6px;
   font-weight: 500;
   margin-bottom: 1rem;
 }
@@ -65,7 +50,7 @@
 }
 
 .document-example .citation {
-    color: var(--aquatic-blue);
+    color: var(--stan-secondary);
 }
 
 .trademark {
@@ -85,7 +70,7 @@
 }
 
 .download-table {
-  margin-bottom: 2em;  
+  margin-bottom: 2em;
 }
 
 .download-table p {
@@ -148,7 +133,7 @@ iframe.reveal-demo {
 
 }
 
-@media (min-width: 576px) { 
+@media (min-width: 576px) {
 .link-cards {
   display: flex;
   flex-direction: row;
@@ -217,7 +202,7 @@ iframe.reveal-demo {
   margin-bottom: 110px;
 }
 
-.carousel .carousel-control-prev-icon, 
+.carousel .carousel-control-prev-icon,
 .carousel .carousel-control-next-icon {
   margin-bottom: 110px;
 }
@@ -227,8 +212,8 @@ iframe.reveal-demo {
 }
 
 .btn-action-primary {
-  color: var(--palest-biege);
-  background-color: var(--aquatic-blue) !important;
+  color: var(--stan-bg);
+  background-color: var(--stan-secondary) !important;
 }
 
 .btn-action-primary:hover {
@@ -269,20 +254,20 @@ iframe.reveal-demo {
   font-size: 0.8em;
 }
 
-.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link, 
-.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link.active, 
+.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link,
+.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link.active,
 .panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-item.show .nav-link {
-  border: 1px solid  var(--warm-beige);
+  border: 1px solid  var(--stan-highlight);
   border-radius: 10px;
 }
 .panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link:hover {
-   border-color: var(--aquatic-blue);
+   border-color: var(--stan-secondary);
    border-width: 1px;
-} 
+}
 
-.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link.active, 
+.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link.active,
 .panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-item.show .nav-link {
-  border-color: var(--aquatic-blue);
+  border-color: var(--stan-secondary);
   border-width: 2px;
 }
 
@@ -295,7 +280,7 @@ iframe.reveal-demo {
 
 .download-text {
   font-size: 1.1em;
-  font-weight: 500;  
+  font-weight: 500;
 }
 
 .preview-image-grid {
@@ -318,16 +303,16 @@ iframe.reveal-demo {
 
 /* stan-dev custom styling */
 
-a { color: var(--aquatic-blue);}
-a.nav-link.active { color: var(--aquatic-blue);}
-a.sidebar-item-text.sidebar-link.active { color: var(--aquatic-blue); }
-.sidebar.a.active { color: var(--aquatic-blue); }
+a { color: var(--stan-secondary);}
+a.nav-link.active { color: var(--stan-secondary);}
+a.sidebar-item-text.sidebar-link.active { color: var(--stan-secondary); }
+.sidebar.a.active { color: var(--stan-secondary); }
 
 
 div [data-bs-target^="#quarto-sidebar-section"] {
     font-weight: bold;
     font-style: italic;
-    color: var(--aquatic-blue);
+    color: var(--stan-secondary);
 }
 
 p.big-top {
@@ -337,42 +322,47 @@ p.big-top {
 code {
     white-space: inherit;
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 code:not(.sourceCode) {
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 p code {
     white-space: inherit;
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 p code:not(.sourceCode) {
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 li code:not(.sourceCode) {
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
+}
+
+td code:not(.sourceCode) {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
 }
 
 pre {
     word-break: normal;
     word-wrap: normal;
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 pre code {
     white-space: inherit;
     margin: 0;
     padding: 0;
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 pre.sourceCode {
@@ -380,12 +370,12 @@ pre.sourceCode {
     margin: 0;
     padding: 0;
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 #quarto-appendix {
     color: var(--bs-dark-warm);
-    background-color: var(--palest-beige);
+    background-color: var(--stan-bg);
 }
 
 .align-equations {
@@ -410,7 +400,7 @@ pre.sourceCode {
     font-weight: bold;
     font-style: italic;
     font-size: 1.3em;
-    color: var(--aquatic-blue);
+    color: var(--stan-secondary);
     margin-above: -20em;
     padding: 0;
 }
diff --git a/docs/404.html b/docs/404.html
index b4c201dfd..1401c7548 100644
--- a/docs/404.html
+++ b/docs/404.html
@@ -70,10 +70,12 @@
 <script src="/docs/site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="/docs/site_libs/quarto-html/anchor.min.js"></script>
 <link href="/docs/site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="/docs/site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="/docs/site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="/docs/site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="/docs/site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="/docs/site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="/docs/site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -102,7 +104,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="/docs/quarto_styles.css">
+<link rel="stylesheet" href="/docs/theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -217,6 +219,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -266,25 +269,6 @@ <h1 class="title">Page Not Found</h1>
 
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -305,6 +289,150 @@ <h1 class="title">Page Not Found</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/cmdstan-guide/bib.html b/docs/cmdstan-guide/bib.html
index b9f265429..812356f72 100644
--- a/docs/cmdstan-guide/bib.html
+++ b/docs/cmdstan-guide/bib.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/cmdstan-guide/command_line_options.html b/docs/cmdstan-guide/command_line_options.html
index 2c38642f5..316feb57d 100644
--- a/docs/cmdstan-guide/command_line_options.html
+++ b/docs/cmdstan-guide/command_line_options.html
@@ -31,17 +31,19 @@
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
 <link href="../cmdstan-guide/mcmc_config.html" rel="next">
-<link href="../cmdstan-guide/generate_quantities_intro.html" rel="prev">
+<link href="../cmdstan-guide/parallelization.html" rel="prev">
 <link href="../img/logo_tm.png" rel="icon" type="image/png">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Command-Line Interface Overview</a></li></ol></nav>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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@@ -278,32 +281,8 @@
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- <span class="menu-text">MCMC Sampling</span></a>
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- <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference using Pathfinder</span></a>
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- <span class="menu-text">Variational Inference using ADVI</span></a>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -332,7 +311,7 @@
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
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  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
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+  const disableStylesheet = (stylesheets) => {
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+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // Hack to workaround the fact that safari doesn't
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+      return storageValue != null ? storageValue : localAlternateSentinel;
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+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
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+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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@@ -971,8 +1069,8 @@ <h2 class="anchored" data-anchor-id="error-messages-and-return-codes">Error mess
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/generate_quantities_intro.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span>
+      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Parallelization</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/cmdstan-guide/compiling_stan_programs.html b/docs/cmdstan-guide/compiling_stan_programs.html
index 7cf03c839..495e4b249 100644
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+++ b/docs/cmdstan-guide/compiling_stan_programs.html
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-<link href="../cmdstan-guide/mcmc_sampling_intro.html" rel="next">
+<link href="../cmdstan-guide/parallelization.html" rel="next">
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -133,7 +135,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -261,7 +264,7 @@
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Quickstart Guide</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/compiling_stan_programs.html">Compiling a Stan Program</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/compiling_stan_programs.html">Compiling a Stan Program</a></li></ol></nav>
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@@ -286,7 +289,7 @@
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -312,32 +315,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
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@@ -345,7 +324,7 @@
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -366,7 +345,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -378,13 +357,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -404,12 +383,6 @@
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-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -605,25 +578,6 @@ <h3 class="anchored" data-anchor-id="optimizing-by-ignoring-range-checks">Optimi
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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@@ -644,6 +598,150 @@ <h3 class="anchored" data-anchor-id="optimizing-by-ignoring-range-checks">Optimi
     }
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   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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@@ -1018,8 +1116,8 @@ <h3 class="anchored" data-anchor-id="optimizing-by-ignoring-range-checks">Optimi
       </a>          
   </div>
   <div class="nav-page nav-page-next">
-      <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="pagination-link" aria-label="MCMC Sampling">
-        <span class="nav-page-text">MCMC Sampling</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
+        <span class="nav-page-text">Parallelization</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/cmdstan-guide/diagnose_config.html b/docs/cmdstan-guide/diagnose_config.html
index 110eb6e35..9cd0c2f9d 100644
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+<link href="../cmdstan-guide/stanc.html" rel="next">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
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-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/diagnose_config.html">Diagnosing HMC by Comparison of Gradients</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/diagnose_config.html">Diagnosing HMC by Comparison of Gradients</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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@@ -278,32 +281,8 @@
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-  <div class="sidebar-item-container"> 
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-          <li class="sidebar-item">
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-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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 </li>
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+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -332,7 +311,7 @@
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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@@ -344,13 +323,13 @@
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
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   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
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  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
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-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
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@@ -516,25 +489,6 @@ <h1>Diagnosing HMC by Comparison of Gradients</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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@@ -555,6 +509,150 @@ <h1>Diagnosing HMC by Comparison of Gradients</h1>
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+  const toggleColorMode = (alternate) => {
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+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // Hack to workaround the fact that safari doesn't
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+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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@@ -929,8 +1027,8 @@ <h1>Diagnosing HMC by Comparison of Gradients</h1>
       </a>          
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-      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
-        <span class="nav-page-text">Parallelization</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../cmdstan-guide/stanc.html" class="pagination-link" aria-label="`stanc`: Translating Stan to C++">
+        <span class="nav-page-text"><code>stanc</code>: Translating Stan to C++</span> <i class="bi bi-arrow-right-short"></i>
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diff --git a/docs/cmdstan-guide/diagnose_utility.html b/docs/cmdstan-guide/diagnose_utility.html
index 861ab386b..4048bcf1f 100644
--- a/docs/cmdstan-guide/diagnose_utility.html
+++ b/docs/cmdstan-guide/diagnose_utility.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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diff --git a/docs/cmdstan-guide/err_handling_apdx.html b/docs/cmdstan-guide/err_handling_apdx.html
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diff --git a/docs/cmdstan-guide/example_model_data.html b/docs/cmdstan-guide/example_model_data.html
index 8115bd0d2..7bd8f42f8 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/example_model_data.html">Example Model and Data</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
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+ <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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 </li>
           <li class="sidebar-item">
@@ -375,12 +354,6 @@
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 </li>
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@@ -505,25 +478,6 @@ <h1>Example Model and Data</h1>
 <li id="fn1"><p>The model is available with the CmdStan distribution at the path <code>&lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan</code>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
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diff --git a/docs/cmdstan-guide/external_code.html b/docs/cmdstan-guide/external_code.html
index 6bdcca97b..1ef9aac18 100644
--- a/docs/cmdstan-guide/external_code.html
+++ b/docs/cmdstan-guide/external_code.html
@@ -71,10 +71,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -103,7 +105,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -218,6 +220,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -282,32 +285,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -315,7 +294,7 @@
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -336,7 +315,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -348,13 +327,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -374,12 +353,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -614,25 +587,6 @@ <h2 class="anchored" data-anchor-id="special-functions-rngs-distributions-editin
 <li id="fn1"><p>Details of programming in the Stan Math style are omitted from this section, it is presented only as an example<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -653,6 +607,150 @@ <h2 class="anchored" data-anchor-id="special-functions-rngs-distributions-editin
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
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+  const toggleColorMode = (alternate) => {
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+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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+    }
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+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+  } else {
+    toggleColorMode(false);
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diff --git a/docs/cmdstan-guide/gc-intro.html b/docs/cmdstan-guide/gc-intro.html
index f9041321b..00f7cba59 100644
--- a/docs/cmdstan-guide/gc-intro.html
+++ b/docs/cmdstan-guide/gc-intro.html
@@ -2,10 +2,10 @@
 <html lang="en-US">
   <meta charset="utf-8">
   <title>Redirecting&hellip;</title>
-  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_intro.html">
-  <script>location="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_intro.html"</script>
-  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/generate_quantities_intro.html">
+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">
+  <script>location="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html"</script>
+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">
   <meta name="robots" content="noindex">
   <h1>Redirecting&hellip;</h1>
-  <a href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_intro.html">Click here if you are not redirected.</a>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">Click here if you are not redirected.</a>
 </html>
diff --git a/docs/cmdstan-guide/generate_quantities_config.html b/docs/cmdstan-guide/generate_quantities_config.html
index 3567a4f6b..f7263340d 100644
--- a/docs/cmdstan-guide/generate_quantities_config.html
+++ b/docs/cmdstan-guide/generate_quantities_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>Standalone Generate Quantities</title>
+<title>Generating Quantities of Interest from a Fitted Model</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -20,6 +20,40 @@
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 }
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+pre > code.sourceCode > span { line-height: 1.25; }
+pre > code.sourceCode > span:empty { height: 1.2em; }
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+code.sourceCode > span { color: inherit; text-decoration: inherit; }
+div.sourceCode { margin: 1em 0; }
+pre.sourceCode { margin: 0; }
+@media screen {
+div.sourceCode { overflow: auto; }
+}
+@media print {
+pre > code.sourceCode { white-space: pre-wrap; }
+pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
+}
+pre.numberSource code
+  { counter-reset: source-line 0; }
+pre.numberSource code > span
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+pre.numberSource code > span > a:first-child::before
+  { content: counter(source-line);
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+    -khtml-user-select: none; -moz-user-select: none;
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+pre.numberSource { margin-left: 3em;  padding-left: 4px; }
+div.sourceCode
+  {   }
+@media screen {
+pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
+}
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@@ -38,10 +72,12 @@
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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-  <li><a href="#standalone-generate-quantities" id="toc-standalone-generate-quantities" class="nav-link active" data-scroll-target="#standalone-generate-quantities">Standalone Generate Quantities</a></li>
+  <li><a href="#gc-intro" id="toc-gc-intro" class="nav-link active" data-scroll-target="#gc-intro">Generating Quantities of Interest from a Fitted Model</a>
+  <ul class="collapse">
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@@ -440,46 +451,101 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="standalone-generate-quantities" class="level1">
-<h1>Standalone Generate Quantities</h1>
-<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. For an overview of the uses of this feature, see the <a href="../cmdstan-guide/generate_quantities_intro.html">QuickStart Guide section</a> and the Stan User’s Guide section on <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
-<p>This method requires sub-argument <code>fitted_params</code> which takes as its value an existing Stan CSV file that contains a sample from an equivalent model, i.e., a model with the same parameters, transformed parameters, and model blocks, conditioned on the same data.</p>
+<section id="gc-intro" class="level1">
+<h1>Generating Quantities of Interest from a Fitted Model</h1>
+<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the <code>generate_quantities</code> method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block.</p>
+<p>This method requires sub-argument <code>fitted_params</code> which takes as its value an existing <a href="../cmdstan-guide/stan_csv_apdx.html">Stan CSV</a> file that contains a parameter values from an equivalent model, i.e., a model with the same parameters block, conditioned on the same data.</p>
+<p>The <a href="https://mc-stan.org/docs/reference-manual/blocks.html#generated-quantities">generated quantities block</a> computes <em>quantities of interest</em> (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:</p>
+<ul>
+<li>generate simulated data for model testing by forward sampling</li>
+<li>generate predictions for new data</li>
+<li>calculate posterior event probabilities, including multiple comparisons, sign tests, etc.</li>
+<li>calculate posterior expectations</li>
+<li>transform parameters for reporting</li>
+<li>apply full Bayesian decision theory</li>
+<li>calculate log likelihoods, deviances, etc. for model comparison</li>
+</ul>
+<p>For an overview of the uses of this feature, see the Stan User’s Guide section on <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
+<section id="example" class="level2">
+<h2 class="anchored" data-anchor-id="example">Example</h2>
+<p>To illustrate how this works we use the <code>generate_quantities</code> method to do posterior predictive checks using the estimate of <code>theta</code> given the example bernoulli model and data, following the <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#posterior-predictive-simulation-in-stan">posterior predictive simulation</a> procedure in the Stan User’s Guide.</p>
+<p>We write a program <code>bernoulli_ppc.stan</code> which contains the following generated quantities block, with comments to explain the procedure:</p>
+<div class="sourceCode" id="cb1"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
+<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
+<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
+<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
+<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
+<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
+<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>The rest of the program is the same as in <code>bernoulli.stan</code>.</p>
+<p>The <code>generate_method</code> requires the sub-argument <code>fitted_params</code> which takes as its value the name of a Stan CSV file. The per-draw parameter values from the <code>fitted_params</code> file will be used to run the generated quantities block.</p>
 <p>If we run the <code>bernoulli.stan</code> program for a single chain to generate a sample in file <code>bernoulli_fit.csv</code>:</p>
 <pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv</code></pre>
 <p>Then we can run the <code>bernoulli_ppc.stan</code> to carry out the posterior predictive checks:</p>
 <pre><code>&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \
                   data file=bernoulli.data.json \
                   output file=bernoulli_ppc.csv</code></pre>
-<p>The <code>fitted_params</code> file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:</p>
+<p>The output file <code>bernoulli_ppc.csv</code> contains only the values for the variables declared in the <code>generated quantities</code> block, i.e., <code>theta_rep</code> and the elements of <code>y_sim</code>:</p>
+<pre><code># model = bernoulli_ppc_model
+# method = generate_quantities
+#   generate_quantities
+#     fitted_params = bernoulli_fit.csv
+# id = 1 (Default)
+# data
+#   file = bernoulli.data.json
+# init = 2 (Default)
+# random
+#   seed = 2983956445 (Default)
+# output
+#   file = output.csv (Default)
+y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep
+1,1,1,0,0,0,1,1,0,1,0.6
+1,1,0,1,0,0,1,0,1,0,0.5
+1,0,1,1,1,1,1,1,0,1,0.8
+0,1,0,1,0,1,0,1,0,0,0.4
+1,0,0,0,0,0,0,0,0,0,0.1
+0,0,0,0,0,1,1,1,0,0,0.3
+0,0,1,0,1,0,0,0,0,0,0.2
+1,0,1,0,1,1,0,1,1,0,0.6
+...</code></pre>
+<p>Given the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:</p>
+<div class="sourceCode" id="cb5"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
+<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
+<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
+<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
+<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_cp = theta;</span>
+<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
+<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
+<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>Now the output is slightly more interpretable: <code>theta_cp</code> is the same as the <code>theta</code> used to generate the values <code>y_sim[1]</code> through <code>y_sim[1]</code>. Comparing columns <code>theta_cp</code> and <code>theta_rep</code> allows us to see how the uncertainty in our estimate of <code>theta</code> is carried forward into our predictions:</p>
+<pre><code>y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep
+0,1,1,0,1,0,0,1,1,0,0.545679,0.5
+1,1,1,1,1,1,0,1,1,0,0.527164,0.8
+1,1,1,1,0,1,1,1,1,0,0.529116,0.8
+1,0,1,1,1,1,0,0,1,0,0.478844,0.6
+0,1,0,0,0,0,1,0,1,0,0.238793,0.3
+0,0,0,0,0,1,1,0,0,0,0.258294,0.2
+1,1,1,0,0,0,0,0,0,0,0.258465,0.3</code></pre>
+</section>
+<section id="errors" class="level2">
+<h2 class="anchored" data-anchor-id="errors">Errors</h2>
+<p>The <code>fitted_params</code> file must be a <a href="../cmdstan-guide/stan_csv_apdx.html">Stan CSV</a> file; attempts to use a regular CSV file will result an error message of the form:</p>
 <pre><code>Error reading fitted param names from sample csv file &lt;filename.csv&gt;</code></pre>
 <p>The <code>fitted_params</code> file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:</p>
 <pre><code>Error reading fitted param names from sample csv file &lt;filename.csv&gt;</code></pre>
-<p>The parameter values of the <code>fitted_params</code> are on the constrained scale and must obey all constraints. For example, if we modify the contencts of the first reported draw in <code>bernoulli_fit.csv</code> so that the value of <code>theta</code> is outside the declared bounds <code>real&lt;lower=0, upper=1&gt;</code>, the program will return the following error message:</p>
-<pre><code>Exception: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] (in 'bernoulli_ppc.stan', line 5, column 2 to column 30)</code></pre>
+<p>The parameter values of the <code>fitted_params</code> are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in <code>bernoulli_fit.csv</code> so that the value of <code>theta</code> is outside the declared bounds <code>real&lt;lower=0, upper=1&gt;</code>, the program will return the following error message:</p>
+<pre><code>Exception: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \
+(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)</code></pre>
 
 
+</section>
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@@ -869,8 +1079,8 @@ <h1>Standalone Generate Quantities</h1>
 </script>
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-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Variational Inference Algorithm: ADVI</span>
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+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Variational Inference using ADVI</span>
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diff --git a/docs/cmdstan-guide/generate_quantities_intro.html b/docs/cmdstan-guide/generate_quantities_intro.html
index 35168204c..00f7cba59 100644
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-<section id="gc-intro" class="level1">
-<h1>Generating Quantities of Interest from a Fitted Model</h1>
-<p>The <a href="https://mc-stan.org/docs/reference-manual/blocks.html#generated-quantities">generated quantities block</a> computes <em>quantities of interest</em> (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:</p>
-<ul>
-<li>generate simulated data for model testing by forward sampling</li>
-<li>generate predictions for new data</li>
-<li>calculate posterior event probabilities, including multiple comparisons, sign tests, etc.</li>
-<li>calculating posterior expectations</li>
-<li>transform parameters for reporting</li>
-<li>apply full Bayesian decision theory</li>
-<li>calculate log likelihoods, deviances, etc. for model comparison</li>
-</ul>
-<p>The <code>generate_quantities</code> method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the <code>generate_quantities</code> method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block. See the Stan User’s Guide section <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#stand-alone-generated-quantities-and-ongoing-prediction">Stand-alone generated quantities and ongoing prediction</a>.</p>
-<p>To illustrate how this works we use the <code>generate_quantities</code> method to do posterior predictive checks using the estimate of <code>theta</code> given the example bernoulli model and data, following the <a href="https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html#posterior-predictive-simulation-in-stan">posterior predictive simulation</a> procedure in the Stan User’s Guide.</p>
-<p>We write a program <code>bernoulli_ppc.stan</code> which contains the following generated quantities block, with comments to explain the procedure:</p>
-<div class="sourceCode" id="cb1"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
-<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep;</span>
-<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
-<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
-<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
-<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
-<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>  theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
-<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>The rest of the program is the same as in <code>bernoulli.stan</code>.</p>
-<p>The <code>generate_method</code> requires the sub-argument <code>fitted_params</code> which takes as its value the name of a Stan CSV file. The per-draw parameter estimates from the <code>fitted_params</code> file will be used to run the generated quantities block.</p>
-<p>If we run the <code>bernoulli.stan</code> program for a single chain to generate a sample in file <code>bernoulli_fit.csv</code>:</p>
-<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv</code></pre>
-<p>Then we can run the <code>bernoulli_ppc.stan</code> to carry out the posterior predictive checks:</p>
-<pre><code>&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \
-                  data file=bernoulli.data.json \
-                  output file=bernoulli_ppc.csv</code></pre>
-<p>The output file <code>bernoulli_ppc.csv</code> consists of just the values for the variables declared in the generated quantities block, i.e., <code>theta_rep</code> and the elements of <code>y_sim</code>:</p>
-<pre><code># model = bernoulli_ppc_model
-# method = generate_quantities
-#   generate_quantities
-#     fitted_params = bernoulli_fit.csv
-# id = 0 (Default)
-# data
-#   file = bernoulli.data.json
-# init = 2 (Default)
-# random
-#   seed = 2135140492 (Default)
-# output
-#   file = bernoulli_ppc.csv
-#   diagnostic_file =  (Default)
-#   refresh = 100 (Default)
-theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10
-0.2,0,0,1,0,0,0,0,0,1,0
-0.3,1,0,0,1,0,1,0,0,0,0
-0.8,1,0,1,1,1,1,1,1,1,0
-0.1,0,0,0,0,0,1,0,0,0,0
-0.3,0,0,0,0,0,0,1,1,1,0</code></pre>
-<p><em>Note</em>: the only relevant analysis of the resulting CSV output is computing per-column statistics; this can easily be done in Python, R, Excel or similar, or you can use the CmdStanPy and CmdStanR interfaces which provide a better user experience for this workflow.</p>
-<p>Given the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:</p>
-<div class="sourceCode" id="cb5"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
-<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_cp = theta;</span>
-<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; theta_rep;</span>
-<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">array</span>[N] <span class="dt">int</span> y_sim;</span>
-<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// use current estimate of theta to generate new sample</span></span>
-<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>    y_sim[n] = bernoulli_rng(theta);</span>
-<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>  <span class="co">// estimate theta_rep from new sample</span></span>
-<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a>  theta_rep = sum(y_sim) * <span class="fl">1.0</span> / N;</span>
-<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>Now the output is slightly more interpretable: <code>theta_cp</code> is the same as the <code>theta</code> used to generate the values <code>y_sim[1]</code> through <code>y_sim[1]</code>. Comparing columns <code>theta_cp</code> and <code>theta_rep</code> allows us to see how the uncertainty in our estimate of <code>theta</code> is carried forward into our predictions:</p>
-<pre><code>theta_cp,theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10
-0.102391,0,0,0,0,0,0,0,0,0,0,0
-0.519567,0.2,0,1,0,0,1,0,0,0,0,0
-0.544634,0.6,1,0,0,0,0,1,1,1,1,1
-0.167651,0,0,0,0,0,0,0,0,0,0,0
-0.167651,0.1,1,0,0,0,0,0,0,0,0,0</code></pre>
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-<footer class="footer"><div class="nav-footer"><div class="nav-footer-center"><div class="toc-actions d-sm-block d-md-none"><ul><li><a href="https://github.com/stan-dev/docs/edit/master/src/cmdstan-guide/generate_quantities_intro.qmd" class="toc-action"><i class="bi bi-github"></i>Edit this page</a></li><li><a href="https://github.com/stan-dev/docs/issues/new" class="toc-action"><i class="bi empty"></i>Report an issue</a></li></ul></div></div></div></footer></body></html>
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+<html lang="en-US">
+  <meta charset="utf-8">
+  <title>Redirecting&hellip;</title>
+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">
+  <script>location="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html"</script>
+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">
+  <meta name="robots" content="noindex">
+  <h1>Redirecting&hellip;</h1>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/generate_quantities_config.html">Click here if you are not redirected.</a>
+</html>
diff --git a/docs/cmdstan-guide/index.html b/docs/cmdstan-guide/index.html
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
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- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
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-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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@@ -302,7 +281,7 @@
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   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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 </li>
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   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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 </li>
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@@ -340,12 +319,6 @@
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- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
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@@ -450,7 +423,7 @@ <h1 class="title">CmdStan User’s Guide</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This document is a user’s guide for CmdStan, the command-line interface to the Stan statistical modeling language. CmdStan provides the programs and tools to compile Stan programs into C++ executables that can be run directly from the command line, together with a few utilities to check and summarize the resulting outputs.</p>
+<p>This document is a user’s guide for CmdStan, the command-line interface to the <a href="https://mc-stan.org/">Stan</a> statistical modeling language. CmdStan provides the programs and tools to compile Stan programs into C++ executables that can be run directly from the command line, together with a few utilities to check and summarize the resulting outputs.</p>
 <p>In CmdStan, statistical models written in the Stan probabilistic programming language are translated into a C++ program which is then compiled together with the CmdStan routines that provide the logic needed to manage all user inputs and program outputs and the Stan inference algorithms and math library. The resulting command line executable program can be used to</p>
 <ul>
 <li><p>do inference on data, producing an exact or approximate estimate of the posterior;</p></li>
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diff --git a/docs/cmdstan-guide/installation.html b/docs/cmdstan-guide/installation.html
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">Getting Started</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/installation.html">CmdStan Installation</a></li></ol></nav>
         <a class="flex-grow-1" role="button" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">      
         </a>
     </div>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -496,7 +469,7 @@ <h1>CmdStan Installation</h1>
 </ul>
 <section id="conda-install" class="level2">
 <h2 class="anchored" data-anchor-id="conda-install">Installation via <code>conda</code></h2>
-<p>With conda, you can install CmdStan from the <a href="https://conda-forge.org/">conda-forge channel</a>. This will install a pre-built version of CmdStan along with the required dependencies (i.e.&nbsp;a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command <code>cmdstan_model</code> to activate the CmdStan makefile from anywhere.</p>
+<p>With conda, you can install CmdStan from the <a href="https://conda-forge.org/">conda-forge channel</a>. This will install a pre-built version of CmdStan along with the required dependencies (i.e.&nbsp;a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command <code>cmdstan_model</code> to activate the CmdStan makefile from anywhere.</p>
 <p><em>Note</em>: This requires that conda has been installed already on your machine. You can either install <a href="https://docs.conda.io/en/latest/miniconda.html">miniconda</a>, a free, minimal installer for conda or you can get the full <a href="https://docs.continuum.io/anaconda/">Anaconda</a> system which provides graphical installer wizards for <a href="https://www.anaconda.com/products/distribution#macos">MacOS</a> and <a href="https://www.anaconda.com/products/distribution#windows">Windows</a> users.</p>
 <p>We recommend installing CmdStan in a new conda environment:</p>
 <pre><code> conda create -n stan -c conda-forge cmdstan</code></pre>
@@ -570,7 +543,7 @@ <h2 class="anchored" data-anchor-id="checking-the-stan-compiler">Checking the St
 &gt; make examples/bernoulli/bernoulli
 
 # fit to provided data (results of 10 trials, 2 out of 10 successes)
-&gt; ./examples/bernoulli/bernoulli sample\ 
+&gt; ./examples/bernoulli/bernoulli sample\
   data file=examples/bernoulli/bernoulli.data.json
 
 # default output written to file `output.csv`,
@@ -741,7 +714,7 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
     4. Build the diagnose utility bin/diagnose
     5. Build all libraries and object files compile and link an executable Stan program
 
-    Note: to build using multiple cores, use the -j option to make, e.g., 
+    Note: to build using multiple cores, use the -j option to make, e.g.,
     for 4 cores:
     &gt; make build -j4
 
@@ -798,25 +771,6 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
 <li id="fn1"><p>To open a Windows command shell, first open the <strong>Start Menu</strong>, (usually in the lower left of the screen), select option <strong>All Programs</strong>, then option <strong>Accessories</strong>, then program <strong>Command Prompt</strong>. Alternatively, enter <code>[Windows+r]</code> (both keys together on the keyboard), and enter <code>cmd</code> into the text field that pops up in the Run window, then press <code>[Return]</code> on the keyboard to run.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -837,6 +791,150 @@ <h2 class="anchored" data-anchor-id="gnu-make">Using GNU Make</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
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+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
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+  } else {
+    toggleColorMode(false);
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diff --git a/docs/cmdstan-guide/json_apdx.html b/docs/cmdstan-guide/json_apdx.html
index 12a1949c2..f6b4763d3 100644
--- a/docs/cmdstan-guide/json_apdx.html
+++ b/docs/cmdstan-guide/json_apdx.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -286,7 +289,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -312,32 +315,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -345,7 +324,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -366,7 +345,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -378,13 +357,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -404,12 +383,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -687,25 +660,6 @@ <h3 class="anchored" data-anchor-id="empty-arrays-in-json">Empty arrays in JSON<
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -726,6 +680,150 @@ <h3 class="anchored" data-anchor-id="empty-arrays-in-json">Empty arrays in JSON<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    i.classList.add('bi');
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diff --git a/docs/cmdstan-guide/laplace_sample_config.html b/docs/cmdstan-guide/laplace_sample_config.html
index 473c935c6..f42ba54f9 100644
--- a/docs/cmdstan-guide/laplace_sample_config.html
+++ b/docs/cmdstan-guide/laplace_sample_config.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -70,7 +72,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
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-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/laplace_sample_config.html">Laplace sampling</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/laplace_sample_config.html">Laplace sampling</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
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-  <div class="sidebar-item-container"> 
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-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -447,7 +420,7 @@ <h2 id="toc-title">On this page</h2>
 
 <section id="laplace-sampling" class="level1">
 <h1>Laplace sampling</h1>
-<p>The <code>laplace</code> method produces a sample from a normal approximation centered at the mode of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See <a href="https://avehtari.github.io/casestudies/Jacobian/jacobian.html">this case study</a> for a visual illustration.)</p>
+<p>The <code>laplace</code> method produces a sample from a normal approximation centered at the <a href="../cmdstan-guide/optimize_config.html">mode</a> of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See <a href="https://avehtari.github.io/casestudies/Jacobian/jacobian.html">this case study</a> for a visual illustration.)</p>
 <p>This is computationally inexpensive compared to exact Bayesian inference with MCMC. The goodness of this estimate depends on both the estimate of the mode and how much the true posterior in the unconstrained space resembles a Gaussian.</p>
 <section id="configuration" class="level2">
 <h2 class="anchored" data-anchor-id="configuration">Configuration</h2>
@@ -518,25 +491,6 @@ <h2 class="anchored" data-anchor-id="example">Example</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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@@ -557,6 +511,150 @@ <h2 class="anchored" data-anchor-id="example">Example</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+          theme: newTheme
+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
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+  const setStyleSentinel = (alternate) => {
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+      return storageValue != null ? storageValue : localAlternateSentinel;
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
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@@ -926,8 +1024,8 @@ <h2 class="anchored" data-anchor-id="example">Example</h2>
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Standalone Generate Quantities">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Standalone Generate Quantities</span>
+      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/cmdstan-guide/log_prob_config.html b/docs/cmdstan-guide/log_prob_config.html
index f0996f429..ed65b3aba 100644
--- a/docs/cmdstan-guide/log_prob_config.html
+++ b/docs/cmdstan-guide/log_prob_config.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -104,7 +106,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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 <body class="nav-sidebar floating nav-fixed">
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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+ <span class="menu-text">Optimization</span></a>
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+ <span class="menu-text">Variational Inference using ADVI</span></a>
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+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -375,12 +354,6 @@
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-          <li class="sidebar-item">
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- <span class="menu-text">Parallelization</span></a>
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diff --git a/docs/cmdstan-guide/mcmc-intro.html b/docs/cmdstan-guide/mcmc-intro.html
index 0d3af21af..8e5d5a5e2 100644
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-  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/mcmc_sampling_intro.html#running-the-sampler">
+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/mcmc_config.html">
+  <script>location="https://mc-stan.org/docs/cmdstan-guide/mcmc_config.html"</script>
+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/mcmc_config.html">
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-  <a href="https://mc-stan.org/docs/cmdstan-guide/mcmc_sampling_intro.html#running-the-sampler">Click here if you are not redirected.</a>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/mcmc_config.html">Click here if you are not redirected.</a>
 </html>
diff --git a/docs/cmdstan-guide/mcmc_config.html b/docs/cmdstan-guide/mcmc_config.html
index 114955e9b..18ceb1d81 100644
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-<title>MCMC Sampling Configuration</title>
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/mcmc_config.html">MCMC Sampling using Hamiltonian Monte Carlo</a></li></ol></nav>
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+ <span class="menu-text">Getting Started</span></a>
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@@ -298,32 +301,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
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- <span class="menu-text">MCMC Sampling</span></a>
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-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
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- <span class="menu-text">Optimization</span></a>
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-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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 </li>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
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@@ -352,7 +331,7 @@
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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           <li class="sidebar-item">
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   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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 </li>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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 </li>
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- <span class="menu-text">Parallelization</span></a>
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 </li>
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   <ul class="collapse">
+  <li><a href="#running-the-sampler" id="toc-running-the-sampler" class="nav-link" data-scroll-target="#running-the-sampler">Running the sampler</a></li>
+  <li><a href="#mcmc_output_csv" id="toc-mcmc_output_csv" class="nav-link" data-scroll-target="#mcmc_output_csv">Stan CSV output file</a></li>
   <li><a href="#iterations" id="toc-iterations" class="nav-link" data-scroll-target="#iterations">Iterations</a></li>
   <li><a href="#adaptation" id="toc-adaptation" class="nav-link" data-scroll-target="#adaptation">Adaptation</a>
   <ul class="collapse">
@@ -493,8 +468,12 @@ <h2 id="toc-title">On this page</h2>
   <li><a href="#hmc-samplers" id="toc-hmc-samplers" class="nav-link" data-scroll-target="#hmc-samplers">HMC samplers</a></li>
   </ul></li>
   <li><a href="#sampler-diag-file" id="toc-sampler-diag-file" class="nav-link" data-scroll-target="#sampler-diag-file">Sampler diagnostic file</a></li>
-  <li><a href="#sampler-num-chains" id="toc-sampler-num-chains" class="nav-link" data-scroll-target="#sampler-num-chains">Multiple chains in one executable</a></li>
-  <li><a href="#examples---older-parallelism" id="toc-examples---older-parallelism" class="nav-link" data-scroll-target="#examples---older-parallelism">Examples - older parallelism</a>
+  <li><a href="#multi-chain-sampling" id="toc-multi-chain-sampling" class="nav-link" data-scroll-target="#multi-chain-sampling">Running multiple chains</a>
+  <ul class="collapse">
+  <li><a href="#sampler-num-chains" id="toc-sampler-num-chains" class="nav-link" data-scroll-target="#sampler-num-chains">Using the num_chains argument to run multiple chains</a></li>
+  </ul></li>
+  <li><a href="#summarizing-sampler-outputs-with-stansummary" id="toc-summarizing-sampler-outputs-with-stansummary" class="nav-link" data-scroll-target="#summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></a></li>
+  <li><a href="#old-multichain" id="toc-old-multichain" class="nav-link" data-scroll-target="#old-multichain">Examples - older parallelism</a>
   <ul class="collapse">
   <li><a href="#running-multiple-chains-with-a-specified-rng-seed" id="toc-running-multiple-chains-with-a-specified-rng-seed" class="nav-link" data-scroll-target="#running-multiple-chains-with-a-specified-rng-seed">Running multiple chains with a specified RNG seed</a></li>
   <li><a href="#changing-the-default-warmup-and-sampling-iterations" id="toc-changing-the-default-warmup-and-sampling-iterations" class="nav-link" data-scroll-target="#changing-the-default-warmup-and-sampling-iterations">Changing the default warmup and sampling iterations</a></li>
@@ -520,8 +499,78 @@ <h2 id="toc-title">On this page</h2>
 <section id="mcmc-config" class="level1">
 <h1>MCMC Sampling using Hamiltonian Monte Carlo</h1>
 <p>The <code>sample</code> method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on <a href="https://mc-stan.org/docs/reference-manual/mcmc.html">MCMC Sampling</a>.</p>
-<p>The full set of configuration options available for the <code>sample</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default sample method configuration options:</p>
-<pre><code># method = sample (Default)
+<section id="running-the-sampler" class="level2">
+<h2 class="anchored" data-anchor-id="running-the-sampler">Running the sampler</h2>
+<p>To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument <code>sample</code> or <code>method=sample</code> together with the input data. The executable can be run from any directory.</p>
+<p>The full set of configuration options available for the <code>sample</code> method is available by using the <code>sample help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.</p>
+<p>Here, we run it in the directory which contains the Stan program and input data, <code>&lt;cmdstan-home&gt;/examples/bernoulli</code>:</p>
+<pre><code>&gt; cd examples/bernoulli
+&gt; ls
+  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan</code></pre>
+<p>To execute sampling of the model under Linux or Mac, use:</p>
+<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json</code></pre>
+<p>In Windows, the <code>./</code> prefix is not needed:</p>
+<pre><code>&gt; bernoulli.exe sample data file=bernoulli.data.json</code></pre>
+<p>The output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:</p>
+<pre><code>method = sample (Default)
+  sample
+    num_samples = 1000 (Default)
+    num_warmup = 1000 (Default)
+    save_warmup = 0 (Default)
+    thin = 1 (Default)
+    adapt
+      engaged = 1 (Default)
+      gamma = 0.050000000000000003 (Default)
+      delta = 0.80000000000000004 (Default)
+      kappa = 0.75 (Default)
+      t0 = 10 (Default)
+      init_buffer = 75 (Default)
+      term_buffer = 50 (Default)
+      window = 25 (Default)
+      save_metric = 0 (Default)
+    algorithm = hmc (Default)
+      hmc
+        engine = nuts (Default)
+          nuts
+            max_depth = 10 (Default)
+        metric = diag_e (Default)
+        metric_file =  (Default)
+        stepsize = 1 (Default)
+        stepsize_jitter = 0 (Default)
+    num_chains = 1 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 3252652196 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>After the configuration has been displayed, a short timing message is given.</p>
+<pre><code>Gradient evaluation took 1.2e-05 seconds
+1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
+Adjust your expectations accordingly!</code></pre>
+<p>Next, the sampler reports the iteration number, reporting the percentage complete.</p>
+<pre><code>Iteration:    1 / 2000 [  0%]  (Warmup)
+...
+Iteration: 2000 / 2000 [100%]  (Sampling)</code></pre>
+<p>Finally, the sampler reports timing information:</p>
+<pre><code> Elapsed Time: 0.007 seconds (Warm-up)
+               0.017 seconds (Sampling)
+               0.024 seconds (Total)</code></pre>
+</section>
+<section id="mcmc_output_csv" class="level2">
+<h2 class="anchored" data-anchor-id="mcmc_output_csv">Stan CSV output file</h2>
+<p>Each execution of the model results in draws from a single Markov chain being written to a file in <a href="../cmdstan-guide/stan_csv_apdx.html">comma-separated value (CSV) format</a>. The default name of the output file is <code>output.csv</code>.</p>
+<p>The first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (<code>#</code>)).</p>
+<p>When the example model <code>bernoulli.stan</code> is run via the command line with all default arguments, the following configuration is displayed:</p>
+<pre><code># stan_version_major = 2
+# stan_version_minor = 23
+# stan_version_patch = 0
+# model = bernoulli_model
+# method = sample (Default)
 #   sample
 #     num_samples = 1000 (Default)
 #     num_warmup = 1000 (Default)
@@ -546,7 +595,44 @@ <h1>MCMC Sampling using Hamiltonian Monte Carlo</h1>
 #         metric_file =  (Default)
 #         stepsize = 1.000000 (Default)
 #         stepsize_jitter = 0.000000 (Default)
-#     num_chains = 1 (Default)</code></pre>
+#     num_chains = 1 (Default)
+# output
+#   file = output.csv (Default)
+#   diagnostic_file =  (Default)
+#   refresh = 100 (Default)</code></pre>
+<p>This is followed by a CSV header indicating the names of the values sampled.</p>
+<pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
+<p>The first output columns report the HMC sampler information:</p>
+<ul>
+<li><code>lp__</code> - the total log probability density (up to an additive constant) at each sample</li>
+<li><code>accept_stat__</code> - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory</li>
+<li><code>stepsize__</code> - integrator step size</li>
+<li><code>treedepth__</code> - depth of tree used by NUTS (NUTS sampler)</li>
+<li><code>n_leapfrog__</code> - number of leapfrog calculations (NUTS sampler)</li>
+<li><code>divergent__</code> - has value <code>1</code> if trajectory diverged, otherwise <code>0</code>. (NUTS sampler)</li>
+<li><code>energy__</code> - value of the Hamiltonian</li>
+<li><code>int_time__</code> - total integration time (static HMC sampler)</li>
+</ul>
+<p>Because the above header is from the NUTS sampler, it has columns <code>treedepth__</code>, <code>n_leapfrog__</code>, and <code>divergent__</code> and doesn’t have column <code>int_time__</code>. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, <code>theta</code>.</p>
+<p>The header line is written to the output file before warmup begins. If option <code>save_warmup</code> is set to <code>1</code>, the warmup draws are output directly after the header. The total number of warmup draws saved is <code>num_warmup</code> divided by <code>thin</code>, rounded up (i.e., <code>ceiling</code>).</p>
+<p>Following the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:</p>
+<pre><code># Adaptation terminated
+# Step size = 0.884484
+# Diagonal elements of inverse mass matrix:
+# 0.535006</code></pre>
+<p>The default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter <code>theta</code>.</p>
+<p>Draws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.</p>
+<pre><code>-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853
+-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295
+-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299
+-6.88712,1,0.884484,1,1,0,7.02101,0.188229
+-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596
+...</code></pre>
+<p>The output ends with timing details:</p>
+<pre><code>#  Elapsed Time: 0.007 seconds (Warm-up)
+#                0.017 seconds (Sampling)
+#                0.024 seconds (Total)</code></pre>
+</section>
 <section id="iterations" class="level2">
 <h2 class="anchored" data-anchor-id="iterations">Iterations</h2>
 <p>At every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters <code>metric</code> and <code>stepsize</code> in order to efficiently sample from <em>typical set</em>, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.</p>
@@ -652,32 +738,86 @@ <h4 class="anchored" data-anchor-id="integration-time">Integration time</h4>
 <h2 class="anchored" data-anchor-id="sampler-diag-file">Sampler diagnostic file</h2>
 <p>The <code>output</code> keyword sub-argument <code>diagnostic_file=&lt;filepath&gt;</code> specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.</p>
 </section>
-<section id="sampler-num-chains" class="level2">
-<h2 class="anchored" data-anchor-id="sampler-num-chains">Multiple chains in one executable</h2>
-<p>As described in the <a href="../cmdstan-guide/mcmc_sampling_intro.html#multi-chain-sampling">quickstart section on parallelism</a>, the preferred way to run multiple chains is to use the <code>num_chains</code> argument.</p>
-<p>This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with <code>STAN_THREADS=true</code>.</p>
+<section id="multi-chain-sampling" class="level2">
+<h2 class="anchored" data-anchor-id="multi-chain-sampling">Running multiple chains</h2>
+<p>A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.</p>
+<p>The preferred way of using multiple chains is to run them all from the same executable using the <code>num_chains</code> argument. There is also the option to use the <a href="#old-multichain">Unix or DOS shell to run multiple executables</a>.</p>
+<section id="sampler-num-chains" class="level3">
+<h3 class="anchored" data-anchor-id="sampler-num-chains">Using the num_chains argument to run multiple chains</h3>
+<p>The <code>num_chains</code> argument can be used for all of Stan’s samplers with the exception of the <code>static</code> HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with <code>STAN_THREADS=true</code>, these will either run in parallel or one after the other.</p>
 <p>The <code>num_chains</code> argument changes the meanings of several other arguments when it is greater than <code>1</code> (the default). Many arguments are now interpreted as a “template” which is used for each chain.</p>
 <p>For example, when <code>num_chains=2</code>, the argument <code>output file=foo.csv</code> <em>no longer produces a file <code>foo.csv</code></em>, but instead produces two files, <code>foo_1.csv</code> and <code>foo_2.csv</code>. If you also supply <code>id=5</code>, the files produced will be <code>foo_5.csv</code> and <code>foo_6.csv</code> – <code>id=5</code> gives the id of the first chain, and the remaining chains are sequential from there.</p>
 <p>This also applies to input files, like those used for initialization. For example, if <code>num_chains=3</code> and <code>init=bar.json</code> will first look for <code>bar_1.json</code>. If it exists, it will use <code>bar_1.json</code> for the first chain, <code>bar_2.json</code> for the second, and so on. If <code>bar_1.json</code> does not exist, it falls back to looking for <code>bar.json</code>, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the <code>id</code> argument, which defaults to <code>1</code>.</p>
+<p>For example, this will run 4 chains:</p>
+<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv</code></pre>
+<p>This will produce samples in <code>output_1.csv</code>, <code>output_2.csv</code>, <code>output_3.csv</code>, <code>output_4.csv</code>. A suffix with the chain id is appended to the provided output filename (<code>output.csv</code> in the above command).</p>
+<p>If the model was not compiled with <code>STAN_THREADS=true</code>, the above command will run 4 chains sequentially.</p>
+<p>If the model was compiled with <code>STAN_THREADS=true</code>, the chains can run in parallel, with the <code>num_threads</code> argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the below command will run 4 chains in parallel, provided there are cores available:</p>
+<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4</code></pre>
+<p>If the model uses within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the <a href="https://www.intel.com/content/www/us/en/docs/onetbb/developer-guide-api-reference/2021-11/how-task-scheduler-works.html">Threading Building Blocks scheduler</a>.</p>
+<pre><code>./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16</code></pre>
 </section>
-<section id="examples---older-parallelism" class="level2">
-<h2 class="anchored" data-anchor-id="examples---older-parallelism">Examples - older parallelism</h2>
+</section>
+<section id="summarizing-sampler-outputs-with-stansummary" class="level2">
+<h2 class="anchored" data-anchor-id="summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></h2>
+<p>The <a href="../cmdstan-guide/stansummary.html"><code>stansummary</code> utility</a> processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file <code>stansummary</code> reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and <span class="math inline">\(\hat{R}\)</span> values.</p>
+<p>To run <code>stansummary</code> on the output files generated by the for loop above, by the above run of the <code>bernoulli</code> model on Mac or Linux:</p>
+<pre><code>&lt;cmdstan-home&gt;/bin/stansummary output_*.csv</code></pre>
+<p>On Windows, use backslashes to call the <code>stansummary.exe</code>.</p>
+<pre><code>&lt;cmdstan-home&gt;\bin\stansummary.exe output_*.csv</code></pre>
+<p>The stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable <code>theta</code>:</p>
+<pre><code>Inference for Stan model: bernoulli_model
+4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.
+
+Warmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total
+Sampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total
+
+                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat
+lp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00
+accept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00
+stepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13
+treedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00
+n_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00
+divergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan
+energy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00
+theta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00
+
+Samples were drawn using hmc with nuts.
+For each parameter, N_Eff is a crude measure of effective sample size,
+and R_hat is the potential scale reduction factor on split chains (at
+convergence, R_hat=1).</code></pre>
+<p>In this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for <code>theta</code> reflect the uncertainty in our estimate, due to the small amount of data, given the prior of <code>beta(1, 1)</code></p>
+</section>
+<section id="old-multichain" class="level2">
+<h2 class="anchored" data-anchor-id="old-multichain">Examples - older parallelism</h2>
 <p><strong>Note</strong>: Many of these examples can be simplified by using the <a href="#sampler-num-chains"><code>num_chains</code> argument</a>.</p>
-<p>The Quickstart Guide <a href="../cmdstan-guide/mcmc_sampling_intro.html">MCMC Sampling chapter</a> section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.</p>
-<p>This creates multiple copies of the model process which will all load the data.</p>
-<p>To run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:</p>
+<p>When the <code>num_chains</code> argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.</p>
+<p>To run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell <code>for</code> loop to set up index variables needed to identify each chain and its outputs.</p>
+<p>On MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:</p>
+<pre><code>for NAME [in LIST]; do COMMANDS; done</code></pre>
+<p>The list can be a simple sequence of numbers, or you can use the shell expansion syntax <code>{1..N}</code> which expands to the sequence from <span class="math inline">\(1\)</span> to <span class="math inline">\(N\)</span>, e.g.&nbsp;<code>{1..4}</code> expands to <code>1 2 3 4</code>. Note that the expression <code>{1..N}</code> cannot contain spaces.</p>
+<p>To run 4 chains for the example bernoulli model on MacOS or Linux:</p>
+<pre><code>&gt; for i in {1..4}
+    do
+      ./bernoulli sample data file=bernoulli.data.json \
+      output file=output_${i}.csv
+    done</code></pre>
+<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. To run chains in parallel, put an ampersand (<code>&amp;</code>) at the end of the nested sampler command:</p>
 <pre><code>&gt; for i in {1..4}
     do
-      ./bernoulli sample data file=my_model.data.json \
-                  output file=output_${i}.csv &amp;
+      ./bernoulli sample data file=bernoulli.data.json \
+      output file=output_${i}.csv &amp;
     done</code></pre>
-<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. The ampersand (<code>&amp;</code>) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
-<p>On Windows the corresponding loop is:</p>
+<p>This pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
+<p>On Windows, the DOS <a href="https://www.windows-commandline.com/windows-for-loop-examples/">for-loop syntax</a> is one of:</p>
+<pre><code>for %i in (SET) do COMMAND COMMAND-ARGUMENTS
+for /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS</code></pre>
+<p>To run 4 chains in parallel on Windows:</p>
 <pre><code>&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^
-                                    data file=my_model.data.json my_data ^
+                                    data file=bernoulli.data.json my_data ^
                                     output file=output_%i.csv</code></pre>
 <p>The caret (<code>^</code>) indicates a line continuation in DOS. The expression <code>%i</code> is the loop index.</p>
-<p>In the following examples, we focus on just the nested sampler command for Unix.</p>
+<p>In the following extended examples, we focus on just the nested sampler command for Unix.</p>
 <section id="running-multiple-chains-with-a-specified-rng-seed" class="level3">
 <h3 class="anchored" data-anchor-id="running-multiple-chains-with-a-specified-rng-seed">Running multiple chains with a specified RNG seed</h3>
 <p>For reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.</p>
@@ -835,25 +975,6 @@ <h3 class="anchored" data-anchor-id="everything-example">Everything example</h3>
 Betancourt, Michael. 2017. <span>“A Conceptual Introduction to <span>H</span>amiltonian <span>M</span>onte <span>C</span>arlo.”</span> <em>arXiv</em> 1701.02434. <a href="https://arxiv.org/abs/1701.02434">https://arxiv.org/abs/1701.02434</a>.
 </div>
 </div></section></div></main> <!-- /main -->
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-<section id="mcmc-intro" class="level1">
-<h1>MCMC Sampling</h1>
-<section id="running-the-sampler" class="level2">
-<h2 class="anchored" data-anchor-id="running-the-sampler">Running the sampler</h2>
-<p>To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument <code>sample</code> or <code>method=sample</code> together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, <code>&lt;cmdstan-home&gt;/examples/bernoulli</code>:</p>
-<pre><code>&gt; cd examples/bernoulli</code></pre>
-<p>To execute sampling of the model under Linux or Mac, use:</p>
-<pre><code>&gt; ./bernoulli sample data file=bernoulli.data.json</code></pre>
-<p>In Windows, the <code>./</code> prefix is not needed:</p>
-<pre><code>&gt; bernoulli.exe sample data file=bernoulli.data.json</code></pre>
-<p>The output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:</p>
-<pre><code>method = sample (Default)
-  sample
-    num_samples = 1000 (Default)
-    num_warmup = 1000 (Default)
-    save_warmup = 0 (Default)
-    thin = 1 (Default)
-    adapt
-      engaged = 1 (Default)
-      gamma = 0.050000000000000003 (Default)
-      delta = 0.80000000000000004 (Default)
-      kappa = 0.75 (Default)
-      t0 = 10 (Default)
-      init_buffer = 75 (Default)
-      term_buffer = 50 (Default)
-      window = 25 (Default)
-      save_metric = 0 (Default)
-    algorithm = hmc (Default)
-      hmc
-        engine = nuts (Default)
-          nuts
-            max_depth = 10 (Default)
-        metric = diag_e (Default)
-        metric_file =  (Default)
-        stepsize = 1 (Default)
-        stepsize_jitter = 0 (Default)
-    num_chains = 1 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 3252652196 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>After the configuration has been displayed, a short timing message is given.</p>
-<pre><code>Gradient evaluation took 1.2e-05 seconds
-1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
-Adjust your expectations accordingly!</code></pre>
-<p>Next, the sampler reports the iteration number, reporting the percentage complete.</p>
-<pre><code>Iteration:    1 / 2000 [  0%]  (Warmup)
-....
-Iteration: 2000 / 2000 [100%]  (Sampling)</code></pre>
-<p>Finally, the sampler reports timing information:</p>
-<pre><code> Elapsed Time: 0.007 seconds (Warm-up)
-               0.017 seconds (Sampling)
-               0.024 seconds (Total)</code></pre>
-</section>
-<section id="multi-chain-sampling" class="level2">
-<h2 class="anchored" data-anchor-id="multi-chain-sampling">Running multiple chains</h2>
-<p>A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.</p>
-<p>There are two different ways of running multiple chains, with the <code>num_chains</code> argument using a single executable and by using the Unix and DOS shell to run multiple executables.</p>
-<section id="using-the-num_chains-argument-to-run-multiple-chains" class="level3">
-<h3 class="anchored" data-anchor-id="using-the-num_chains-argument-to-run-multiple-chains">Using the num_chains argument to run multiple chains</h3>
-<p>The <code>num_chains</code> argument can be used for all of Stan’s samplers with the exception of the <code>static</code> HMC engine.</p>
-<p>Example that will run 4 chains:</p>
-<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv</code></pre>
-<p>If the model was not compiled with <code>STAN_THREADS=true</code>, the above command will run 4 chains sequentially and will produce the sample in <code>output_1.csv</code>, <code>output_2.csv</code>, <code>output_3.csv</code>, <code>output_4.csv</code>. A suffix with the chain id is appended to the provided output filename (<code>output.csv</code> in the above command).</p>
-<p>If the model was compiled with <code>STAN_THREADS=true</code>, the chains can run in parallel, with the <code>num_threads</code> argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the below command will run 4 chains in parallel, provided there are cores available:</p>
-<pre><code>./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4</code></pre>
-<p>If the model uses within-chain parallelization (<code>map_rect</code> or <code>reduce_sum</code> calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the <a href="https://www.intel.com/content/www/us/en/docs/onetbb/developer-guide-api-reference/2021-11/how-task-scheduler-works.html">Threading Building Blocks scheduler</a>.</p>
-<pre><code>./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16</code></pre>
-</section>
-<section id="using-shell-for-running-multiple-chains" class="level3">
-<h3 class="anchored" data-anchor-id="using-shell-for-running-multiple-chains">Using shell for running multiple chains</h3>
-<p>To run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell <code>for</code> loop to set up index variables needed to identify each chain and its outputs.</p>
-<p>On MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:</p>
-<pre><code>for NAME [in LIST]; do COMMANDS; done</code></pre>
-<p>The list can be a simple sequence of numbers, or you can use the shell expansion syntax <code>{1..N}</code> which expands to the sequence from <span class="math inline">\(1\)</span> to <span class="math inline">\(N\)</span>, e.g.&nbsp;<code>{1..4}</code> expands to <code>1 2 3 4</code>. Note that the expression <code>{1..N}</code> cannot contain spaces.</p>
-<p>To run 4 chains for the example bernoulli model on MacOS or Linux:</p>
-<pre><code>&gt; for i in {1..4}
-    do
-      ./bernoulli sample data file=bernoulli.data.json \
-      output file=output_${i}.csv
-    done</code></pre>
-<p>The backslash (<code>\</code>) indicates a line continuation in Unix. The expression <code>${i}</code> substitutes in the value of loop index variable <code>i</code>. To run chains in parallel, put an ampersand (<code>&amp;</code>) at the end of the nested sampler command:</p>
-<pre><code>&gt; for i in {1..4}
-    do
-      ./bernoulli sample data file=bernoulli.data.json \
-      output file=output_${i}.csv &amp;
-    done</code></pre>
-<p>This pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.</p>
-<p>On Windows, the DOS <a href="https://www.windows-commandline.com/windows-for-loop-examples/">for-loop syntax</a> is one of:</p>
-<pre><code>for %i in (SET) do COMMAND COMMAND-ARGUMENTS
-for /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS</code></pre>
-<p>To run 4 chains in parallel on Windows:</p>
-<pre><code>&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^
-                                    data file=bernoulli.data.json my_data ^
-                                    output file=output_%i.csv</code></pre>
-<p>The caret (<code>^</code>) indicates a line continuation in DOS.</p>
-</section>
-</section>
-<section id="mcmc_output_csv" class="level2">
-<h2 class="anchored" data-anchor-id="mcmc_output_csv">Stan CSV output file</h2>
-<p>Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is <code>output.csv</code>.</p>
-<p>The first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (<code>#</code>)).</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = sample (Default)
-#   sample
-#     num_samples = 1000 (Default)
-#     num_warmup = 1000 (Default)
-...
-# output
-#   file = output.csv (Default)
-#   diagnostic_file =  (Default)
-#   refresh = 100 (Default)</code></pre>
-<p>This is followed by a CSV header indicating the names of the values sampled.</p>
-<pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
-<p>The first output columns report the HMC sampler information:</p>
-<ul>
-<li><code>lp__</code> - the total log probability density (up to an additive constant) at each sample</li>
-<li><code>accept_stat__</code> - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory</li>
-<li><code>stepsize__</code> - integrator step size</li>
-<li><code>treedepth__</code> - depth of tree used by NUTS (NUTS sampler)</li>
-<li><code>n_leapfrog__</code> - number of leapfrog calculations (NUTS sampler)</li>
-<li><code>divergent__</code> - has value <code>1</code> if trajectory diverged, otherwise <code>0</code>. (NUTS sampler)</li>
-<li><code>energy__</code> - value of the Hamiltonian</li>
-<li><code>int_time__</code> - total integration time (static HMC sampler)</li>
-</ul>
-<p>Because the above header is from the NUTS sampler, it has columns <code>treedepth__</code>, <code>n_leapfrog__</code>, and <code>divergent__</code> and doesn’t have column <code>int_time__</code>. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, <code>theta</code>.</p>
-<p>The header line is written to the output file before warmup begins. If option <code>save_warmup</code> is set to <code>1</code>, the warmup draws are output directly after the header. The total number of warmup draws saved is <code>num_warmup</code> divided by <code>thin</code>, rounded up (i.e., <code>ceiling</code>).</p>
-<p>Following the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:</p>
-<pre><code># Adaptation terminated
-# Step size = 0.884484
-# Diagonal elements of inverse mass matrix:
-# 0.535006</code></pre>
-<p>The default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter <code>theta</code>.</p>
-<p>Draws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.</p>
-<pre><code>-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853
--6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295
--7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299
--6.88712,1,0.884484,1,1,0,7.02101,0.188229
--7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596
-...</code></pre>
-<p>The output ends with timing details:</p>
-<pre><code>#  Elapsed Time: 0.007 seconds (Warm-up)
-#                0.017 seconds (Sampling)
-#                0.024 seconds (Total)</code></pre>
-</section>
-<section id="summarizing-sampler-outputs-with-stansummary" class="level2">
-<h2 class="anchored" data-anchor-id="summarizing-sampler-outputs-with-stansummary">Summarizing sampler output(s) with <code>stansummary</code></h2>
-<p>The <code>stansummary</code> utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file <code>stansummary</code> reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and <span class="math inline">\(\hat{R}\)</span> values.</p>
-<p>To run <code>stansummary</code> on the output files generated by the for loop above, by the above run of the <code>bernoulli</code> model on Mac or Linux:</p>
-<pre><code>&lt;cmdstan-home&gt;/bin/stansummary output_*.csv</code></pre>
-<p>On Windows, use backslashes to call the <code>stansummary.exe</code>.</p>
-<pre><code>&lt;cmdstan-home&gt;\bin\stansummary.exe output_*.csv</code></pre>
-<p>The stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable <code>theta</code>:</p>
-<pre><code>Inference for Stan model: bernoulli_model
-4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.
-
-Warmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total
-Sampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total
-
-                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat
-lp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00
-accept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00
-stepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13
-treedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00
-n_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00
-divergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan
-energy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00
-theta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00
-
-Samples were drawn using hmc with nuts.
-For each parameter, N_Eff is a crude measure of effective sample size,
-and R_hat is the potential scale reduction factor on split chains (at
-convergence, R_hat=1).</code></pre>
-<p>In this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for <code>theta</code> reflect the uncertainty in our estimate, due to the small amount of data, given the prior of <code>beta(1, 1)</code></p>
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-<section id="optimization" class="level1">
-<h1>Optimization</h1>
-<p>The CmdStan executable can run Stan’s optimization algorithms which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability.</p>
-<p>The executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli optimize data file=bernoulli.data.json</code></pre>
-<p>Executing this command prints both output to the console and to a csv file.</p>
-<p>The first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:</p>
-<pre><code> ./bernoulli optimize data file=bernoulli.data.json
-method = optimize
-  optimize
-    algorithm = lbfgs (Default)
-      lbfgs
-        init_alpha = 0.001 (Default)
-        tol_obj = 1e-12 (Default)
-        tol_rel_obj = 10000 (Default)
-        tol_grad = 1e-08 (Default)
-        tol_rel_grad = 10000000 (Default)
-        tol_param = 1e-08 (Default)
-        history_size = 5 (Default)
-    iter = 2000 (Default)
-    save_iterations = 0 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 87122538 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>The second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The <code>alpha</code> value is for step size used. In the final state the change in parameters was roughly <span class="math inline">\(0.002\)</span> and the length of the gradient roughly 3e-05 (<span class="math inline">\(0.00003\)</span>).</p>
-<pre><code>Initial log joint probability = -6.85653
-    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes 
-       5      -5.00402    0.00184936   3.35074e-05           1           1        8   
-Optimization terminated normally: 
-  Convergence detected: relative gradient magnitude is below tolerance</code></pre>
-<p>The output from optimization is written into the file <code>output.csv</code> by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = optimize
-#   optimize
-#     algorithm = lbfgs (Default)
-...</code></pre>
-<p>Following the config information, are two lines of output: the CSV headers and the recorded values:</p>
-<pre><code>lp__,theta
--5.00402,0.200003</code></pre>
-<p>Note that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with <code>lp__</code> and the model parameter <code>theta</code>. The maximum log probability is -5.0 and the posterior mode for <code>theta</code> is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.</p>
-<p>All of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument <code>save_iterations</code> specifies whether or not to save the intermediate iterations to the output file. Allowed values are <span class="math inline">\(0\)</span> or <span class="math inline">\(1\)</span>, corresponding to <code>False</code> and <code>True</code> respectively. The default value is <span class="math inline">\(0\)</span>, i.e., intermediate iterations are not saved to the output file. Running the optimizer with <code>save_iterations=1</code> writes both the initial log joint probability and values for all iterations to the output CSV file.</p>
-<p>Running the example model with option <code>save_iterations=1</code>, i.e., the command</p>
-<pre><code>&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json</code></pre>
-<p>produces CSV file output rows:</p>
-<pre><code>lp__,theta
--6.85653,0.493689
--6.10128,0.420936
--5.02953,0.22956
--5.00517,0.206107
--5.00403,0.200299
--5.00402,0.200003</code></pre>
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diff --git a/docs/cmdstan-guide/optimize_config.html b/docs/cmdstan-guide/optimize_config.html
index 1f5b56d7b..2a3ab13e5 100644
--- a/docs/cmdstan-guide/optimize_config.html
+++ b/docs/cmdstan-guide/optimize_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>Optimization Configuration</title>
+<title>Optimization</title>
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 span.smallcaps{font-variant: small-caps;}
@@ -38,10 +38,12 @@
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
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@@ -99,7 +101,7 @@
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -214,6 +216,7 @@
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+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/optimize_config.html">Optimization</a></li></ol></nav>
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@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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@@ -278,32 +281,8 @@
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- <span class="menu-text">MCMC Sampling</span></a>
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- <span class="menu-text">Optimization</span></a>
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- <span class="menu-text">Variational Inference using Pathfinder</span></a>
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- <span class="menu-text">Variational Inference using ADVI</span></a>
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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@@ -311,7 +290,7 @@
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+ <span class="menu-text">Optimization</span></a>
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@@ -344,13 +323,13 @@
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -370,12 +349,6 @@
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@@ -459,7 +432,7 @@
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   <ul>
-  <li><a href="#maximum-likelihood-estimation" id="toc-maximum-likelihood-estimation" class="nav-link active" data-scroll-target="#maximum-likelihood-estimation">Maximum Likelihood Estimation</a>
+  <li><a href="#optimization" id="toc-optimization" class="nav-link active" data-scroll-target="#optimization">Optimization</a>
   <ul class="collapse">
   <li><a href="#jacobian-adjustments" id="toc-jacobian-adjustments" class="nav-link" data-scroll-target="#jacobian-adjustments">Jacobian adjustments</a></li>
   <li><a href="#optimization-algorithms" id="toc-optimization-algorithms" class="nav-link" data-scroll-target="#optimization-algorithms">Optimization algorithms</a></li>
@@ -475,11 +448,49 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="maximum-likelihood-estimation" class="level1">
-<h1>Maximum Likelihood Estimation</h1>
-<p>The <code>optimize</code> method finds the mode of the posterior distribution, assuming that there is one. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability. For optimization, the mode is calculated without the Jacobian adjustment for constrained variables, which shifts the mode due to the change of variables. Thus modes correspond to modes of the model as written.</p>
-<p>The full set of configuration options available for the <code>optimize</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=optimize</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
-<pre><code># model = bernoulli_model
+<section id="optimization" class="level1">
+<h1>Optimization</h1>
+<p>The CmdStan executable can run Stan’s optimization algorithms, which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global optimum as opposed to a local optimum of log probability.</p>
+<p>The full set of configuration options available for the <code>optimize</code> method is available by using the <code>optimize help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the optimizer console output and in the output CSV file’s comments.</p>
+<p>The executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli optimize data file=bernoulli.data.json</code></pre>
+<p>Executing this command prints both output to the console and to a CSV file.</p>
+<p>The first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:</p>
+<pre><code> ./bernoulli optimize data file=bernoulli.data.json
+method = optimize
+  optimize
+    algorithm = lbfgs (Default)
+      lbfgs
+        init_alpha = 0.001 (Default)
+        tol_obj = 1e-12 (Default)
+        tol_rel_obj = 10000 (Default)
+        tol_grad = 1e-08 (Default)
+        tol_rel_grad = 10000000 (Default)
+        tol_param = 1e-08 (Default)
+        history_size = 5 (Default)
+    iter = 2000 (Default)
+    save_iterations = 0 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 87122538 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>The second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The <code>alpha</code> value is for step size used. In the final state the change in parameters was roughly <span class="math inline">\(0.002\)</span> and the length of the gradient roughly 3e-05 (<span class="math inline">\(0.00003\)</span>).</p>
+<pre><code>Initial log joint probability = -6.85653
+    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes
+       5      -5.00402    0.00184936   3.35074e-05           1           1        8
+Optimization terminated normally:
+  Convergence detected: relative gradient magnitude is below tolerance</code></pre>
+<p>The output from optimization is written into the file <code>output.csv</code> by default. The output follows the <a href="../cmdstan-guide/stan_csv_apdx.html">same pattern</a> as the output for sampling, first dumping the entire set of parameters used as comment lines:</p>
+<pre><code># stan_version_major = 2
+# stan_version_minor = 23
+# stan_version_patch = 0
+# model = bernoulli_model
 # method = optimize
 #   optimize
 #     algorithm = lbfgs (Default)
@@ -493,10 +504,29 @@ <h1>Maximum Likelihood Estimation</h1>
 #         history_size = 5 (Default)
 #     jacobian = 0 (Default)
 #     iter = 2000 (Default)
-#     save_iterations = 0 (Default)</code></pre>
+#     save_iterations = 0 (Default)
+...</code></pre>
+<p>Following the config information are two lines of output, the CSV headers and the recorded values:</p>
+<pre><code>lp__,theta
+-5.00402,0.200003</code></pre>
+<p>Note that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with <code>lp__</code> and the model parameter <code>theta</code>. The maximum log probability is -5.0 and the posterior mode for <code>theta</code> is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.</p>
+<p>All of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument <code>save_iterations</code> specifies whether or not to save the intermediate iterations to the output file. Allowed values are <span class="math inline">\(0\)</span> or <span class="math inline">\(1\)</span>, corresponding to <code>False</code> and <code>True</code> respectively. The default value is <span class="math inline">\(0\)</span>, i.e., intermediate iterations are not saved to the output file. Running the optimizer with <code>save_iterations=1</code> writes both the initial log joint probability and values for all iterations to the output CSV file.</p>
+<p>Running the example model with option <code>save_iterations=1</code>, i.e., the command</p>
+<pre><code>&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json</code></pre>
+<p>produces CSV file output rows:</p>
+<pre><code>lp__,theta
+-6.85653,0.493689
+-6.10128,0.420936
+-5.02953,0.22956
+-5.00517,0.206107
+-5.00403,0.200299
+-5.00402,0.200003</code></pre>
 <section id="jacobian-adjustments" class="level2">
 <h2 class="anchored" data-anchor-id="jacobian-adjustments">Jacobian adjustments</h2>
-<p>The <code>jacobian</code> argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms. Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\)</span>, the value which maximizes the likelihood of the data given the parameters, (including prior terms). Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\,p(\theta)\)</span>. By default this value is <code>0</code> (false), do not include the Jacobian adjustment.</p>
+<p>The <code>jacobian</code> argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of <a href="https://mc-stan.org/docs/reference-manual/transforms.html">inverse parameter transforms</a>.</p>
+<p>Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\)</span>, the value which maximizes the likelihood of the data given the parameters, (including prior terms).</p>
+<p>Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, <span class="math inline">\(\mathrm{argmax}_{\theta}\ p(y | \theta)\,p(\theta)\)</span>.</p>
+<p>By default this value is <code>0</code> (false), do not include the Jacobian adjustment.</p>
 </section>
 <section id="optimization-algorithms" class="level2">
 <h2 class="anchored" data-anchor-id="optimization-algorithms">Optimization algorithms</h2>
@@ -534,25 +564,6 @@ <h2 class="anchored" data-anchor-id="the-newton-optimizer">The Newton optimizer<
 </section>
 
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diff --git a/docs/cmdstan-guide/parallelization.html b/docs/cmdstan-guide/parallelization.html
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -185,6 +187,7 @@
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+ <span class="menu-text">Getting Started</span></a>
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@@ -249,32 +252,8 @@
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+ <span class="menu-text">Parallelization</span></a>
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+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -341,12 +320,6 @@
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@@ -484,7 +457,7 @@ <h3 class="anchored" data-anchor-id="running">Running</h3>
 <p>Generally, this number should not exceed the number of available cores for best performance.</p>
 <p>Example:</p>
 <pre><code>./model sample data file=data.json num_threads=4 ...</code></pre>
-<p>When the model is compiled with <code>STAN_THREADS</code> we can sample with multiple chains with a single executable (see section <span id="multi-chain-sampling">running multiple chains</span> for cases when this is available). When running multiple chains <code>num_threads</code> is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:</p>
+<p>When the model is compiled with <code>STAN_THREADS</code> we can sample with multiple chains with a single executable (see section <a href="../cmdstan-guide/mcmc_config.html#multi-chain-sampling">running multiple chains</a> for cases when this is available). When running multiple chains <code>num_threads</code> is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:</p>
 <pre><code>./model sample num_chains=2 data file=data.json num_threads=8 ...</code></pre>
 </section>
 </section>
@@ -576,25 +549,6 @@ <h3 class="anchored" data-anchor-id="running-2">Running</h3>
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+  // Dark / light mode switch
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@@ -984,13 +1082,13 @@ <h3 class="anchored" data-anchor-id="running-2">Running</h3>
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+      <a href="../cmdstan-guide/compiling_stan_programs.html" class="pagination-link" aria-label="Compiling a Stan Program">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Compiling a Stan Program</span>
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+      <a href="../cmdstan-guide/command_line_options.html" class="pagination-link" aria-label="Command-Line Interface Overview">
+        <span class="nav-page-text">Command-Line Interface Overview</span> <i class="bi bi-arrow-right-short"></i>
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diff --git a/docs/cmdstan-guide/pathfinder-intro.html b/docs/cmdstan-guide/pathfinder-intro.html
index 0ef5dce08..4f3244d4a 100644
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+++ b/docs/cmdstan-guide/pathfinder-intro.html
@@ -2,10 +2,10 @@
 <html lang="en-US">
   <meta charset="utf-8">
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+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">
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+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">
   <meta name="robots" content="noindex">
   <h1>Redirecting&hellip;</h1>
-  <a href="https://mc-stan.org/docs/cmdstan-guide/pathfinder_intro.html">Click here if you are not redirected.</a>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">Click here if you are not redirected.</a>
 </html>
diff --git a/docs/cmdstan-guide/pathfinder_config.html b/docs/cmdstan-guide/pathfinder_config.html
index 7c6136f7f..5cf450850 100644
--- a/docs/cmdstan-guide/pathfinder_config.html
+++ b/docs/cmdstan-guide/pathfinder_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>Pathfinder Configuration</title>
+<title>Pathfinder Method for Approximate Bayesian Inference</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -20,7 +20,27 @@
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   vertical-align: middle;
 }
-</style>
+/* CSS for citations */
+div.csl-bib-body { }
+div.csl-entry {
+  clear: both;
+  margin-bottom: 0em;
+}
+.hanging-indent div.csl-entry {
+  margin-left:2em;
+  text-indent:-2em;
+}
+div.csl-left-margin {
+  min-width:2em;
+  float:left;
+}
+div.csl-right-inline {
+  margin-left:2em;
+  padding-left:1em;
+}
+div.csl-indent {
+  margin-left: 2em;
+}</style>
 
 
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@@ -38,10 +58,12 @@
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@@ -99,7 +121,7 @@
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -214,6 +236,7 @@
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+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -227,7 +250,7 @@
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+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/pathfinder_config.html">Pathfinder Method for Approximate Bayesian Inference</a></li></ol></nav>
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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@@ -278,32 +301,8 @@
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-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
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- <span class="menu-text">Variational Inference using Pathfinder</span></a>
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
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@@ -311,7 +310,7 @@
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             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
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+ <span class="menu-text">Running CmdStan</span></a>
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@@ -332,7 +331,7 @@
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- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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@@ -344,13 +343,13 @@
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -370,12 +369,6 @@
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@@ -477,9 +470,14 @@ <h2 id="toc-title">On this page</h2>
 
 <section id="pathfinder-config" class="level1">
 <h1>Pathfinder Method for Approximate Bayesian Inference</h1>
-<p>The Pathfinder algorithm is described in section <a href="../cmdstan-guide/pathfinder_intro.html">Pathfinder overview</a>.</p>
+<p>The CmdStan method <code>pathfinder</code> uses the Pathfinder algorithm of <span class="citation" data-cites="zhang_pathfinder:2022">Zhang et al. (<a href="#ref-zhang_pathfinder:2022" role="doc-biblioref">2022</a>)</span>, which is further described in the <a href="https://mc-stan.org/docs/reference-manual/pathfinder.html">Stan Reference Manual</a>.</p>
+<p>A single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes.</p>
 <p>The <code>pathfinder</code> method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument <code>num_paths</code> specifies the number of single-path Pathfinders, the default is <span class="math inline">\(4\)</span>. If <code>num_paths</code> is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.</p>
-<p>The full set of configuration options available for the <code>pathfinder</code> method is reported at the beginning of the pathfinder output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=pathfinder</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
+<p>The full set of configuration options available for the <code>pathfinder</code> method is available by using the <code>pathfinder help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.</p>
+<p>The following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli pathfinder data file=bernoulli.data.R</code></pre>
+<p>Executing this command prints both output to the console and csv files.</p>
+<p>The first part of the console output reports on the configuration used.</p>
 <pre><code>method = pathfinder
   pathfinder
     init_alpha = 0.001 (Default)
@@ -496,7 +494,39 @@ <h1>Pathfinder Method for Approximate Bayesian Inference</h1>
     save_single_paths = 0 (Default)
     max_lbfgs_iters = 1000 (Default)
     num_draws = 1000 (Default)
-    num_elbo_draws = 25 (Default)</code></pre>
+    num_elbo_draws = 25 (Default)
+id = 1 (Default)
+data
+  file = examples/bernoulli/bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 1995513073 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)
+  sig_figs = -1 (Default)
+  profile_file = profile.csv (Default)
+num_threads = 1 (Default)</code></pre>
+<p>The rest of the output describes the progression of the algorithm.</p>
+<p>By default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, then uses importance resampling on the set of returned draws to produce the specified number of draws.</p>
+<pre><code>Path [1] :Initial log joint density = -11.543343
+Path [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00
+Path [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)
+Path [2] :Initial log joint density = -7.443345
+Path [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00
+Path [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)
+Path [3] :Initial log joint density = -18.986308
+Path [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00
+Path [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)
+Path [4] :Initial log joint density = -8.304453
+Path [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
+              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00
+Path [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)
+Total log probability function evaluations:8404</code></pre>
 <section id="pathfinder-configuration" class="level2">
 <h2 class="anchored" data-anchor-id="pathfinder-configuration">Pathfinder Configuration</h2>
 <ul>
@@ -512,18 +542,43 @@ <h2 class="anchored" data-anchor-id="pathfinder-configuration">Pathfinder Config
 </section>
 <section id="l-bfgs-configuration" class="level2">
 <h2 class="anchored" data-anchor-id="l-bfgs-configuration">L-BFGS Configuration</h2>
-<p>Arguments <code>init_alpha</code> through <code>history_size</code> are the full set of arguments to the L-BFGS optimizer and have the same defaults for <a href="../cmdstan-guide/optimize_config.html">optimization</a>.</p>
+<p>Arguments <code>init_alpha</code> through <code>history_size</code> are the full set of arguments to the L-BFGS optimizer and have the same defaults for <a href="../cmdstan-guide/optimize_config.html#the-quasi-newton-optimizers">optimization</a>.</p>
 </section>
 <section id="pathfinder_csv" class="level2">
 <h2 class="anchored" data-anchor-id="pathfinder_csv">Multi-path Pathfinder CSV files</h2>
 <p>By default, the <code>pathfinder</code> method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a <a href="../cmdstan-guide/stan_csv_apdx.html">StanCSV file</a>.</p>
 <p>The CSV files have the following structure:</p>
-<ul>
-<li><p>The full set of configuration options available for the <code>pathfinder</code> method is reported at the beginning of the sampler output file as CSV comments.</p></li>
-<li><p>The CSV header row consists of columns <code>lp_approx__</code>, <code>lp__</code>, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.</p></li>
-<li><p>The data rows contain the draws from the single- or multi-path run.</p></li>
-<li><p>Final comments containing timing information.</p></li>
-</ul>
+<p>The initial CSV comment rows contain the complete set of CmdStan configuration options.</p>
+<pre><code>...
+# method = pathfinder
+#   pathfinder
+#     init_alpha = 0.001 (Default)
+#     tol_obj = 9.9999999999999998e-13 (Default)
+#     tol_rel_obj = 10000 (Default)
+#     tol_grad = 1e-08 (Default)
+#     tol_rel_grad = 10000000 (Default)
+#     tol_param = 1e-08 (Default)
+#     history_size = 5 (Default)
+#     num_psis_draws = 1000 (Default)
+#     num_paths = 4 (Default)
+#     psis_resample = 1 (Default)
+#     calculate_lp = 1 (Default)
+#     save_single_paths = 0 (Default)
+#     max_lbfgs_iters = 1000 (Default)
+#     num_draws = 1000 (Default)
+#     num_elbo_draws = 25 (Default)
+...</code></pre>
+<p>Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs <code>lp_approx__</code>, the log density in the approximating distribution, and <code>lp__</code>, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.</p>
+<pre><code>lp_approx__,lp__,theta
+-2.4973, -8.2951, 0.0811852
+-0.87445, -7.06526, 0.160207
+-0.812285, -7.07124, 0.35819
+...</code></pre>
+<p>The final lines are comment lines which give timing information.</p>
+<pre><code># Elapsed Time: 0.016000 seconds (Pathfinders)
+#               0.003000 seconds (PSIS)
+#               0.019000 seconds (Total)</code></pre>
+<p>Pathfinder provides option <code>save_single_paths</code> which will save output from the single-path Pathfinder runs.</p>
 </section>
 <section id="single-path-pathfinder-outputs." class="level2">
 <h2 class="anchored" data-anchor-id="single-path-pathfinder-outputs.">Single-path Pathfinder Outputs.</h2>
@@ -591,29 +646,15 @@ <h2 class="anchored" data-anchor-id="single-path-pathfinder-outputs.">Single-pat
 <p>Option <code>num_paths=1</code> runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments <code>num_paths=1 save_single_paths=1</code> creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.</p>
 
 
+
 </section>
 </section>
 
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+<a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a><div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography" id="quarto-bibliography"><h2 class="anchored quarto-appendix-heading">References</h2><div id="refs" class="references csl-bib-body hanging-indent" data-entry-spacing="0" role="list">
+<div id="ref-zhang_pathfinder:2022" class="csl-entry" role="listitem">
+Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
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-<section id="pathfinder-intro" class="level1">
-<h1>Variational Inference using Pathfinder</h1>
-<p>The CmdStan method <code>pathfinder</code> uses the Pathfinder algorithm of <span class="citation" data-cites="zhang_pathfinder:2022">Zhang et al. (<a href="#ref-zhang_pathfinder:2022" role="doc-biblioref">2022</a>)</span>. Pathfinder is a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the negative inverse Hessian estimates produced by the L-BFGS optimizer. Pathfinder returns draws from the Gaussian approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior.</p>
-<p>Pathfinder differs from the ADVI method in that it uses quasi-Newton optimization on the log posterior instead of stochastic gradient descent (SGD) on the Monte Carlo computation of the evidence lower bound (ELBO). Pathfinder’s approach is both faster and more stable than that of ADVI. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors.</p>
-<p>A single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes. By default, the <code>pathfinder</code> method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws.</p>
-<p>The following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli pathfinder data file=bernoulli.data.R</code></pre>
-<p>Executing this command prints both output to the console and csv files.</p>
-<p>The first part of the console output reports on the configuration used.</p>
-<pre><code>method = pathfinder
-  pathfinder
-    init_alpha = 0.001 (Default)
-    tol_obj = 9.9999999999999998e-13 (Default)
-    tol_rel_obj = 10000 (Default)
-    tol_grad = 1e-08 (Default)
-    tol_rel_grad = 10000000 (Default)
-    tol_param = 1e-08 (Default)
-    history_size = 5 (Default)
-    num_psis_draws = 1000 (Default)
-    num_paths = 4 (Default)
-    psis_resample = 1 (Default)
-    calculate_lp = 1 (Default)
-    save_single_paths = 0 (Default)
-    max_lbfgs_iters = 1000 (Default)
-    num_draws = 1000 (Default)
-    num_elbo_draws = 25 (Default)
-id = 1 (Default)
-data
-  file = examples/bernoulli/bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 1995513073 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)
-  sig_figs = -1 (Default)
-  profile_file = profile.csv (Default)
-num_threads = 1 (Default)</code></pre>
-<p>The rest of the output describes the progression of the algorithm.</p>
-<p>By default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, the uses importance resampling on the set of returned draws to produce the specified number of draws.</p>
-<pre><code>Path [1] :Initial log joint density = -11.543343
-Path [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00
-Path [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)
-Path [2] :Initial log joint density = -7.443345
-Path [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00
-Path [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)
-Path [3] :Initial log joint density = -18.986308
-Path [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00
-Path [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)
-Path [4] :Initial log joint density = -8.304453
-Path [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes
-              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00
-Path [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)
-Total log probability function evaluations:8404</code></pre>
-<p>Pathfinder outputs a <a href="../cmdstan-guide/pathfinder_config.html#pathfinder_csv">StanCSV file</a> file which contains the importance resampled draws from multi-path Pathfinder. The initial CSV comment rows contain the complete set of CmdStan configuration options. Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs <code>lp_approx__</code>, the log density in the approximating distribution, and <code>lp__</code>, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.</p>
-<pre><code>lp_approx__,lp__,theta
--2.4973, -8.2951, 0.0811852
--0.87445, -7.06526, 0.160207
--0.812285, -7.07124, 0.35819
-...</code></pre>
-<p>The final lines are comment lines which give timing information.</p>
-<pre><code># Elapsed Time: 0.016000 seconds (Pathfinders)
-#               0.003000 seconds (PSIS)
-#               0.019000 seconds (Total)</code></pre>
-<p>Pathfinder provides option <code>save_single_paths</code> which will save output from the single-path Pathfinder runs. See section <a href="../cmdstan-guide/pathfinder_config.html">Pathfinder Method</a> for details.</p>
-
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-<div id="ref-zhang_pathfinder:2022" class="csl-entry" role="listitem">
-Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
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-  <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/optimization_intro.html" class="pagination-link" aria-label="Optimization">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Optimization</span>
-      </a>          
-  </div>
-  <div class="nav-page nav-page-next">
-      <a href="../cmdstan-guide/variational_intro.html" class="pagination-link" aria-label="Variational Inference using ADVI">
-        <span class="nav-page-text">Variational Inference using ADVI</span> <i class="bi bi-arrow-right-short"></i>
-      </a>
-  </div>
-</nav>
-</div> <!-- /content -->
-
-
-
-
-<footer class="footer"><div class="nav-footer"><div class="nav-footer-center"><div class="toc-actions d-sm-block d-md-none"><ul><li><a href="https://github.com/stan-dev/docs/edit/master/src/cmdstan-guide/pathfinder_intro.qmd" class="toc-action"><i class="bi bi-github"></i>Edit this page</a></li><li><a href="https://github.com/stan-dev/docs/issues/new" class="toc-action"><i class="bi empty"></i>Report an issue</a></li></ul></div></div></div></footer></body></html>
\ No newline at end of file
+<html lang="en-US">
+  <meta charset="utf-8">
+  <title>Redirecting&hellip;</title>
+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">
+  <script>location="https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html"</script>
+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">
+  <meta name="robots" content="noindex">
+  <h1>Redirecting&hellip;</h1>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/pathfinder_config.html">Click here if you are not redirected.</a>
+</html>
diff --git a/docs/cmdstan-guide/print.html b/docs/cmdstan-guide/print.html
index 59a934c48..f8b75a86c 100644
--- a/docs/cmdstan-guide/print.html
+++ b/docs/cmdstan-guide/print.html
@@ -38,10 +38,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -223,7 +226,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -282,7 +261,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -303,7 +282,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -315,13 +294,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -448,25 +421,6 @@ <h1><code>print</code> (deprecated): MCMC Output Analysis</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-
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@@ -487,6 +441,150 @@ <h1><code>print</code> (deprecated): MCMC Output Analysis</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+        }
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+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
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+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
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+    } else {
+      return false;
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+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  // Switch to dark mode if need be
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diff --git a/docs/cmdstan-guide/rdump_apdx.html b/docs/cmdstan-guide/rdump_apdx.html
index e2bdbddc4..0504edc45 100644
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+++ b/docs/cmdstan-guide/rdump_apdx.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -99,7 +101,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
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     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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   <i class="bi"></i>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -704,25 +677,6 @@ <h2 class="anchored" data-anchor-id="bnf-grammar-for-dump-data">BNF grammar for
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
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-
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 <script id="quarto-html-after-body" type="application/javascript">
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   const toggleBodyColorMode = (bsSheetEl) => {
@@ -743,6 +697,150 @@ <h2 class="anchored" data-anchor-id="bnf-grammar-for-dump-data">BNF grammar for
     }
   }
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+      changeGiscusTheme();
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+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
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diff --git a/docs/cmdstan-guide/stan_csv_apdx.html b/docs/cmdstan-guide/stan_csv_apdx.html
index 94d260a78..15a31cda0 100644
--- a/docs/cmdstan-guide/stan_csv_apdx.html
+++ b/docs/cmdstan-guide/stan_csv_apdx.html
@@ -38,10 +38,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -570,7 +543,7 @@ <h3 class="anchored" data-anchor-id="sampler-stan-csv-output-file">Sampler Stan
 #   refresh = 100 (Default)</code></pre>
 <p>Note that when running multi-threaded programs which use <code>reduce_sum</code> for <a href="../cmdstan-guide/parallelization.html">high-level parallelization</a>, the number of threads used will also be included in this initial comment header.</p>
 <p><strong>Column headers</strong></p>
-<p>The CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the <a href="../cmdstan-guide/mcmc_sampling_intro.html#mcmc_output_csv">output file</a> section of the Quickstart Guide chapter on MCMC Sampling. The example model <code>bernoulli.stan</code> only contains one parameter <code>theta</code>, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:</p>
+<p>The CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the <a href="../cmdstan-guide/mcmc_config.html#mcmc_output_csv">output file</a> section of the chapter on MCMC Sampling. The example model <code>bernoulli.stan</code> only contains one parameter <code>theta</code>, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:</p>
 <pre><code>lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta</code></pre>
 <p>As a second example, we show the output of the <code>eight_schools.stan</code> model on run on example dataset. This model has 3 parameters: <code>mu</code>, <code>theta</code> a vector whose length is dependent on the input data, here <code>N = 8</code>, and <code>tau</code>. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:</p>
 <pre><code>mu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau</code></pre>
@@ -693,25 +666,6 @@ <h2 class="anchored" data-anchor-id="diagnose-method-outputs">Diagnose method ou
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -732,6 +686,150 @@ <h2 class="anchored" data-anchor-id="diagnose-method-outputs">Diagnose method ou
     }
   }
   toggleBodyColorPrimary();  
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+      stylesheet.rel = 'stylesheet';
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+  }
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+    const els = window.document.querySelectorAll(selector);
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+        } else {
+          toggle.classList.remove("alternate");
+        }
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
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+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
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+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/cmdstan-guide/stanc.html b/docs/cmdstan-guide/stanc.html
index c40d78df8..c59f92f4f 100644
--- a/docs/cmdstan-guide/stanc.html
+++ b/docs/cmdstan-guide/stanc.html
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 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
 <link href="../cmdstan-guide/stansummary.html" rel="next">
-<link href="../cmdstan-guide/parallelization.html" rel="prev">
+<link href="../cmdstan-guide/diagnose_config.html" rel="prev">
 <link href="../img/logo_tm.png" rel="icon" type="image/png">
 <script src="../site_libs/quarto-html/quarto.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -70,7 +72,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -185,6 +187,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -249,32 +252,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
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- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
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-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
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 </li>
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- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -303,7 +282,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
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@@ -315,13 +294,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -341,12 +320,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
       </ul>
   </li>
@@ -478,25 +451,6 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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@@ -517,6 +471,150 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
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+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+    }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
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+    }
+    const changeGiscusTheme = () => {
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
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+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    // Hack to workaround the fact that safari doesn't
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+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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+      window.scrollTo(0, 1);
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+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
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+  const hasAlternateSentinel = () => {  
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+  const setStyleSentinel = (alternate) => {
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+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
@@ -886,8 +984,8 @@ <h2 class="anchored" data-anchor-id="the-stan-compiler-program">The Stan compile
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/parallelization.html" class="pagination-link" aria-label="Parallelization">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Parallelization</span>
+      <a href="../cmdstan-guide/diagnose_config.html" class="pagination-link" aria-label="Diagnosing HMC by Comparison of Gradients">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Diagnosing HMC by Comparison of Gradients</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/cmdstan-guide/stansummary.html b/docs/cmdstan-guide/stansummary.html
index 63a4aa3e6..339920e0a 100644
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+++ b/docs/cmdstan-guide/stansummary.html
@@ -38,10 +38,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -252,7 +255,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -278,32 +281,8 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/optimization_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Optimization</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/pathfinder_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using Pathfinder</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/variational_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference using ADVI</span></a>
-  </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
       </ul>
@@ -311,7 +290,7 @@
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
- <span class="menu-text">Reference Manual</span></a>
+ <span class="menu-text">Running CmdStan</span></a>
           <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true" aria-label="Toggle section">
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           </a> 
@@ -332,7 +311,7 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -344,13 +323,13 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/variational_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../cmdstan-guide/generate_quantities_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
   </div>
 </li>
           <li class="sidebar-item">
@@ -370,12 +349,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
   </div>
-</li>
-          <li class="sidebar-item">
-  <div class="sidebar-item-container"> 
-  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Parallelization</span></a>
-  </div>
 </li>
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@@ -599,25 +572,6 @@ <h2 class="anchored" data-anchor-id="command-line-options">Command-line options<
 </section>
 
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diff --git a/docs/cmdstan-guide/variational-inference-using-advi.html b/docs/cmdstan-guide/variational-inference-using-advi.html
index 98ded8b83..40b498424 100644
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+++ b/docs/cmdstan-guide/variational-inference-using-advi.html
@@ -2,10 +2,10 @@
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+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/variational_config.html">
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   <meta name="robots" content="noindex">
   <h1>Redirecting&hellip;</h1>
-  <a href="https://mc-stan.org/docs/cmdstan-guide/variational_intro.html">Click here if you are not redirected.</a>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/variational_config.html">Click here if you are not redirected.</a>
 </html>
diff --git a/docs/cmdstan-guide/variational_config.html b/docs/cmdstan-guide/variational_config.html
index 09f3a71b1..3ed610e23 100644
--- a/docs/cmdstan-guide/variational_config.html
+++ b/docs/cmdstan-guide/variational_config.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 
-<title>ADVI Configuration</title>
+<title>ADVI for Variational Inference</title>
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@@ -58,10 +58,12 @@
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@@ -119,7 +121,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -234,6 +236,7 @@
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     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -247,7 +250,7 @@
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-        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Reference Manual</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/variational_config.html">Variational Inference Algorithm: ADVI</a></li></ol></nav>
+        <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="../cmdstan-guide/command_line_options.html">Running CmdStan</a></li><li class="breadcrumb-item"><a href="../cmdstan-guide/variational_config.html">Variational Inference using ADVI</a></li></ol></nav>
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@@ -272,7 +275,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="true">
- <span class="menu-text">Quickstart Guide</span></a>
+ <span class="menu-text">Getting Started</span></a>
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@@ -298,32 +301,8 @@
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-  <a href="../cmdstan-guide/mcmc_sampling_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">MCMC Sampling</span></a>
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- <span class="menu-text">Variational Inference using Pathfinder</span></a>
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-  <a href="../cmdstan-guide/generate_quantities_intro.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
+  <a href="../cmdstan-guide/parallelization.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Parallelization</span></a>
   </div>
 </li>
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@@ -331,7 +310,7 @@
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-3" aria-expanded="true">
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+ <span class="menu-text">Running CmdStan</span></a>
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@@ -352,7 +331,7 @@
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   <a href="../cmdstan-guide/optimize_config.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Maximum Likelihood Estimation</span></a>
+ <span class="menu-text">Optimization</span></a>
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@@ -364,13 +343,13 @@
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   <div class="sidebar-item-container"> 
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- <span class="menu-text">Variational Inference Algorithm: ADVI</span></a>
+ <span class="menu-text">Variational Inference using ADVI</span></a>
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- <span class="menu-text">Standalone Generate Quantities</span></a>
+ <span class="menu-text">Generating Quantities of Interest from a Fitted Model</span></a>
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@@ -390,12 +369,6 @@
   <a href="../cmdstan-guide/diagnose_config.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Diagnosing HMC by Comparison of Gradients</span></a>
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-  </div>
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@@ -479,7 +452,7 @@
     <h2 id="toc-title">On this page</h2>
    
   <ul>
-  <li><a href="#variational-inference-algorithm-advi" id="toc-variational-inference-algorithm-advi" class="nav-link active" data-scroll-target="#variational-inference-algorithm-advi">Variational Inference Algorithm: ADVI</a>
+  <li><a href="#variational-inference-using-advi" id="toc-variational-inference-using-advi" class="nav-link active" data-scroll-target="#variational-inference-using-advi">Variational Inference using ADVI</a>
   <ul class="collapse">
   <li><a href="#variational-algorithms" id="toc-variational-algorithms" class="nav-link" data-scroll-target="#variational-algorithms">Variational algorithms</a></li>
   <li><a href="#configuration" id="toc-configuration" class="nav-link" data-scroll-target="#configuration">Configuration</a></li>
@@ -494,33 +467,81 @@ <h2 id="toc-title">On this page</h2>
 
 
 
-<section id="variational-inference-algorithm-advi" class="level1">
-<h1>Variational Inference Algorithm: ADVI</h1>
-<p>CmdStan can approximate the posterior distribution using variational inference. The approximation is a Gaussian in the unconstrained variable space.</p>
+<section id="variational-inference-using-advi" class="level1">
+<h1>Variational Inference using ADVI</h1>
 <p>Stan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) <span class="citation" data-cites="kucukelbir:2017">Kucukelbir et al. (<a href="#ref-kucukelbir:2017" role="doc-biblioref">2017</a>)</span>. ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using <a href="https://mc-stan.org/docs/reference-manual/variational.html#stochastic-gradient-ascent">stochastic gradient ascent</a>. The measures of convergence are similar to the relative tolerance scheme of Stan’s <a href="https://mc-stan.org/docs/reference-manual/optimization.html">optimization algorithms</a>.</p>
 <p>The algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.</p>
-<p>The full set of configuration options available for the <code>variational</code> method is reported at the beginning of the sampler output file as CSV comments. When the example model <code>bernoulli.stan</code> is run with <code>method=variational</code> via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:</p>
-<pre><code># method = variational
-#   variational
-#     algorithm = meanfield (Default)
-#       meanfield
-#     iter = 10000 (Default)
-#     grad_samples = 1 (Default)
-#     elbo_samples = 100 (Default)
-#     eta = 1 (Default)
-#     adapt
-#       engaged = 1 (Default)
-#       iter = 50 (Default)
-#     tol_rel_obj = 0.01 (Default)
-#     eval_elbo = 100 (Default)
-#     output_samples = 1000 (Default)</code></pre>
-<p>The console output includes a notice that this algorithm is considered to be experimental:</p>
-<pre><code>EXPERIMENTAL ALGORITHM:
+<p>The full set of configuration options available for the <code>variational</code> method is available by using the <code>variational help-all</code> subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.</p>
+<p>The following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.</p>
+<pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
+<p>Executing this command prints both output to the console and to a csv file.</p>
+<p>The first part of the console output reports on the configuration used: the default option <code>algorithm=meanfield</code> and the default tolerances for monitoring the algorithm’s convergence.</p>
+<pre><code>method = variational
+  variational
+    algorithm = meanfield (Default)
+      meanfield
+    iter = 10000 (Default)
+    grad_samples = 1 (Default)
+    elbo_samples = 100 (Default)
+    eta = 1 (Default)
+    adapt
+      engaged = 1 (Default)
+      iter = 50 (Default)
+    tol_rel_obj = 0.01 (Default)
+    eval_elbo = 100 (Default)
+    output_samples = 1000 (Default)
+id = 0 (Default)
+data
+  file = bernoulli.data.json
+init = 2 (Default)
+random
+  seed = 3323783840 (Default)
+output
+  file = output.csv (Default)
+  diagnostic_file =  (Default)
+  refresh = 100 (Default)</code></pre>
+<p>After the configuration has been displayed, informational and timing messages are output:</p>
+<pre><code>------------------------------------------------------------
+EXPERIMENTAL ALGORITHM:
   This procedure has not been thoroughly tested and may be unstable
-  or buggy. The interface is subject to change.</code></pre>
+  or buggy. The interface is subject to change.
+------------------------------------------------------------
+
+Gradient evaluation took 2.1e-05 seconds
+1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
+Adjust your expectations accordingly!</code></pre>
+<p>The rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in <span class="math inline">\(\sim\)</span> 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.</p>
+<pre><code>Begin eta adaptation.
+Iteration:   1 / 250 [  0%]  (Adaptation)
+Iteration:  50 / 250 [ 20%]  (Adaptation)
+Iteration: 100 / 250 [ 40%]  (Adaptation)
+Iteration: 150 / 250 [ 60%]  (Adaptation)
+Iteration: 200 / 250 [ 80%]  (Adaptation)
+Success! Found best value [eta = 1] earlier than expected.
+
+Begin stochastic gradient ascent.
+  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes
+   100           -6.131             1.000            1.000
+   200           -6.458             0.525            1.000
+   300           -6.300             0.359            0.051
+   400           -6.137             0.276            0.051
+   500           -6.243             0.224            0.027
+   600           -6.305             0.188            0.027
+   700           -6.289             0.162            0.025
+   800           -6.402             0.144            0.025
+   900           -6.103             0.133            0.025
+  1000           -6.314             0.123            0.027
+  1100           -6.348             0.024            0.025
+  1200           -6.244             0.020            0.018
+  1300           -6.293             0.019            0.017
+  1400           -6.250             0.017            0.017
+  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED
+
+Drawing a sample of size 1000 from the approximate posterior...
+COMPLETED.</code></pre>
 <section id="variational-algorithms" class="level2">
 <h2 class="anchored" data-anchor-id="variational-algorithms">Variational algorithms</h2>
-<p>Stan implements two variational algorithms. The <code>algorithm</code> argument specifies the variational algorithm.</p>
+<p>Stan implements two variational algorithms. They differ in the approximating distribution used in the unconstrained variable space. By default, ADVI uses option <code>algorithm=meanfield</code>. The <code>algorithm</code> argument specifies the variational algorithm.</p>
 <ul>
 <li><p><code>algorithm=meanfield</code> - Use a fully factorized Gaussian for the approximation. This is the default algorithm.</p></li>
 <li><p><code>algorithm=fullrank</code> Use a Gaussian with a full-rank covariance matrix for the approximation.</p></li>
@@ -555,7 +576,8 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 </ul>
 <p>To illustrate, we call Stan’s variational inference on the example model and data:</p>
 <pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
-<p>By default, the output file is <code>output.csv</code>. Lines 1 - 28 contain configuration information:</p>
+<p>By default, the output file is <code>output.csv</code>.</p>
+<p>The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:</p>
 <pre><code># stan_version_major = 2
 # stan_version_minor = 23
 # stan_version_patch = 0
@@ -575,19 +597,21 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 #     eval_elbo = 100 (Default)
 #     output_samples = 1000 (Default)
 ...</code></pre>
-<p>The column header row is:</p>
+<p>Next, the column header row:</p>
 <pre><code>lp__,log_p__,log_g__,theta</code></pre>
-<p>The stepsize adaptation information is:</p>
+<p>Additional comments provide stepsize adaptation information:</p>
 <pre><code># Stepsize adaptation complete.
 # eta = 1</code></pre>
-<p>The reported mean variational approximations information is:</p>
+<p>Followed by the data rows. The first line is special — it is the mean of the variational approximation.</p>
 <pre><code>0,0,0,0.214911</code></pre>
 <p>That is, the estimate for <code>theta</code> given the data is <code>0.2</code>.</p>
+<p>The rest of the output contains <code>output_samples</code> number of samples drawn from the variational approximation.</p>
 <p>The following is a sample based on this approximation:</p>
 <pre><code>0,-14.0252,-5.21718,0.770397
 0,-7.05063,-0.10025,0.162061
 0,-6.75031,-0.0191099,0.241606
 ...</code></pre>
+<p>The header indicates the unnormalized log probability with <code>lp__</code>. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.</p>
 
 
 
@@ -599,25 +623,6 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
 Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
 </div>
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@@ -1012,8 +1161,8 @@ <h2 class="anchored" data-anchor-id="csv-output">CSV output</h2>
       </a>          
   </div>
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-      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Standalone Generate Quantities">
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+      <a href="../cmdstan-guide/generate_quantities_config.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
+        <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/cmdstan-guide/variational_intro.html b/docs/cmdstan-guide/variational_intro.html
index 56b433951..40b498424 100644
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-<section id="variational-inference-using-advi" class="level1">
-<h1>Variational Inference using ADVI</h1>
-<p>The CmdStan method <code>variational</code> uses the Automatic Differentiation Variational Inference (ADVI) algorithm of <span class="citation" data-cites="kucukelbir:2017">Kucukelbir et al. (<a href="#ref-kucukelbir:2017" role="doc-biblioref">2017</a>)</span> to provide an approximate posterior distribution of the model conditioned on the data. The approximating distribution it uses is a Gaussian in the unconstrained variable space, either a fully factorized Gaussian approximation, specified by argument <code>algorithm=meanfield</code> option, or a Gaussian approximation using a full-rank covariance matrix, specified by argument <code>algorithm=fullrank</code>. By default, ADVI uses option <code>algorithm=meanfield</code>.</p>
-<p>The following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.</p>
-<pre><code>&gt; ./bernoulli variational data file=bernoulli.data.R</code></pre>
-<p>Executing this command prints both output to the console and to a csv file.</p>
-<p>The first part of the console output reports on the configuration used: the default option <code>algorithm=meanfield</code> and the default tolerances for monitoring the algorithm’s convergence.</p>
-<pre><code>method = variational
-  variational
-    algorithm = meanfield (Default)
-      meanfield
-    iter = 10000 (Default)
-    grad_samples = 1 (Default)
-    elbo_samples = 100 (Default)
-    eta = 1 (Default)
-    adapt
-      engaged = 1 (Default)
-      iter = 50 (Default)
-    tol_rel_obj = 0.01 (Default)
-    eval_elbo = 100 (Default)
-    output_samples = 1000 (Default)
-id = 0 (Default)
-data
-  file = bernoulli.data.json
-init = 2 (Default)
-random
-  seed = 3323783840 (Default)
-output
-  file = output.csv (Default)
-  diagnostic_file =  (Default)
-  refresh = 100 (Default)</code></pre>
-<p>After the configuration has been displayed, informational and timing messages are output:</p>
-<pre><code>------------------------------------------------------------
-EXPERIMENTAL ALGORITHM:
-  This procedure has not been thoroughly tested and may be unstable
-  or buggy. The interface is subject to change.
-------------------------------------------------------------
-
-Gradient evaluation took 2.1e-05 seconds
-1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
-Adjust your expectations accordingly!</code></pre>
-<p>The rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter <code>eta</code>. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in <span class="math inline">\(\sim\)</span> 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.</p>
-<pre><code>Begin eta adaptation.
-Iteration:   1 / 250 [  0%]  (Adaptation)
-Iteration:  50 / 250 [ 20%]  (Adaptation)
-Iteration: 100 / 250 [ 40%]  (Adaptation)
-Iteration: 150 / 250 [ 60%]  (Adaptation)
-Iteration: 200 / 250 [ 80%]  (Adaptation)
-Success! Found best value [eta = 1] earlier than expected.
-
-Begin stochastic gradient ascent.
-  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes 
-   100           -6.131             1.000            1.000
-   200           -6.458             0.525            1.000
-   300           -6.300             0.359            0.051
-   400           -6.137             0.276            0.051
-   500           -6.243             0.224            0.027
-   600           -6.305             0.188            0.027
-   700           -6.289             0.162            0.025
-   800           -6.402             0.144            0.025
-   900           -6.103             0.133            0.025
-  1000           -6.314             0.123            0.027
-  1100           -6.348             0.024            0.025
-  1200           -6.244             0.020            0.018
-  1300           -6.293             0.019            0.017
-  1400           -6.250             0.017            0.017
-  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED
-
-Drawing a sample of size 1000 from the approximate posterior... 
-COMPLETED.</code></pre>
-<p>The output from variational is written into the file <code>output.csv</code> by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:</p>
-<pre><code># stan_version_major = 2
-# stan_version_minor = 23
-# stan_version_patch = 0
-# model = bernoulli_model
-# method = variational
-#   variational
-#     algorithm = meanfield (Default)
-#       meanfield
-#     iter = 10000 (Default)
-#     grad_samples = 1 (Default)
-#     elbo_samples = 100 (Default)
-#     eta = 1 (Default)
-#     adapt
-#       engaged = 1 (Default)
-#       iter = 50 (Default)
-#     tol_rel_obj = 0.01 (Default)
-#     eval_elbo = 100 (Default)
-#     output_samples = 1000 (Default)
-...</code></pre>
-<p>Next is the column header line, followed more CSV comments reporting the adapted value for the stepsize, followed by the values. The first line is special: it is the mean of the variational approximation. The rest of the output contains <code>output_samples</code> number of samples drawn from the variational approximation.</p>
-<pre><code>lp__,log_p__,log_g__,theta
-# Stepsize adaptation complete.
-# eta = 1
-0,0,0,0.236261
-0,-6.82318,-0.0929121,0.300415
-0,-6.89701,-0.158687,0.321982
-0,-6.99391,-0.23916,0.343643
-0,-7.35801,-0.51787,0.401554
-0,-7.4668,-0.539473,0.123081
-...</code></pre>
-<p>The header indicates the unnormalized log probability with <code>lp__</code>. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.</p>
-
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-<div id="ref-kucukelbir:2017" class="csl-entry" role="listitem">
-Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
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-  <div class="nav-page nav-page-previous">
-      <a href="../cmdstan-guide/pathfinder_intro.html" class="pagination-link" aria-label="Variational Inference using Pathfinder">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Variational Inference using Pathfinder</span>
-      </a>          
-  </div>
-  <div class="nav-page nav-page-next">
-      <a href="../cmdstan-guide/generate_quantities_intro.html" class="pagination-link" aria-label="Generating Quantities of Interest from a Fitted Model">
-        <span class="nav-page-text">Generating Quantities of Interest from a Fitted Model</span> <i class="bi bi-arrow-right-short"></i>
-      </a>
-  </div>
-</nav>
-</div> <!-- /content -->
-
-
-
-
-<footer class="footer"><div class="nav-footer"><div class="nav-footer-center"><div class="toc-actions d-sm-block d-md-none"><ul><li><a href="https://github.com/stan-dev/docs/edit/master/src/cmdstan-guide/variational_intro.qmd" class="toc-action"><i class="bi bi-github"></i>Edit this page</a></li><li><a href="https://github.com/stan-dev/docs/issues/new" class="toc-action"><i class="bi empty"></i>Report an issue</a></li></ul></div></div></div></footer></body></html>
\ No newline at end of file
+<html lang="en-US">
+  <meta charset="utf-8">
+  <title>Redirecting&hellip;</title>
+  <link rel="canonical" href="https://mc-stan.org/docs/cmdstan-guide/variational_config.html">
+  <script>location="https://mc-stan.org/docs/cmdstan-guide/variational_config.html"</script>
+  <meta http-equiv="refresh" content="0; url=https://mc-stan.org/docs/cmdstan-guide/variational_config.html">
+  <meta name="robots" content="noindex">
+  <h1>Redirecting&hellip;</h1>
+  <a href="https://mc-stan.org/docs/cmdstan-guide/variational_config.html">Click here if you are not redirected.</a>
+</html>
diff --git a/docs/functions-reference/array_operations.html b/docs/functions-reference/array_operations.html
index 95ce8ae37..a822c158b 100644
--- a/docs/functions-reference/array_operations.html
+++ b/docs/functions-reference/array_operations.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -880,25 +883,6 @@ <h2 class="anchored" data-anchor-id="reversing-functions">Reversing functions</h
 <li id="fn1"><p>Dividing by <span class="math inline">\(N\)</span> rather than <span class="math inline">\((N-1)\)</span> produces a maximum likelihood estimate of variance, which is biased to underestimate variance.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
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@@ -919,6 +903,150 @@ <h2 class="anchored" data-anchor-id="reversing-functions">Reversing functions</h
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
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diff --git a/docs/functions-reference/binary_distributions.html b/docs/functions-reference/binary_distributions.html
index 11519b89c..be66aa289 100644
--- a/docs/functions-reference/binary_distributions.html
+++ b/docs/functions-reference/binary_distributions.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -703,25 +706,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan Functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -742,6 +726,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan Functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/bounded_continuous_distributions.html b/docs/functions-reference/bounded_continuous_distributions.html
index 499c6f71b..00a1498b0 100644
--- a/docs/functions-reference/bounded_continuous_distributions.html
+++ b/docs/functions-reference/bounded_continuous_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -569,25 +572,6 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -608,6 +592,150 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/bounded_discrete_distributions.html b/docs/functions-reference/bounded_discrete_distributions.html
index 4c2f1b515..d418d5988 100644
--- a/docs/functions-reference/bounded_discrete_distributions.html
+++ b/docs/functions-reference/bounded_discrete_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1140,25 +1143,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1179,6 +1163,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/circular_distributions.html b/docs/functions-reference/circular_distributions.html
index d97541370..58464c502 100644
--- a/docs/functions-reference/circular_distributions.html
+++ b/docs/functions-reference/circular_distributions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -620,25 +623,6 @@ <h3 class="anchored" data-anchor-id="numerical-stability">Numerical stability</h
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -659,6 +643,150 @@ <h3 class="anchored" data-anchor-id="numerical-stability">Numerical stability</h
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/complex-valued_basic_functions.html b/docs/functions-reference/complex-valued_basic_functions.html
index 5814d44f6..35ec0abe9 100644
--- a/docs/functions-reference/complex-valued_basic_functions.html
+++ b/docs/functions-reference/complex-valued_basic_functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -916,25 +919,6 @@ <h2 class="anchored" data-anchor-id="complex-h-trig">Complex hyperbolic trigonom
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -955,6 +939,150 @@ <h2 class="anchored" data-anchor-id="complex-h-trig">Complex hyperbolic trigonom
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/complex_matrix_operations.html b/docs/functions-reference/complex_matrix_operations.html
index 921b2106b..01e52f20e 100644
--- a/docs/functions-reference/complex_matrix_operations.html
+++ b/docs/functions-reference/complex_matrix_operations.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1479,25 +1482,6 @@ <h2 class="anchored" data-anchor-id="reverse-functions-for-complex-matrices">Rev
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1518,6 +1502,150 @@ <h2 class="anchored" data-anchor-id="reverse-functions-for-complex-matrices">Rev
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/compound_arithmetic_and_assignment.html b/docs/functions-reference/compound_arithmetic_and_assignment.html
index 804d67c8c..0076bc5b8 100644
--- a/docs/functions-reference/compound_arithmetic_and_assignment.html
+++ b/docs/functions-reference/compound_arithmetic_and_assignment.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -549,25 +552,6 @@ <h2 class="anchored" data-anchor-id="compound-elementwise-division-and-assignmen
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -588,6 +572,150 @@ <h2 class="anchored" data-anchor-id="compound-elementwise-division-and-assignmen
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/continuous_distributions_on_0_1.html b/docs/functions-reference/continuous_distributions_on_0_1.html
index bd879dd66..06a043783 100644
--- a/docs/functions-reference/continuous_distributions_on_0_1.html
+++ b/docs/functions-reference/continuous_distributions_on_0_1.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -620,25 +623,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -659,6 +643,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/conventions_for_probability_functions.html b/docs/functions-reference/conventions_for_probability_functions.html
index 8f4c64d26..11186c168 100644
--- a/docs/functions-reference/conventions_for_probability_functions.html
+++ b/docs/functions-reference/conventions_for_probability_functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -751,25 +754,6 @@ <h4 class="anchored" data-anchor-id="return-type">Return type</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -790,6 +774,150 @@ <h4 class="anchored" data-anchor-id="return-type">Return type</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/functions-reference/correlation_matrix_distributions.html b/docs/functions-reference/correlation_matrix_distributions.html
index 009a7845a..3f52b0530 100644
--- a/docs/functions-reference/correlation_matrix_distributions.html
+++ b/docs/functions-reference/correlation_matrix_distributions.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -680,25 +683,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
 <li id="fn1"><p>It is possible to build up a valid <code>L</code> within Stan, but that would then require Jacobian adjustments to imply the intended posterior.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -719,6 +703,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-1">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/functions-reference/covariance_matrix_distributions.html b/docs/functions-reference/covariance_matrix_distributions.html
index e6fe7f383..1b5cbd87e 100644
--- a/docs/functions-reference/covariance_matrix_distributions.html
+++ b/docs/functions-reference/covariance_matrix_distributions.html
@@ -38,10 +38,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -684,25 +687,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-3">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -723,6 +707,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-3">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/functions-reference/deprecated_functions.html b/docs/functions-reference/deprecated_functions.html
index a048564b6..ab815ffc5 100644
--- a/docs/functions-reference/deprecated_functions.html
+++ b/docs/functions-reference/deprecated_functions.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -562,7 +565,7 @@ <h2 id="toc-title">On this page</h2>
 <section id="deprecated-functions" class="level1">
 <h1>Deprecated Functions</h1>
 <p>This appendix lists currently deprecated functionality along with how to replace it.</p>
-<p>Starting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of <code>get_lp</code> or <code>multiply_log</code>) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Functions]. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> these on the behalf of the user for the entire deprecation window and at least one version following the removal.</p>
+<p>Starting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of <code>get_lp</code> or <code>multiply_log</code>) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in <a href="../functions-reference/removed_functions.html">Removed Functions</a>. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> these on the behalf of the user for the entire deprecation window and at least one version following the removal.</p>
 <section id="integer-division-with-operator" class="level2">
 <h2 class="anchored" data-anchor-id="integer-division-with-operator">Integer division with <code>operator/</code></h2>
 <p><em>Deprecated</em>: Using <code>/</code> with two integer arguments is interpreted as integer floor division, such that</p>
@@ -729,25 +732,6 @@ <h4 class="anchored" data-anchor-id="sizes-and-parallel-arrays-1">Sizes and para
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -768,6 +752,150 @@ <h4 class="anchored" data-anchor-id="sizes-and-parallel-arrays-1">Sizes and para
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
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+      const sendMessage = (message) => {
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      changeGiscusTheme();
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+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
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+    }
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
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+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
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+  // Switch to dark mode if need be
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+    toggleColorMode(true);
+  } else {
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diff --git a/docs/functions-reference/distributions_over_unbounded_vectors.html b/docs/functions-reference/distributions_over_unbounded_vectors.html
index f991c8da3..61c77adbf 100644
--- a/docs/functions-reference/distributions_over_unbounded_vectors.html
+++ b/docs/functions-reference/distributions_over_unbounded_vectors.html
@@ -58,10 +58,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -996,25 +999,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-7">Stan functions</h3>
 Durbin, J., and S. J. Koopman. 2001. <em>Time Series Analysis by State Space Methods</em>. New York: Oxford University Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
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-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1035,6 +1019,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-7">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      const sendMessage = (message) => {
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+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      for (const sheetNode of alternateStylesheets) {
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+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
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+  }
+  const isFileUrl = () => { 
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
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+    }
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+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
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+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/functions-reference/functions_index.html b/docs/functions-reference/functions_index.html
index ae9014a08..024648881 100644
--- a/docs/functions-reference/functions_index.html
+++ b/docs/functions-reference/functions_index.html
@@ -37,10 +37,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
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@@ -3773,25 +3776,6 @@ <h2 class="anchored" data-anchor-id="z">Z</h2>
 </section>
 
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-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -3812,6 +3796,150 @@ <h2 class="anchored" data-anchor-id="z">Z</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/hidden_markov_models.html b/docs/functions-reference/hidden_markov_models.html
index 3f0184954..684d9abf7 100644
--- a/docs/functions-reference/hidden_markov_models.html
+++ b/docs/functions-reference/hidden_markov_models.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -550,25 +553,6 @@ <h2 class="anchored" data-anchor-id="hmm-stan-functions">Stan functions</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -589,6 +573,150 @@ <h2 class="anchored" data-anchor-id="hmm-stan-functions">Stan functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/higher-order_functions.html b/docs/functions-reference/higher-order_functions.html
index 0a4b8f31e..65a29180c 100644
--- a/docs/functions-reference/higher-order_functions.html
+++ b/docs/functions-reference/higher-order_functions.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1036,25 +1039,6 @@ <h3 class="anchored" data-anchor-id="rectangular-map">Rectangular map</h3>
 Tanaka, Ken’ichiro, Masaaki Sugihara, Kazuo Murota, and Masatake Mori. 2009. <span>“Function Classes for Double Exponential Integration Formulas.”</span> <em>Numerische Mathematik</em> 111 (4): 631–55. <a href="https://doi.org/10.1007/s00211-008-0195-1">https://doi.org/10.1007/s00211-008-0195-1</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1075,6 +1059,150 @@ <h3 class="anchored" data-anchor-id="rectangular-map">Rectangular map</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
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+    a.classList.add('quarto-color-scheme-toggle');
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+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/functions-reference/index.html b/docs/functions-reference/index.html
index 5d1162140..6a1cee9e0 100644
--- a/docs/functions-reference/index.html
+++ b/docs/functions-reference/index.html
@@ -37,10 +37,12 @@
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -491,7 +494,7 @@ <h1 class="title">Stan Functions Reference</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the reference for the functions defined in the Stan math library and available in the Stan programming language.</p>
+<p>This is the reference for the functions defined in the <a href="https://github.com/stan-dev/math">Stan math library</a> and available in the <a href="https://mc-stan.org/">Stan</a> programming language.</p>
 <p>For more information the Stan language and inference engines and how to use Stan for Bayesian inference, see</p>
 <ul>
 <li><p>the <a href="https://mc-stan.org/docs/stan-users-guide/index.html"><strong>Stan User’s Guide</strong></a>. The Stan user’s guide provides example models and programming techniques for coding statistical models in Stan. It also serves as an example-driven introduction to Bayesian modeling and inference:</p></li>
@@ -517,25 +520,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -556,6 +540,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
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+    }
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+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // properly recolor the scrollbar when toggling (#1455)
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+        window.scrollTo(0, 0);
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diff --git a/docs/functions-reference/integer-valued_basic_functions.html b/docs/functions-reference/integer-valued_basic_functions.html
index b78ab5cbf..72326a0ee 100644
--- a/docs/functions-reference/integer-valued_basic_functions.html
+++ b/docs/functions-reference/integer-valued_basic_functions.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -99,7 +101,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -633,7 +636,7 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
 <p></p>
 <p><code>int</code> <strong><code>to_int</code></strong><code>(data real x)</code><br></p>
 <p>Return the value <code>x</code> truncated to an integer. This will throw an error if the value of <code>x</code> is too big to represent as a 32-bit signed integer.</p>
-<p>This is similar to <code>trunc</code> (see [Rounding functions]) but the return type is of type <code>int</code>. For example, <code>to_int(3.9)</code> is <code>3</code>, and <code>to_int(-3.9)</code> is <code>-3</code>.</p>
+<p>This is similar to <code>trunc</code> (see <a href="../functions-reference/real-valued_basic_functions.html#rounding-functions">Rounding functions</a>) but the return type is of type <code>int</code>. For example, <code>to_int(3.9)</code> is <code>3</code>, and <code>to_int(-3.9)</code> is <code>-3</code>.</p>
 <small><i>Available since 2.31</i></small>
 <a id="index-entry-04b9411b8a7b904eccd2c168519bf4ccca638602" class="index-entry"></a> <!-- int; to_int; (data real x); -->
 <p></p>
@@ -646,25 +649,6 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -685,6 +669,150 @@ <h2 class="anchored" data-anchor-id="casting-functions">Casting functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  }
+  const enableStylesheet = (stylesheets) => {
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
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+    const els = window.document.querySelectorAll(selector);
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+    // Switch the stylesheets
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // Hack to workaround the fact that safari doesn't
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+      manageTransitions("body", false);
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+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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diff --git a/docs/functions-reference/mathematical_functions.html b/docs/functions-reference/mathematical_functions.html
index 439154ee6..0342c4e4e 100644
--- a/docs/functions-reference/mathematical_functions.html
+++ b/docs/functions-reference/mathematical_functions.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -99,7 +101,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -550,25 +553,6 @@ <h2 class="anchored" data-anchor-id="digamma-appendix">Digamma</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -589,6 +573,150 @@ <h2 class="anchored" data-anchor-id="digamma-appendix">Digamma</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/matrix_operations.html b/docs/functions-reference/matrix_operations.html
index 332a60138..ea9d1b0ec 100644
--- a/docs/functions-reference/matrix_operations.html
+++ b/docs/functions-reference/matrix_operations.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -2014,25 +2017,6 @@ <h2 class="anchored" data-anchor-id="reverse-functions">Reverse functions</h2>
 = 1\)</span> and <span class="math inline">\(\theta_m = 0\)</span> for <span class="math inline">\(m \neq n\)</span>, thus providing a “soft” version of the maximum function.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -2053,6 +2037,150 @@ <h2 class="anchored" data-anchor-id="reverse-functions">Reverse functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/mixed_operations.html b/docs/functions-reference/mixed_operations.html
index e62055dd9..e15f7e695 100644
--- a/docs/functions-reference/mixed_operations.html
+++ b/docs/functions-reference/mixed_operations.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -750,25 +753,6 @@ <h1>Mixed Operations</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -789,6 +773,150 @@ <h1>Mixed Operations</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/multivariate_discrete_distributions.html b/docs/functions-reference/multivariate_discrete_distributions.html
index 8fcdc5034..1556b6f38 100644
--- a/docs/functions-reference/multivariate_discrete_distributions.html
+++ b/docs/functions-reference/multivariate_discrete_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -645,25 +648,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -684,6 +668,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-2">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/positive_continuous_distributions.html b/docs/functions-reference/positive_continuous_distributions.html
index 2cf5d4db9..792bbcf22 100644
--- a/docs/functions-reference/positive_continuous_distributions.html
+++ b/docs/functions-reference/positive_continuous_distributions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1070,25 +1073,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1109,6 +1093,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/positive_lower-bounded_distributions.html b/docs/functions-reference/positive_lower-bounded_distributions.html
index baa17dce5..8439cd3e1 100644
--- a/docs/functions-reference/positive_lower-bounded_distributions.html
+++ b/docs/functions-reference/positive_lower-bounded_distributions.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -698,25 +701,6 @@ <h3 class="anchored" data-anchor-id="boundaries">Boundaries</h3>
 Vandekerckhove, Joachim, and Dominik Wabersich. 2014. <span>“The <span>R</span><span>W</span>iener Package: An <span>R</span> Package Providing Distribution Functions for the <span>W</span>iener Diffusion Model.”</span> <em>The R Journal</em> 6/1. <a href="http://journal.r-project.org/archive/2014-1/vandekerckhove-wabersich.pdf">http://journal.r-project.org/archive/2014-1/vandekerckhove-wabersich.pdf</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -737,6 +721,150 @@ <h3 class="anchored" data-anchor-id="boundaries">Boundaries</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/real-valued_basic_functions.html b/docs/functions-reference/real-valued_basic_functions.html
index 1aaebcdbf..8722e902b 100644
--- a/docs/functions-reference/real-valued_basic_functions.html
+++ b/docs/functions-reference/real-valued_basic_functions.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1579,25 +1582,6 @@ <h2 class="anchored" data-anchor-id="special-functions">Special functions</h2>
 Wichura, Michael J. 1988. <span>“Algorithm <span>AS</span> 241: The Percentage Points of the Normal Distribution.”</span> <em>Journal of the Royal Statistical Society. Series C (Applied Statistics)</em> 37 (3): 477–84. <a href="http://www.jstor.org/stable/2347330">http://www.jstor.org/stable/2347330</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1618,6 +1602,150 @@ <h2 class="anchored" data-anchor-id="special-functions">Special functions</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/references.html b/docs/functions-reference/references.html
index 989877854..b2208f997 100644
--- a/docs/functions-reference/references.html
+++ b/docs/functions-reference/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/removed_functions.html b/docs/functions-reference/removed_functions.html
index d00785035..2da919b82 100644
--- a/docs/functions-reference/removed_functions.html
+++ b/docs/functions-reference/removed_functions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -581,25 +584,6 @@ <h2 class="anchored" data-anchor-id="real-arguments-to-logical-operators-operato
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -620,6 +604,150 @@ <h2 class="anchored" data-anchor-id="real-arguments-to-logical-operators-operato
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/simplex_distributions.html b/docs/functions-reference/simplex_distributions.html
index bc3dfe75e..8b27dbc57 100644
--- a/docs/functions-reference/simplex_distributions.html
+++ b/docs/functions-reference/simplex_distributions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -628,25 +631,6 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -667,6 +651,150 @@ <h3 class="anchored" data-anchor-id="stan-functions">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/sparse_matrix_operations.html b/docs/functions-reference/sparse_matrix_operations.html
index 04372d93e..d40c24f11 100644
--- a/docs/functions-reference/sparse_matrix_operations.html
+++ b/docs/functions-reference/sparse_matrix_operations.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -589,25 +592,6 @@ <h3 class="anchored" data-anchor-id="sparse-matrix-multiplication">Sparse matrix
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -628,6 +612,150 @@ <h3 class="anchored" data-anchor-id="sparse-matrix-multiplication">Sparse matrix
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/unbounded_continuous_distributions.html b/docs/functions-reference/unbounded_continuous_distributions.html
index 59cc3496b..35ec48ad4 100644
--- a/docs/functions-reference/unbounded_continuous_distributions.html
+++ b/docs/functions-reference/unbounded_continuous_distributions.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1176,25 +1179,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
 Ding, Peng, and Joseph K. Blitzstein. 2018. <span>“On the Gaussian Mixture Representation of the Laplace Distribution.”</span> <em>The American Statistician</em> 72 (2): 172–74. <a href="https://doi.org/10.1080/00031305.2017.1291448">https://doi.org/10.1080/00031305.2017.1291448</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1215,6 +1199,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-10">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/unbounded_discrete_distributions.html b/docs/functions-reference/unbounded_discrete_distributions.html
index 1c20b1ea7..eddc5546b 100644
--- a/docs/functions-reference/unbounded_discrete_distributions.html
+++ b/docs/functions-reference/unbounded_discrete_distributions.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -934,25 +937,6 @@ <h3 class="anchored" data-anchor-id="stan-functions-6">Stan functions</h3>
 Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. <em>Bayesian Data Analysis</em>. Third Edition. London: Chapman &amp; Hall / CRC Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -973,6 +957,150 @@ <h3 class="anchored" data-anchor-id="stan-functions-6">Stan functions</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/functions-reference/void_functions.html b/docs/functions-reference/void_functions.html
index b21010d4f..871d6a0f1 100644
--- a/docs/functions-reference/void_functions.html
+++ b/docs/functions-reference/void_functions.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -511,25 +514,6 @@ <h2 class="anchored" data-anchor-id="reject-statement">Reject statement</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -550,6 +534,150 @@ <h2 class="anchored" data-anchor-id="reject-statement">Reject statement</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
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+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/index.html b/docs/index.html
index 9e277b867..fead176e3 100644
--- a/docs/index.html
+++ b/docs/index.html
@@ -37,10 +37,12 @@
 <script src="site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="site_libs/quarto-html/anchor.min.js"></script>
 <link href="site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="quarto_styles.css">
+<link rel="stylesheet" href="theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -230,7 +233,7 @@ <h1 class="title">Stan Documentation</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official documentation for Stan.</p>
+<p>This is the official documentation for <a href="https://mc-stan.org/">Stan</a>.</p>
 <ul>
 <li><p>The <a href="stan-users-guide/index.html"><strong>Stan User’s Guide</strong></a> (<a href="https://mc-stan.org/docs/2_34/stan-users-guide-2_34.pdf">pdf</a>) provides example models and programming techniques for coding statistical models in Stan.</p></li>
 <li><p>The <a href="reference-manual/index.html"><strong>Stan Reference Manual</strong></a> (<a href="https://mc-stan.org/docs/2_34/reference-manual-2_34.pdf">pdf</a>) specifies the Stan programming language and inference algorithms.</p></li>
@@ -256,25 +259,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -295,6 +279,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
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diff --git a/docs/reference-manual/analysis.html b/docs/reference-manual/analysis.html
index 18c53e280..6c7dc7af4 100644
--- a/docs/reference-manual/analysis.html
+++ b/docs/reference-manual/analysis.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -713,25 +716,6 @@ <h3 class="unnumbered anchored" data-anchor-id="thinning-samples">Thinning sampl
 <li id="fn3"><p>The <code>lp__</code> value also represents the potential energy in the Hamiltonian system and is rate bounded by the randomly supplied kinetic energy each iteration, which follows a Chi-square distribution in the number of parameters.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -752,6 +736,150 @@ <h3 class="unnumbered anchored" data-anchor-id="thinning-samples">Thinning sampl
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/blocks.html b/docs/reference-manual/blocks.html
index 83b5370a7..d7c0d2ae2 100644
--- a/docs/reference-manual/blocks.html
+++ b/docs/reference-manual/blocks.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -648,7 +651,7 @@ <h3 class="unnumbered anchored" data-anchor-id="variable-read-write-and-definiti
 <td style="text-align: left;">write / sample</td>
 </tr>
 <tr class="even">
-<td style="text-align: left;"><code>\slshape (initialization)</code></td>
+<td style="text-align: left;"><em><code>(initialization)</code></em></td>
 <td style="text-align: center;">n/a</td>
 <td style="text-align: left;">read, transform / chain</td>
 </tr>
@@ -802,7 +805,7 @@ <h2 class="anchored" data-anchor-id="statistical-variable-taxonomy">Statistical
 <span id="cb2-29"><a href="#cb2-29" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> variance_y;       <span class="co">// derived quantity (transform)</span></span>
 <span id="cb2-30"><a href="#cb2-30" aria-hidden="true" tabindex="-1"></a>  variance_y = sigma_y * sigma_y;</span>
 <span id="cb2-31"><a href="#cb2-31" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>In this example, <code>y[N]</code> is a modeled data vector. Although it is specified in the <code>data</code> block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.</p>
+<p>In this example, <code>y</code> is an array of modeled data. Although it is specified in the <code>data</code> block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.</p>
 <p>The variable <code>N</code> is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.</p>
 <p>The other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the <code>data</code> block may be specified on a per-model execution basis as part of the data read. In the above model, <code>mu_mu</code> and <code>sigma_mu</code> are configurable unmodeled parameters.</p>
 <p>Unmodeled parameters that are hard coded in the model must be declared in the <code>transformed data</code> block. For example, the unmodeled parameters <code>alpha</code> and <code>beta</code> are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the <code>data</code> block, like the variables <code>mu_mu</code> and <code>sigma_mu</code>.</p>
@@ -934,25 +937,6 @@ <h2 class="anchored" data-anchor-id="program-block-generated-quantities">Program
 <li id="fn3"><p>With multiple threads, or even running chains sequentially in a single thread, data could be read only once per set of chains. Stan was designed to be thread safe and future versions will provide a multithreading option for Markov chains.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -973,6 +957,150 @@ <h2 class="anchored" data-anchor-id="program-block-generated-quantities">Program
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/comments.html b/docs/reference-manual/comments.html
index 6fd2ba64e..753ceacf2 100644
--- a/docs/reference-manual/comments.html
+++ b/docs/reference-manual/comments.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -469,25 +472,6 @@ <h2 class="anchored" data-anchor-id="bracketed-comments">Bracketed comments</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -508,6 +492,150 @@ <h2 class="anchored" data-anchor-id="bracketed-comments">Bracketed comments</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/deprecations.html b/docs/reference-manual/deprecations.html
index 6f5108bfe..b9172cc10 100644
--- a/docs/reference-manual/deprecations.html
+++ b/docs/reference-manual/deprecations.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -452,7 +455,7 @@ <h2 id="toc-title">On this page</h2>
 <section id="deprecated-features" class="level1">
 <h1>Deprecated Features</h1>
 <p>This appendix lists currently deprecated functionality along with how to replace it.</p>
-<p>Starting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Features]. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> many of these on the behalf of the user for at least one version after they are removed.</p>
+<p>Starting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in <a href="../reference-manual/removals.html">Removed Features</a>. The Stan compiler can <a href="https://mc-stan.org/docs/stan-users-guide/using-stanc.html#stanc-pretty-printing">automatically update</a> many of these on the behalf of the user for at least one version after they are removed.</p>
 <p>Any feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).</p>
 <section id="lkj_cov-distribution" class="level2">
 <h2 class="anchored" data-anchor-id="lkj_cov-distribution"><code>lkj_cov</code> distribution</h2>
@@ -482,25 +485,6 @@ <h2 class="anchored" data-anchor-id="deprecated-functions">Deprecated Functions<
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -521,6 +505,150 @@ <h2 class="anchored" data-anchor-id="deprecated-functions">Deprecated Functions<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/diagnostics.html b/docs/reference-manual/diagnostics.html
index 5159736ae..aece8cae8 100644
--- a/docs/reference-manual/diagnostics.html
+++ b/docs/reference-manual/diagnostics.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -494,7 +497,7 @@ <h3 class="unnumbered anchored" data-anchor-id="unconstrained-scale">Unconstrain
 </section>
 <section id="includes-jacobian" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="includes-jacobian">Includes Jacobian</h3>
-<p>The log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.</p>
+<p>The log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.</p>
 </section>
 </section>
 <section id="configuration-options" class="level2">
@@ -510,25 +513,6 @@ <h2 class="anchored" data-anchor-id="speed-warning-and-data-trimming">Speed warn
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -549,6 +533,150 @@ <h2 class="anchored" data-anchor-id="speed-warning-and-data-trimming">Speed warn
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/encoding.html b/docs/reference-manual/encoding.html
index 21dea1990..f4d855dae 100644
--- a/docs/reference-manual/encoding.html
+++ b/docs/reference-manual/encoding.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -444,25 +447,6 @@ <h2 class="anchored" data-anchor-id="string-literals">String literals</h2>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -483,6 +467,150 @@ <h2 class="anchored" data-anchor-id="string-literals">String literals</h2>
     }
   }
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
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+    const els = window.document.querySelectorAll(selector);
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+        el.classList.remove('notransition');
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+      }
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/reference-manual/execution.html b/docs/reference-manual/execution.html
index 157e155e9..b553daa79 100644
--- a/docs/reference-manual/execution.html
+++ b/docs/reference-manual/execution.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -543,7 +546,7 @@ <h3 class="unnumbered anchored" data-anchor-id="define-transformed-data">Define
 </section>
 <section id="initialization" class="level2">
 <h2 class="anchored" data-anchor-id="initialization">Initialization</h2>
-<p>Initialization is the same for sampling, optimization, and diagnosis</p>
+<p>Initialization is the same for all of Stan’s algorithms.</p>
 <section id="user-supplied-initial-values" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="user-supplied-initial-values">User-supplied initial values</h3>
 <p>If there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.</p>
@@ -641,25 +644,6 @@ <h3 class="unnumbered anchored" data-anchor-id="write">Write</h3>
 <li id="fn2"><p>In the R version of Stan, the values may either be written to a CSV file or directly back to R’s memory.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -680,6 +664,150 @@ <h3 class="unnumbered anchored" data-anchor-id="write">Write</h3>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
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+        el.classList.remove('notransition');
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+        el.classList.add('notransition');
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
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+        }
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
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+      return styleSentinel === "alternate";
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+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
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+      return storageValue != null ? storageValue : localAlternateSentinel;
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+    }
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+  // Dark / light mode switch
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+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
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+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
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+    toggleColorMode(true);
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diff --git a/docs/reference-manual/expressions.html b/docs/reference-manual/expressions.html
index 65a57a8e7..a20b159ab 100644
--- a/docs/reference-manual/expressions.html
+++ b/docs/reference-manual/expressions.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -622,8 +625,7 @@ <h4 class="unnumbered anchored" data-anchor-id="reserved-words-from-stan-languag
 cholesky_factor_corr, cholesky_factor_cov,
 corr_matrix, cov_matrix, array</code></pre>
 <p>The following built in functions are also reserved and cannot be used as variable names:</p>
-<pre><code>print, reject, profile, get_lp, increment_log_prob,
-target</code></pre>
+<pre><code>print, reject, profile, target</code></pre>
 <p>The following block identifiers are reserved and cannot be used as variable names:</p>
 <pre><code>functions, model, data, parameters, quantities,
 transformed, generated</code></pre>
@@ -1384,6 +1386,10 @@ <h3 class="unnumbered anchored" data-anchor-id="promotion">Promotion</h3>
 <li><code>array[...] U</code> may be promoted to <code>array[...] T</code> if <code>U</code> can be promoted to <code>T</code>.</li>
 </ol>
 <p>For example, this means <code>array[,] int</code> may be used where <code>array [,] real</code> or <code>array [,] complex</code> is required; as another example, <code>array[] real</code> may be used anywhere <code>array[] complex</code> is required.</p>
+<p>Tuples have the natural extension of the above rules, applied to all sub-types at once</p>
+<ol start="8" type="1">
+<li>A <code>tuple(U1, ..., UN)</code> may be promoted to a <code>tuple(T1, ..., TN)</code> if every <code>Un</code> can be promoted to <code>Tn</code> for <code>n</code> in <code>1:N</code></li>
+</ol>
 <section id="literals" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="literals">Literals</h4>
 <p>An integer literal expression such as <code>42</code> is of type <code>int</code>. Real literals such as <code>42.0</code> are of type <code>real</code>. Imaginary literals such as <code>-17i</code> are of type <code>complex</code>. the expression <code>7 - 2i</code> acts like a complex literal, but technically it combines a real literal <code>7</code> and an imaginary literal <code>2i</code> through subtraction.</p>
@@ -1619,25 +1625,6 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-problems-with-derivat
 <li id="fn2"><p>Internally, they are implemented as their own expression types because Stan doesn’t have object-level functional types (yet).<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1658,6 +1645,150 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-problems-with-derivat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
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+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
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+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
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+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/includes.html b/docs/reference-manual/includes.html
index 3dd17252c..fdaa12503 100644
--- a/docs/reference-manual/includes.html
+++ b/docs/reference-manual/includes.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -501,11 +504,21 @@ <h3 class="unnumbered anchored" data-anchor-id="comments-after-includes">Comment
 </section>
 <section id="recursive-includes" class="level2">
 <h2 class="anchored" data-anchor-id="recursive-includes">Recursive includes</h2>
-<p>Recursive includes will be ignored. For example, suppose <code>a.stan</code> contains</p>
+<p>Recursive includes will lead to a compiler error. For example, suppose <code>a.stan</code> contains</p>
 <div class="sourceCode" id="cb6"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co">#include b.stan</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>and <code>b.stan</code> contains</p>
 <div class="sourceCode" id="cb7"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co">#include a.stan</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>The result of processing this file will be empty, because <code>a.stan</code> will include <code>b.stan</code>, from which the include of <code>a.stan</code> is ignored and a warning printed.</p>
+<p>This will result in an error explaining the circular dependency:</p>
+<pre><code>Syntax error in './b.stan', line 1, column 0, included from
+'./a.stan', line 1, column 0, included from
+'./b.stan', line 1, column 0, included from
+'a.stan', line 1, column 0, include error:
+   -------------------------------------------------
+     1:  #include a.stan
+         ^
+   -------------------------------------------------
+
+File a.stan recursively included itself.</code></pre>
 </section>
 <section id="include-paths" class="level2">
 <h2 class="anchored" data-anchor-id="include-paths">Include paths</h2>
@@ -520,25 +533,6 @@ <h3 class="unnumbered anchored" data-anchor-id="slashes-in-include-paths">Slashe
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -559,6 +553,150 @@ <h3 class="unnumbered anchored" data-anchor-id="slashes-in-include-paths">Slashe
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
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+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/index.html b/docs/reference-manual/index.html
index dc1eb0da7..146d9e4d4 100644
--- a/docs/reference-manual/index.html
+++ b/docs/reference-manual/index.html
@@ -37,10 +37,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -421,7 +424,7 @@ <h1 class="title">Stan Reference Manual</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official reference manual for Stan’s <em>programming language</em> for coding probability models, <em>inference algorithms</em> for fitting models and making predictions, and <em>posterior analysis</em> tools for evaluating the results. This manual applies to all Stan interfaces.</p>
+<p>This is the official reference manual for <a href="https://mc-stan.org/">Stan</a>’s <em>programming language</em> for coding probability models, <em>inference algorithms</em> for fitting models and making predictions, and <em>posterior analysis</em> tools for evaluating the results. This manual applies to all Stan interfaces.</p>
 <p>The first part of the reference manual provides a full specification of the Stan programming language. The language is responsible for defining a log density function conditioned on data. Typically, this is a Bayesian posterior, but it may also be a penalized likelihood function. The second part of the manual specifies the inference algorithms and posterior inference tools. The third part provides auxiliary information about the use of Stan.</p>
 <p><a href="https://mc-stan.org/docs/2_34/reference-manual-2_34.pdf">Download the pdf version of this manual</a>.</p>
 <section id="copyright-and-trademark" class="level4 unnumbered">
@@ -443,25 +446,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -482,6 +466,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/laplace.html b/docs/reference-manual/laplace.html
index 6758dc2a1..7d5f03a6b 100644
--- a/docs/reference-manual/laplace.html
+++ b/docs/reference-manual/laplace.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -456,25 +459,6 @@ <h1>Laplace Approximation</h1>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -495,6 +479,150 @@ <h1>Laplace Approximation</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/licenses.html b/docs/reference-manual/licenses.html
index 4d05cdac4..c5c5b7bc7 100644
--- a/docs/reference-manual/licenses.html
+++ b/docs/reference-manual/licenses.html
@@ -37,10 +37,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -406,6 +409,7 @@ <h2 id="toc-title">On this page</h2>
   <li><a href="#boost-license" id="toc-boost-license" class="nav-link" data-scroll-target="#boost-license">Boost license</a></li>
   <li><a href="#eigen-license" id="toc-eigen-license" class="nav-link" data-scroll-target="#eigen-license">Eigen license</a></li>
   <li><a href="#sundials-license" id="toc-sundials-license" class="nav-link" data-scroll-target="#sundials-license">SUNDIALS license</a></li>
+  <li><a href="#threaded-building-blocks-tbb-license" id="toc-threaded-building-blocks-tbb-license" class="nav-link" data-scroll-target="#threaded-building-blocks-tbb-license">Threaded Building Blocks (TBB) License</a></li>
   <li><a href="#google-test-license" id="toc-google-test-license" class="nav-link" data-scroll-target="#google-test-license">Google test license</a></li>
   </ul></li>
   </ul>
@@ -454,6 +458,14 @@ <h2 class="anchored" data-anchor-id="sundials-license">SUNDIALS license</h2>
 </ul>
 <p>The copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.</p>
 </section>
+<section id="threaded-building-blocks-tbb-license" class="level2">
+<h2 class="anchored" data-anchor-id="threaded-building-blocks-tbb-license">Threaded Building Blocks (TBB) License</h2>
+<p>Stan uses the Threaded Building Blocks (TBB) library for parallel computations. TBB is distributed under the</p>
+<ul>
+<li><a href="https://opensource.org/license/apache-2-0">Apache License, version 2</a></li>
+</ul>
+<p>The copyright of TBB is owned by Intel Corporation.</p>
+</section>
 <section id="google-test-license" class="level2">
 <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h2>
 <p>Stan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the</p>
@@ -473,25 +485,6 @@ <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h
 <li id="fn1"><p>Universities or companies often own the copyright of computer programs developed by their employees.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -512,6 +505,150 @@ <h2 class="anchored" data-anchor-id="google-test-license">Google test license</h
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/mcmc.html b/docs/reference-manual/mcmc.html
index c929f089b..665fcbf27 100644
--- a/docs/reference-manual/mcmc.html
+++ b/docs/reference-manual/mcmc.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -929,25 +932,6 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-and-eliminating-diver
 <li id="fn5"><p>The current default threshold is a factor of <span class="math inline">\(10^3\)</span>, whereas when the leapfrog integrator is working properly, the divergences will be around <span class="math inline">\(10^{-7}\)</span> and do not compound due to the symplectic nature of the leapfrog integrator.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -968,6 +952,150 @@ <h3 class="unnumbered anchored" data-anchor-id="diagnosing-and-eliminating-diver
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/optimization.html b/docs/reference-manual/optimization.html
index 0678acdd2..06c273ef2 100644
--- a/docs/reference-manual/optimization.html
+++ b/docs/reference-manual/optimization.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -591,25 +594,6 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-vs.-unconstrained-pa
 Nocedal, Jorge, and Stephen J. Wright. 2006. <em>Numerical Optimization</em>. Second. Berlin: Springer-<span>V</span>erlag.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -630,6 +614,150 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-vs.-unconstrained-pa
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/pathfinder.html b/docs/reference-manual/pathfinder.html
index f54c256bc..2e23c273d 100644
--- a/docs/reference-manual/pathfinder.html
+++ b/docs/reference-manual/pathfinder.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -90,7 +92,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -205,6 +207,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -445,25 +448,6 @@ <h1>Pathfinder</h1>
 Zhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. <span>“Pathfinder: Parallel Quasi-Newton Variational Inference.”</span> <em>Journal of Machine Learning Research</em> 23 (306): 1–49. <a href="http://jmlr.org/papers/v23/21-0889.html">http://jmlr.org/papers/v23/21-0889.html</a>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -484,6 +468,150 @@ <h1>Pathfinder</h1>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/references.html b/docs/reference-manual/references.html
index 14c964857..220aec734 100644
--- a/docs/reference-manual/references.html
+++ b/docs/reference-manual/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/removals.html b/docs/reference-manual/removals.html
index 1944fb1cb..ec06d517b 100644
--- a/docs/reference-manual/removals.html
+++ b/docs/reference-manual/removals.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -581,25 +584,6 @@ <h2 class="anchored" data-anchor-id="real-values-in-conditionals">Real values in
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -620,6 +604,150 @@ <h2 class="anchored" data-anchor-id="real-values-in-conditionals">Real values in
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/reproducibility.html b/docs/reference-manual/reproducibility.html
index 43ff31826..6121adea7 100644
--- a/docs/reference-manual/reproducibility.html
+++ b/docs/reference-manual/reproducibility.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -90,7 +92,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -205,6 +207,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -460,25 +463,6 @@ <h1>Reproducibility</h1>
 Corden, Martyn J., and David Kreitzer. 2014. <span>“Consistency of Floating-Point Results Using the <span>I</span>ntel Compiler or <span>W</span>hy Doesn’t My Application Always Give the Same Answer?”</span> Intel Corporation. <a href="https://software.intel.com/en-us/articles/consistency-of-floating-point-results-using-the-intel-compiler">https://software.intel.com/en-us/articles/consistency-of-floating-point-results-using-the-intel-compiler</a>.
 </div>
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diff --git a/docs/reference-manual/statements.html b/docs/reference-manual/statements.html
index 143478c89..4d4bb44c4 100644
--- a/docs/reference-manual/statements.html
+++ b/docs/reference-manual/statements.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
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@@ -133,7 +135,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -562,10 +565,11 @@ <h3 class="anchored" data-anchor-id="promotion">Promotion</h3>
 </section>
 <section id="lvalue-summary" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="lvalue-summary">Lvalue summary</h3>
-<p>The expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,</p>
+<p>The expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,</p>
 <ul>
 <li>a variable, or</li>
-<li>a variable with one or more indices.</li>
+<li>a variable with one or more indices, or</li>
+<li>a comma separated list of lvalues surrounded by <code>(</code> and <code>)</code></li>
 </ul>
 <p>To be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.</p>
 </section>
@@ -1061,21 +1065,20 @@ <h3 class="unnumbered anchored" data-anchor-id="local-variable-declarations">Loc
 </section>
 <section id="no-constraints-on-local-variables" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="no-constraints-on-local-variables">No constraints on local variables</h3>
-<p>Local variables may not have constraints on their declaration. The only types that may be used are</p>
-<pre><code>int, real, vector[K], row_vector[K], matrix[M, N].</code></pre>
+<p>Local variables may not have constraints on their declaration. The only types that may be used are listed in the <a href="../reference-manual/types.html#id:constrained-types.figure">types table</a> under “local”.</p>
 </section>
 <section id="blocks-within-blocks" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="blocks-within-blocks">Blocks within blocks</h3>
 <p>A block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following</p>
-<div class="sourceCode" id="cb55"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (m <span class="cf">in</span> <span class="dv">1</span>:M) {</span>
-<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>  {</span>
-<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>     <span class="dt">int</span> n = <span class="dv">2</span> * m;</span>
-<span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>     sum += n;</span>
-<span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
-<span id="cb55-7"><a href="#cb55-7" aria-hidden="true" tabindex="-1"></a>    sum += x[m, n];</span>
-<span id="cb55-8"><a href="#cb55-8" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb55-9"><a href="#cb55-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb54"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (m <span class="cf">in</span> <span class="dv">1</span>:M) {</span>
+<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a>  {</span>
+<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a>     <span class="dt">int</span> n = <span class="dv">2</span> * m;</span>
+<span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a>     sum += n;</span>
+<span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb54-6"><a href="#cb54-6" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) {</span>
+<span id="cb54-7"><a href="#cb54-7" aria-hidden="true" tabindex="-1"></a>    sum += x[m, n];</span>
+<span id="cb54-8"><a href="#cb54-8" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb54-9"><a href="#cb54-9" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The variable declaration <code>int n;</code> is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of <code>n</code> are unrelated.</p>
 </section>
 </section>
@@ -1086,56 +1089,56 @@ <h2 class="anchored" data-anchor-id="break-continue-statements">Break and contin
 <section id="break-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="break-statements">Break statements</h3>
 <p>When a <code>break</code> statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:</p>
-<div class="sourceCode" id="cb56"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (<span class="dv">1</span>) {</span>
-<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (n &lt; <span class="dv">0</span>) {</span>
-<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">break</span>;</span>
-<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
-<span id="cb56-6"><a href="#cb56-6" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
-<span id="cb56-7"><a href="#cb56-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb55"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (<span class="dv">1</span>) {</span>
+<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (n &lt; <span class="dv">0</span>) {</span>
+<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>    <span class="cf">break</span>;</span>
+<span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
+<span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
+<span id="cb55-7"><a href="#cb55-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The <code>while~(1)</code> loop is a “forever” loop, because <code>1</code> is the true value, so the test always succeeds. Within the loop, if the value of <code>n</code> is less than 0, the loop terminates, otherwise it executes <code>foo(n)</code> and then decrements <code>n</code>. The statement above does exactly the same thing as</p>
-<div class="sourceCode" id="cb57"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (n &gt;= <span class="dv">0</span>) {</span>
-<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
-<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
-<span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb56"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (n &gt;= <span class="dv">0</span>) {</span>
+<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>  foo(n);</span>
+<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>  n = n - <span class="dv">1</span>;</span>
+<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This case is simply illustrative of the behavior; it is not a case where a <code>break</code> simplifies the loop.</p>
 </section>
 <section id="continue-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="continue-statements">Continue statements</h3>
 <p>The <code>continue</code> statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array <code>x</code>,</p>
+<div class="sourceCode" id="cb57"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
+<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
+<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
+<span id="cb57-4"><a href="#cb57-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &lt;= <span class="dv">0</span>) {</span>
+<span id="cb57-5"><a href="#cb57-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">continue</span>;</span>
+<span id="cb57-6"><a href="#cb57-6" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb57-7"><a href="#cb57-7" aria-hidden="true" tabindex="-1"></a>  sum += x[n];</span>
+<span id="cb57-8"><a href="#cb57-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<p>When the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an <code>x[n]</code> less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,</p>
 <div class="sourceCode" id="cb58"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
 <span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
 <span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
-<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &lt;= <span class="dv">0</span>) {</span>
-<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">continue</span>;</span>
+<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &gt; <span class="dv">0</span>) {</span>
+<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a>    sum += x[n];</span>
 <span id="cb58-6"><a href="#cb58-6" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a>  sum += x[n];</span>
-<span id="cb58-8"><a href="#cb58-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<p>When the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an <code>x[n]</code> less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,</p>
-<div class="sourceCode" id="cb59"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> sum;</span>
-<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a>sum = <span class="dv">0</span>;</span>
-<span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:size(x)) {</span>
-<span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x[n] &gt; <span class="dv">0</span>) {</span>
-<span id="cb59-5"><a href="#cb59-5" aria-hidden="true" tabindex="-1"></a>    sum += x[n];</span>
-<span id="cb59-6"><a href="#cb59-6" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb59-7"><a href="#cb59-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>While the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.</p>
 </section>
 <section id="breaking-and-continuing-nested-loops" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="breaking-and-continuing-nested-loops">Breaking and continuing nested loops</h3>
 <p>If there is a loop nested within a loop, a <code>break</code> or <code>continue</code> statement only breaks out of the inner loop. So</p>
-<div class="sourceCode" id="cb60"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (cond1) {</span>
-<span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb60-3"><a href="#cb60-3" aria-hidden="true" tabindex="-1"></a>  <span class="cf">while</span> (cond2) {</span>
-<span id="cb60-4"><a href="#cb60-4" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
-<span id="cb60-5"><a href="#cb60-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">if</span> (cond3) {</span>
-<span id="cb60-6"><a href="#cb60-6" aria-hidden="true" tabindex="-1"></a>      <span class="cf">break</span>;</span>
-<span id="cb60-7"><a href="#cb60-7" aria-hidden="true" tabindex="-1"></a>    }</span>
-<span id="cb60-8"><a href="#cb60-8" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
-<span id="cb60-9"><a href="#cb60-9" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb60-10"><a href="#cb60-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// execution continues here after break</span></span>
-<span id="cb60-11"><a href="#cb60-11" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb60-12"><a href="#cb60-12" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb59"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> (cond1) {</span>
+<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a>  <span class="cf">while</span> (cond2) {</span>
+<span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
+<span id="cb59-5"><a href="#cb59-5" aria-hidden="true" tabindex="-1"></a>    <span class="cf">if</span> (cond3) {</span>
+<span id="cb59-6"><a href="#cb59-6" aria-hidden="true" tabindex="-1"></a>      <span class="cf">break</span>;</span>
+<span id="cb59-7"><a href="#cb59-7" aria-hidden="true" tabindex="-1"></a>    }</span>
+<span id="cb59-8"><a href="#cb59-8" aria-hidden="true" tabindex="-1"></a>    <span class="co">// ...</span></span>
+<span id="cb59-9"><a href="#cb59-9" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb59-10"><a href="#cb59-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// execution continues here after break</span></span>
+<span id="cb59-11"><a href="#cb59-11" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb59-12"><a href="#cb59-12" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>If the break is triggered by <code>cond3</code> being true, execution will continue after the nested loop.</p>
 <p>As with break statements, continue statements go back to the top of the most deeply nested loop in which the <code>continue</code> appears.</p>
 <p>Although break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.</p>
@@ -1144,32 +1147,32 @@ <h3 class="unnumbered anchored" data-anchor-id="breaking-and-continuing-nested-l
 <section id="print-statements.section" class="level2">
 <h2 class="anchored" data-anchor-id="print-statements.section">Print statements</h2>
 <p>Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.</p>
-<div class="sourceCode" id="cb61"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) { <span class="kw">print</span>(<span class="st">"loop iteration: "</span>, n); ... }</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb60"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:N) { <span class="kw">print</span>(<span class="st">"loop iteration: "</span>, n); ... }</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression <code>n</code>, followed by a new line.</p>
 <section id="print-content" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="print-content">Print content</h3>
 <p>The text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.</p>
 <p>Printing a simple <code>real</code> or <code>int</code> typed variable always prints the variable’s value.<a href="#fn5" class="footnote-ref" id="fnref5" role="doc-noteref"><sup>5</sup></a></p>
 <p>For array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as</p>
-<div class="sourceCode" id="cb62"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb61"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a>[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>and a <span class="math inline">\(2 \times 3\)</span>-matrix as</p>
-<div class="sourceCode" id="cb63"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>], [<span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>]]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb62"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a>[[<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>], [<span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>]]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>Complex numbers print as pairs. For example, the pair of statements</p>
-<div class="sourceCode" id="cb64"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="dt">complex</span> z = to_complex(<span class="fl">1.2</span>, -<span class="fl">3.5</span>);</span>
-<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(z)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb63"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="dt">complex</span> z = to_complex(<span class="fl">1.2</span>, -<span class="fl">3.5</span>);</span>
+<span id="cb63-2"><a href="#cb63-2" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(z)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>will print as <code>(1.2,-3.5)</code>, with no space after the comma or within the parentheses.</p>
 <p>Printing a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.</p>
-<div class="sourceCode" id="cb65"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="kw">transformed data</span> {</span>
-<span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">matrix</span>[<span class="dv">2</span>, <span class="dv">2</span>] u;</span>
-<span id="cb65-3"><a href="#cb65-3" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">1</span>, <span class="dv">1</span>] = <span class="fl">1.0</span>;   u[<span class="dv">1</span>, <span class="dv">2</span>] = <span class="fl">4.0</span>;</span>
-<span id="cb65-4"><a href="#cb65-4" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">2</span>, <span class="dv">1</span>] = <span class="fl">9.0</span>;   u[<span class="dv">2</span>, <span class="dv">2</span>] = <span class="fl">16.0</span>;</span>
-<span id="cb65-5"><a href="#cb65-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:<span class="dv">2</span>) {</span>
-<span id="cb65-6"><a href="#cb65-6" aria-hidden="true" tabindex="-1"></a>    <span class="kw">print</span>(<span class="st">"u["</span>, n, <span class="st">"] = "</span>, u[n]);</span>
-<span id="cb65-7"><a href="#cb65-7" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb65-8"><a href="#cb65-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb64"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="kw">transformed data</span> {</span>
+<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">matrix</span>[<span class="dv">2</span>, <span class="dv">2</span>] u;</span>
+<span id="cb64-3"><a href="#cb64-3" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">1</span>, <span class="dv">1</span>] = <span class="fl">1.0</span>;   u[<span class="dv">1</span>, <span class="dv">2</span>] = <span class="fl">4.0</span>;</span>
+<span id="cb64-4"><a href="#cb64-4" aria-hidden="true" tabindex="-1"></a>  u[<span class="dv">2</span>, <span class="dv">1</span>] = <span class="fl">9.0</span>;   u[<span class="dv">2</span>, <span class="dv">2</span>] = <span class="fl">16.0</span>;</span>
+<span id="cb64-5"><a href="#cb64-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (n <span class="cf">in</span> <span class="dv">1</span>:<span class="dv">2</span>) {</span>
+<span id="cb64-6"><a href="#cb64-6" aria-hidden="true" tabindex="-1"></a>    <span class="kw">print</span>(<span class="st">"u["</span>, n, <span class="st">"] = "</span>, u[n]);</span>
+<span id="cb64-7"><a href="#cb64-7" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb64-8"><a href="#cb64-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This print statement executes twice, printing the following two lines of output.</p>
-<div class="sourceCode" id="cb66"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">1</span>] = [<span class="dv">1</span>, <span class="dv">4</span>]</span>
-<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">2</span>] = [<span class="dv">9</span>, <span class="dv">16</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb65"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">1</span>] = [<span class="dv">1</span>, <span class="dv">4</span>]</span>
+<span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>u[<span class="dv">2</span>] = [<span class="dv">9</span>, <span class="dv">16</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </section>
 <section id="non-void-input" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="non-void-input">Non-void input</h3>
@@ -1188,12 +1191,13 @@ <h3 class="unnumbered anchored" data-anchor-id="string-literals">String literals
 <h3 class="unnumbered anchored" data-anchor-id="debug-by-print">Debug by <code>print</code></h3>
 <p>Because Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.</p>
 <p>It is particularly useful to print the value of the target log density accumulator (through the <code>target()</code> function), as in the following example.</p>
-<div class="sourceCode" id="cb67"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="dt">vector</span>[<span class="dv">2</span>] y;</span>
-<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>y[<span class="dv">1</span>] = <span class="dv">1</span>;</span>
-<span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density before ="</span>, target());</span>
-<span id="cb67-4"><a href="#cb67-4" aria-hidden="true" tabindex="-1"></a>y ~ normal(<span class="dv">0</span>,<span class="dv">1</span>);  <span class="co">// bug!  y[2] not defined</span></span>
-<span id="cb67-5"><a href="#cb67-5" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density after ="</span>, target());</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb66"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="dt">vector</span>[<span class="dv">2</span>] y;</span>
+<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>y[<span class="dv">1</span>] = <span class="dv">1</span>;</span>
+<span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density before ="</span>, target());</span>
+<span id="cb66-4"><a href="#cb66-4" aria-hidden="true" tabindex="-1"></a>y ~ normal(<span class="dv">0</span>,<span class="dv">1</span>);  <span class="co">// bug!  y[2] not defined</span></span>
+<span id="cb66-5"><a href="#cb66-5" aria-hidden="true" tabindex="-1"></a><span class="kw">print</span>(<span class="st">"log density after ="</span>, target());</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The example has a bug in that <code>y[2]</code> is not defined before the vector <code>y</code> is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).</p>
+<p>Note that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.</p>
 </section>
 </section>
 <section id="reject-statements.section" class="level2">
@@ -1201,9 +1205,9 @@ <h2 class="anchored" data-anchor-id="reject-statements.section">Reject statement
 <p>The Stan <code>reject</code> statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.</p>
 <p>Like the <code>print</code> statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.</p>
 <p>Reject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable <code>x</code>’s value is negative.</p>
-<div class="sourceCode" id="cb68"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> (x &lt; <span class="dv">0</span>) {</span>
-<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">reject</span>(<span class="st">"x must not be negative; found x="</span>, x);</span>
-<span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb67"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> (x &lt; <span class="dv">0</span>) {</span>
+<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">reject</span>(<span class="st">"x must not be negative; found x="</span>, x);</span>
+<span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <section id="behavior-of-reject-statements" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="behavior-of-reject-statements">Behavior of reject statements</h3>
 <p>Reject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the <code>normal_lpdf</code> function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.</p>
@@ -1226,26 +1230,26 @@ <h4 class="unnumbered anchored" data-anchor-id="recoverable-rejection-contexts">
 <section id="rejection-is-not-for-constraints" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="rejection-is-not-for-constraints">Rejection is not for constraints</h3>
 <p>Rejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.</p>
-<div class="sourceCode" id="cb69"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
-<span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> a;</span>
-<span id="cb69-3"><a href="#cb69-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a&gt; b;</span>
-<span id="cb69-4"><a href="#cb69-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a, <span class="kw">upper</span>=b&gt; theta;</span>
-<span id="cb69-5"><a href="#cb69-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb69-6"><a href="#cb69-6" aria-hidden="true" tabindex="-1"></a>}</span>
-<span id="cb69-7"><a href="#cb69-7" aria-hidden="true" tabindex="-1"></a><span class="kw">model</span> {</span>
-<span id="cb69-8"><a href="#cb69-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// **wrong** needs explicit truncation</span></span>
-<span id="cb69-9"><a href="#cb69-9" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span>
-<span id="cb69-10"><a href="#cb69-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
-<span id="cb69-11"><a href="#cb69-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb68"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
+<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span> a;</span>
+<span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a&gt; b;</span>
+<span id="cb68-4"><a href="#cb68-4" aria-hidden="true" tabindex="-1"></a>  <span class="dt">real</span>&lt;<span class="kw">lower</span>=a, <span class="kw">upper</span>=b&gt; theta;</span>
+<span id="cb68-5"><a href="#cb68-5" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb68-6"><a href="#cb68-6" aria-hidden="true" tabindex="-1"></a>}</span>
+<span id="cb68-7"><a href="#cb68-7" aria-hidden="true" tabindex="-1"></a><span class="kw">model</span> {</span>
+<span id="cb68-8"><a href="#cb68-8" aria-hidden="true" tabindex="-1"></a>  <span class="co">// **wrong** needs explicit truncation</span></span>
+<span id="cb68-9"><a href="#cb68-9" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span>
+<span id="cb68-10"><a href="#cb68-10" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
+<span id="cb68-11"><a href="#cb68-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>This program is wrong because its truncation bounds on <code>theta</code> depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.</p>
-<div class="sourceCode" id="cb70"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>) <span class="kw">T</span>[a, b];</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb69"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>) <span class="kw">T</span>[a, b];</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>The conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over <span class="math inline">\([a ,b]\)</span>; in these simple univariate cases, we can overcome that with the <code>T[ , ]</code> notation, which essentially divides by whatever the prior integrates to over <span class="math inline">\([a, b]\)</span>.</p>
 <p>This problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.</p>
-<div class="sourceCode" id="cb71"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (theta &lt; a || theta &gt; b) {</span>
-<span id="cb71-2"><a href="#cb71-2" aria-hidden="true" tabindex="-1"></a>    <span class="kw">reject</span>(<span class="st">"theta not in (a, b)"</span>);</span>
-<span id="cb71-3"><a href="#cb71-3" aria-hidden="true" tabindex="-1"></a>  }</span>
-<span id="cb71-4"><a href="#cb71-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// still **wrong**, needs T[a,b]</span></span>
-<span id="cb71-5"><a href="#cb71-5" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb70"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (theta &lt; a || theta &gt; b) {</span>
+<span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a>    <span class="kw">reject</span>(<span class="st">"theta not in (a, b)"</span>);</span>
+<span id="cb70-3"><a href="#cb70-3" aria-hidden="true" tabindex="-1"></a>  }</span>
+<span id="cb70-4"><a href="#cb70-4" aria-hidden="true" tabindex="-1"></a>  <span class="co">// still **wrong**, needs T[a,b]</span></span>
+<span id="cb70-5"><a href="#cb70-5" aria-hidden="true" tabindex="-1"></a>  theta ~ normal(<span class="dv">0</span>, <span class="dv">1</span>);</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>In this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.</p>
 <p>Even if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.</p>
 
@@ -1265,25 +1269,6 @@ <h3 class="unnumbered anchored" data-anchor-id="rejection-is-not-for-constraints
 <li id="fn5"><p>The adjoint component is always zero during execution for the algorithmic differentiation variables used to implement parameters, transformed parameters, and local variables in the model.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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diff --git a/docs/reference-manual/syntax.html b/docs/reference-manual/syntax.html
index f48ba3e91..75861137a 100644
--- a/docs/reference-manual/syntax.html
+++ b/docs/reference-manual/syntax.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -842,7 +845,7 @@ <h3 class="unnumbered anchored" data-anchor-id="probability-function-naming">Pro
 <section id="indexes" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
 <p>Standalone expressions used as indexes must denote either an integer (<code>int</code>) or an integer array (<code>array[] int</code>). Expressions participating in range indexes (e.g., <code>a</code> and <code>b</code> in <code>a : b</code>) must denote integers (<code>int</code>).</p>
-<p>A second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type <code>matrix[ , , ]</code>, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.</p>
+<p>A second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type <code>array[ , , ] matrix</code>, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.</p>
 
 
 </section>
@@ -850,25 +853,6 @@ <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
 </section>
 
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@@ -889,6 +873,150 @@ <h3 class="unnumbered anchored" data-anchor-id="indexes">Indexes</h3>
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diff --git a/docs/reference-manual/transforms.html b/docs/reference-manual/transforms.html
index e1a01d613..f2dc3e762 100644
--- a/docs/reference-manual/transforms.html
+++ b/docs/reference-manual/transforms.html
@@ -58,10 +58,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1405,25 +1408,6 @@ <h3 class="unnumbered anchored" data-anchor-id="absolute-jacobian-determinant-of
 <li id="fn3"><p>For convenience, a summation with no terms, such as <span class="math inline">\(\sum_{j' &lt; 1} x_{i,j'}\)</span>, is defined to be 0. This implies <span class="math inline">\(x_{1,1} = 1\)</span> and that <span class="math inline">\(x_{i,1} = z_{i,1}\)</span> for <span class="math inline">\(i &gt; 1\)</span>.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1444,6 +1428,150 @@ <h3 class="unnumbered anchored" data-anchor-id="absolute-jacobian-determinant-of
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
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+        el.classList.add('notransition');
+      }
+    }
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
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+  }
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diff --git a/docs/reference-manual/types.html b/docs/reference-manual/types.html
index e048bd438..c0910ef4a 100644
--- a/docs/reference-manual/types.html
+++ b/docs/reference-manual/types.html
@@ -72,10 +72,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -600,7 +603,7 @@ <h3 class="unnumbered anchored" data-anchor-id="array-types">Array types</h3>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="dt">array</span>[<span class="dv">6</span>, <span class="dv">7</span>] <span class="dt">matrix</span>[<span class="dv">3</span>, <span class="dv">3</span>] m;</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="dt">array</span>[<span class="dv">12</span>, <span class="dv">8</span>, <span class="dv">15</span>] <span class="dt">complex</span> z;</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>declares <code>x</code> to be a one-dimensional array of size 10 containing real values, declares <code>m</code> to be a two-dimensional array of size <span class="math inline">\(6 \times 7\)</span> containing values that are <span class="math inline">\(3 \times 3\)</span> matrices, and declares <code>z</code> to be a <span class="math inline">\(12 \times 8 \times 15\)</span> array of complex numbers.</p>
-<p>Prior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.</p>
+<p>Prior to 2.26 Stan models used a different syntax which has since been removed. See the <a href="../reference-manual/removals.html">Removed Features</a> chapter for more details.</p>
 </section>
 <section id="tuple-types" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="tuple-types">Tuple types</h3>
@@ -630,8 +633,8 @@ <h2 class="anchored" data-anchor-id="numerical-data-types">Primitive numerical d
 <section id="integers" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="integers">Integers</h3>
 <p>Stan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is <span class="math inline">\(2^{31}-1\)</span>; the minimum value is <span class="math inline">\(-(2^{31})\)</span>.</p>
-<p>When integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.</p>
-<p>Integer arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see <a href="https://mc-stan.org/docs/functions-reference/integer-valued_basic_functions.html#int-arithmetic">the Stan Functions Reference integer-valued arithmetic operators section</a> for more information).</p>
+<p>When integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.</p>
+<p>Integer arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see <a href="https://mc-stan.org/docs/functions-reference/integer-valued_basic_functions.html#int-arithmetic">the Stan Functions Reference integer-valued arithmetic operators section</a> for more information).</p>
 </section>
 <section id="reals" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="reals">Reals</h3>
@@ -721,7 +724,7 @@ <h3 class="unnumbered anchored" data-anchor-id="constrained-real">Constrained re
 <div class="sourceCode" id="cb15"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span>&lt;<span class="kw">lower</span>=-<span class="dv">1</span>, <span class="kw">upper</span>=<span class="dv">1</span>&gt; rho;</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <section id="infinite-constraints" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="infinite-constraints">Infinite constraints</h4>
-<p>Lower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants <code>positive_infinity()</code> and <code>negative_infinity()</code> which may be used for this purpose, or they may be read as data in the dump format.</p>
+<p>Lower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants <code>positive_infinity()</code> and <code>negative_infinity()</code> which may be used for this purpose, or they may be supplied as data.</p>
 </section>
 </section>
 <section id="affinely-transformed-real" class="level3 unnumbered">
@@ -1621,25 +1624,6 @@ <h3 class="unnumbered anchored" data-anchor-id="combining-with-other-features">C
 <li id="fn1"><p>Stan compiles integers to <code>int</code> and reals to <code>double</code> types in C++. Precise details of rounding will depend on the compiler and hardware architecture on which the code is run.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -1660,6 +1644,150 @@ <h3 class="unnumbered anchored" data-anchor-id="combining-with-other-features">C
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/user-functions.html b/docs/reference-manual/user-functions.html
index cf0aca155..98a25d035 100644
--- a/docs/reference-manual/user-functions.html
+++ b/docs/reference-manual/user-functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -657,6 +660,7 @@ <h3 class="unnumbered anchored" data-anchor-id="return-guarantee-required">Retur
 <li>a for loop or while loop qualifies if its body qualifies, and</li>
 <li>a conditional statement qualifies if it has a default else clause and all of its body statements qualify.</li>
 </ul>
+<p>An exception is made for “obviously infinite” loops like <code>while (1)</code>, which contain a <code>return</code> statement and no <code>break</code> statements. The only way to exit such a loop is to return, so they are considered as returning statements.</p>
 <p>These rules disqualify</p>
 <div class="sourceCode" id="cb14"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="dt">real</span> foo(<span class="dt">real</span> x) {</span>
 <span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> (x &gt; <span class="dv">2</span>) {</span>
@@ -727,25 +731,6 @@ <h2 class="anchored" data-anchor-id="forward-declarations.section">Declarations<
 <li id="fn1"><p>Despite being declared constant and appearing to have a pass-by-value syntax in Stan, the implementation of the language passes function arguments by constant reference in C++.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -766,6 +751,150 @@ <h2 class="anchored" data-anchor-id="forward-declarations.section">Declarations<
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
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diff --git a/docs/reference-manual/variational.html b/docs/reference-manual/variational.html
index 6e01b5f83..8b2430d6a 100644
--- a/docs/reference-manual/variational.html
+++ b/docs/reference-manual/variational.html
@@ -58,10 +58,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -119,7 +121,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -234,6 +236,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -505,25 +508,6 @@ <h3 class="unnumbered anchored" data-anchor-id="assessing-convergence">Assessing
 Kucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. <span>“Automatic Differentiation Variational Inference.”</span> <em>Journal of Machine Learning Research</em>.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -544,6 +528,150 @@ <h3 class="unnumbered anchored" data-anchor-id="assessing-convergence">Assessing
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/reference-manual/whitespace.html b/docs/reference-manual/whitespace.html
index 8e164b6c9..f993470dc 100644
--- a/docs/reference-manual/whitespace.html
+++ b/docs/reference-manual/whitespace.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -70,7 +72,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -185,6 +187,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -437,25 +440,6 @@ <h2 class="anchored" data-anchor-id="whitespace-location">Whitespace location</h
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -476,6 +460,150 @@ <h2 class="anchored" data-anchor-id="whitespace-location">Whitespace location</h
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
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diff --git a/docs/search.json b/docs/search.json
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--- a/docs/search.json
+++ b/docs/search.json
@@ -4,7 +4,7 @@
     "href": "stan-users-guide/using-stanc.html",
     "title": "Using the Stan Compiler",
     "section": "",
-    "text": "Stan is used in most of our interfaces through the Stan compiler stanc. Since version 2.22, the Stan compiler has been implemented in OCaml and is referred to as stanc3. The binary name is still simply stanc, so this document uses both stanc and stanc3 interchangeably.\n\n\nThe stanc3 compiler has the following command-line syntax:\n&gt; stanc (options) &lt;model_file&gt;\nwhere &lt;model_file&gt; is a path to either a Stan model file ending in suffix .stan or a Stan functions file which ends in .stanfunctions.\nThe stanc3 options are:\n\n--help - Displays the complete list of stanc3 options, then exits.\n--version - Display stanc version number\n--info - Print information about the model, such as the type information for variables and the list of used distributions.\n--name=&lt;model_name&gt; - Specify the name of the class used for the implementation of the Stan model in the generated C++ code.\n--o=&lt;file_name&gt; - Specify a path to an output file for generated C++ code (default = .hpp) or auto-formatting output (default: no file/print to stdout)\n--allow-undefined - Do not throw a parser error if there is a function in the Stan program that is declared but not defined in the functions block.\n--include_paths=&lt;dir1,...dirN&gt; - Takes a comma-separated list of directories that may contain a file in an #include directive.\n--use-opencl - If set, will use additional Stan OpenCL features enabled in the Stan-to-C++ compiler.\n--auto-format - Pretty prints the program to the console. See more on auto formatting.\n--canonicalize - Make changes to the program before pretty-printing by specifying options in a comma separated list. Options are ‘deprecations’, ‘parentheses’, ‘braces’, ‘includes’.\n--max-line-length=&lt;number&gt; - Set the column number at which formatting with --auto-format attempts to split lines. The default value is 78, which results in most lines being shorter than 80 characters.\n--print-canonical - Synonymous with --auto-format --canonicalize=[all   options].\n--print-cpp - If set, output the generated C++ Stan model class to stdout.\n--standalone-functions - If set, only generate the code for the functions defined in the file. This is the default behavior for .stanfunctions files.\n--O0 (Default) Do not apply optimizations to the Stan code.\n--O1 Apply level 1 compiler optimizations (only basic optimizations).\n--Oexperimental WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Allow the compiler to apply all optimizations to the Stan code.\n--O WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Same as --Oexperimental. Allow the compiler to apply all optimizations to the Stan code.\n--warn-uninitialized - Emit warnings about uninitialized variables to stderr. Currently an experimental feature.\n--warn-pedantic - Emit warnings in Pedantic mode which warns of potential issues in the meaning of your program.\n\nThe compiler also provides a number of debug options which are primarily of interest to stanc3 developers; use the --help option to see the full set.\n\n\n\nDuring model compilation, stanc can produce a variety of errors (issues that prevent the model from being compiled) and warnings (non-fatal issues that should still be considered).\n\n\nEven without the optional --warn-pedantic and --warn-uninitialized command line flags, both of which enable additional warnings, stanc can still produce warnings about your program. In particular, warnings will be produced in two situations\n\nA completely blank Stan program will produce the following warning message\nWarning: Empty file 'empty.stan' detected;\n    this is a valid stan model but likely unintended!\nThe use of any deprecated features will lead to warnings which will look as follows\nWarning in 'deprecated.stan', line 2, column 0: Comments beginning with # are\n deprecated and this syntax will be removed in Stan 2.32.0. Use // to\n begin line comments; this can be done automatically using stanc\n --auto-format\nA single Stan program can produce many warnings during compilation.\n\n\n\n\nErrors differ from warnings in their severity and format. In particular, errors are fatal and stop compilation, so at most one error is displayed per run of stanc.\nThere are five kinds of errors emitted by stanc3\n\nFile errors occur when the file passed to stanc is either missing or cannot be opened (i.e. has permissions issues). They look like\nError: file 'notfound.stan' not found or cannot be opened\nSyntactic errors occur whenever a program violates the Stan language’s syntax requirements. There are three kinds of errors within syntax errors; “lexing” errors mean that the input was unable to be read properly on the character level, “include” errors which occur when the #include directive fails, and “parsing” errors which result when the structure of the program is incorrect.\n\nThe lexing errors occur due to the use of invalid characters in a program. For example, a lexing error due to the use of $ in a variable name will look like the following.\nSyntax error in 'char.stan', line 2, column 6, lexing error:\n-------------------------------------------------\n  1:  data {\n  2:     int $ome_variable;\n             ^\n  3:  }\n-------------------------------------------------\nInvalid character found.\nWhen an include directive is used, it can lead to errors if the included file is not found, or if a file includes itself (including a recursive loop of includes, such as A -&gt; B -&gt; A).\nSyntax error in './incl.stan', line 1, column 0, included from\n'./incl.stan', line 1, column 0, included from\n'incl.stan', line 1, column 0, include error:\n-------------------------------------------------\n  1:  #include &lt;incl.stan&gt;\n      ^\n-------------------------------------------------\nFile incl.stan recursively included itself.\nIt is much more common to see parsing errors, which tend to have more in-depth explanations of the error found. For example, if a user forgets to put a size on a type like vector, as in the following, this raises a parsing (structural) error in the compiler.\nSyntax error in vec.stan', line 3, column 10 to column 11, parsing error:\n-------------------------------------------------\n  1:  data {\n  2:     int&lt;lower=0&gt; N;\n  3:     vector x;\n                ^\n  4:  }\n-------------------------------------------------\n\"[\" expression \"]\" expected for vector size.\n\nSemantic errors (also known as type errors) occur when a program is structured correctly but features an error in the type rules imposed by the language. An example of this is assigning a real value to a variable defined as an integer.\nSemantic error in 'type.stan', line 2, column 3 to column 15:\n-------------------------------------------------\n  1:  transformed data {\n  2:     int x = 1.5;\n         ^\n  3:  }\n-------------------------------------------------\nIll-typed arguments supplied to assignment operator =: lhs has\ntype int and rhs has type real\nThe compiler will raise an error for use of any removed features for at least one version following their removal. The deprecation warnings mentioned above eventually turn into this kind of error to prompt the user to update their model. After the version of removal, these errors will be converted to one of the other types listed here, depending on the feature.\nFinally, the compiler can raise an internal error. These are caused by bugs in the compiler, not your model, and we would appreciate it if you report them on the stanc3 repo with the error message provided. These errors usually say something like “This should never happen,” and we apologize if they do.\n\n\n\n\n\nPedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.\n\n\n\n\n\nIn addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.\n\n\n\n\n\nThe stanc3 compiler can optimize the code of Stan model during compilation. The optimized model code behaves the same as unoptimized code, but it may be faster, more memory efficient, or more numerically stable.\nThis section introduces the available optimization options and describes their effect.\nTo print out a representation of the optimized Stan program, use the stanc3 command-line flag --debug-optimized-mir-pretty. To print an analogous representation of the Stan program prior to optimization, use the flag --debug-transformed-mir-pretty.\n\n\nTo turn optimizations on, the user specifies the desired optimization level. The level specifies the set of optimizations to use. The chosen optimizations are used in a specific order, with some of them applied repeatedly.\nOptimization levels are specified by the numbers 0 and 1 and the ‘experimental’ tag:\n\nO0 No optimizations are applied.\nO1 Optimizations that are simple, do not dramatically change the program, and are unlikely to noticeably slow down compile times are applied.\nOexperimental All optimizations are applied. Some of these are not thorougly tested and may not always improve a programs performance.\n\nO0 is the default setting.\nThe levels include these optimizations:\n\nO0 includes no optimizations.\nO1 includes:\n\nDead code elimination\nCopy propagation\nConstant propagation\n\nOexperimental includes optimizations specified by O1 and also:\n\nAutomatic-differentiation level optimization\nOne step loop unrolling\nExpression propagation\nPartial evaluation\nLazy code motion\nFunction inlining\nStatic loop unrolling\n\n\nIn addition, Oexperimental will apply more repetitions of the optimizations, which may increase compile times.\n\n\n\n\n\nDead code is code that does not affect the behavior of the program. Code is not dead if it affects target, the value of any outside-observable variable like transformed parameters or generated quantities, or side effects such as print statements. Removing dead code can speed up a program by avoiding unnecessary computations.\nExample Stan program:\nmodel {\n  int i;\n  i = 5;\n  for (j in 1:10);\n  if (0) {\n    print(\"Dead code\");\n  } else {\n    print(\"Hi!\");\n  }\n}\nCompiler representation of program before dead code elimination (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  int i = 5;\n  for(j in 1:10) {\n    ;\n  }\n  if(0) {\n    FnPrint__(\"Dead code\");\n  } else {\n    FnPrint__(\"Hi!\");\n  }\n}\nCompiler representation of program after dead code elimination (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i;\n  FnPrint__(\"Hi!\");\n}\n\n\n\nConstant propagation replaces uses of a variable which is known to have a constant value C with that constant C. This removes the overhead of looking up the variable, and also makes many other optimizations possible (such as static loop unrolling and partial evaluation).\nExample Stan program:\ntransformed data {\n  int n = 100;\n  int a[n];\n  for (i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before constant propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, n] a;\n  for(i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after constant propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, 100] a;\n  for(i in 1:100) {\n    a[i] = i;\n  }\n}\n\n\n\nCopy propagation is similar to expression propagation, but only propagates variables rather than arbitrary expressions. This can reduce the complexity of the code for other optimizations such as expression propagation.\nExample Stan program:\nmodel {\n  int i = 1;\n  int j = i;\n  int k = i + j;\n}\nCompiler representation of program before copy propagation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n    int i = 1;\n    int j = i;\n    int k = (i + j);\n}\nCompiler representation of program after copy propagation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i = 1;\n  int j = i;\n  int k = (i + i);\n}\n\n\n\nPartial evaluation searches for expressions that we can replace with a faster, simpler, more memory efficient, or more numerically stable expression with the same meaning.\nExample Stan program:\nmodel {\n  real a = 1 + 1;\n  real b = log(1 - a);\n  real c = a + b * 5;\n}\nCompiler representation of program before partial evaluation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real a = (1 + 1);\n  real b = log((1 - a));\n  real c = (a + (b * 5));\n}\nCompiler representation of program after partial evaluation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  real a = 2;\n  real b = log1m(a);\n  real c = fma(b, 5, a);\n}\n\n\n\n\n\n\nStan variables can have two auto-differentiation (AD) levels: AD or non-AD. AD variables carry gradient information with them, which allows Stan to calculate the log-density gradient, but they also have more overhead than non-AD variables. It is therefore inefficient for a variable to be AD unnecessarily. AD-level optimization sets every variable to be a floating point type unless its gradient is necessary.\nExample Stan program:\ndata {\n  real y;\n}\nmodel {\n  real x = y + 1;\n}\nCompiler representation of program before AD-level optimization (simplified from the output of --debug-transformed-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  real x = (y + 1);\n}\nCompiler representation of program after AD-level optimization (simplified from the output of --debug-optimized-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  data real x = (y + 1);\n}\n\n\n\nOne step loop unrolling is similar to static loop unrolling. However, this optimization only ‘unrolls’ the first loop iteration, and can therefore work even when the total number of iterations is not predictable. This can speed up a program by providing more opportunities for further optimizations such as partial evaluation and lazy code motion.\nExample Stan program:\ndata {\n  int n;\n}\ntransformed data {\n  int x = 0;\n  for (i in 1:n) {\n    x += i;\n  }\n}\nCompiler representation of program before one step static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  data int x = 0;\n  for(i in 1:n) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after one step static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  int x = 0;\n  if((n &gt;= 1)) {\n    x = (x + 1);\n    for(i in (1 + 1):n) {\n      x = (x + i);\n    }\n  }\n}\n\n\n\nConstant propagation replaces the uses of a variable which is known to have a constant value E with that constant E. This often results in recalculating the expression, but provides more opportunities for further optimizations such as partial evaluation. Expression propagation is always followed by lazy code motion to avoid unnecessarily recomputing expressions.\nExample Stan program:\ndata {\n  int m;\n}\ntransformed data {\n  int n = m+1;\n  int a[n];\n  for (i in 1:n-1) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before expression propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, n] a;\n  for(i in 1:(n - 1)) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after expression propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, (m + 1)] a;\n  for(i in 1:((m + 1) - 1)) {\n    a[i] = i;\n  }\n}\n\n\n\nLazy code motion rearranges the statements and expressions in a program with the goals of:\n\nAvoiding computing expressions more than once, and\nComputing expressions as late as possible (to minimize the strain on the working memory set).\n\nTo accomplish these goals, lazy code motion will perform optimizations such as:\n\nMoving a repeatedly calculated expression to its own variable (also referred to as common-subexpression elimination)\nMoving an expression outside of a loop if it does not need to be in the loop (also referred to as loop-invariant code motion)\n\nLazy code motion can make some programs significantly more efficient by avoiding redundant or early computations.\nExample Stan program:\nmodel {\n  real x;\n  real y;\n  real z;\n\n  for (i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program before lazy code motion (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real x;\n  real y;\n  real z;\n  for(i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program after lazy code motion (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  data real lcm_sym4__;\n  data real lcm_sym3__;\n  real x;\n  real y;\n  lcm_sym4__ = sqrt(10);\n  real z;\n  for(i in 1:10) {\n    x = lcm_sym4__;\n    y = sqrt(i);\n  }\n  z = lcm_sym4__;\n}\n\n\n\nFunction inlining replaces each function call to each user-defined function f with the body of f. It does this by copying the function body to the call site and doing appropriately renaming the argument variables. This optimization can speed up a program by avoiding the overhead of a function call and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return x + y;\n  }\n}\ntransformed data {\n  int a = 2;\n  int b = incr(a);\n}\nCompiler representation of program before function inlining (simplified from the output of --debug-transformed-mir-pretty):\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return (x + y);\n  }\n}\n\nprepare_data {\n  data int a = 2;\n  data int b = incr(a);\n}\nCompiler representation of program after function inlining (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int a;\n  a = 2;\n  data int b;\n  data int inline_sym1__;\n  data int inline_sym3__;\n  inline_sym3__ = 0;\n  for(inline_sym4__ in 1:1) {\n    int inline_sym2__;\n    inline_sym2__ = 1;\n    inline_sym3__ = 1;\n    inline_sym1__ = (a + inline_sym2__);\n    break;\n  }\n  b = inline_sym1__;\n}\nIn this code, the for loop and break is used to simulate the behavior of a return statement. The value to be returned is held in inline_sym1__. The flag variable inline_sym3__ indicates whether a return has occurred and is necessary to handle return statements nested inside loops within the function body.\n\n\n\nStatic loop unrolling takes a loop with a predictable number of iterations X and replaces it by writing out the loop body X times. The loop index in each repeat is replaced with the appropriate constant. This optimization can speed up a program by avoiding the overhead of a loop and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\ntransformed data {\n  int x = 0;\n  for (i in 1:4) {\n    x += i;\n  }\n}\nCompiler representation of program before static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int x = 0;\n  for(i in 1:4) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int x;\n  x = 0;\n  x = (x + 1);\n  x = (x + 2);\n  x = (x + 3);\n  x = (x + 4);\n}",
+    "text": "Stan is used in most of our interfaces through the Stan compiler stanc. Since version 2.22, the Stan compiler has been implemented in OCaml and is referred to as stanc3. The binary name is still simply stanc, so this document uses both stanc and stanc3 interchangeably.\n\n\nThe stanc3 compiler has the following command-line syntax:\n&gt; stanc (options) &lt;model_file&gt;\nwhere &lt;model_file&gt; is a path to either a Stan model file ending in suffix .stan or a Stan functions file which ends in .stanfunctions.\nThe stanc3 options are:\n\n--help - Displays the complete list of stanc3 options, then exits.\n--version - Display stanc version number\n--info - Print information about the model, such as the type information for variables and the list of used distributions.\n--name=&lt;model_name&gt; - Specify the name of the class used for the implementation of the Stan model in the generated C++ code.\n--o=&lt;file_name&gt; - Specify a path to an output file for generated C++ code (default = .hpp) or auto-formatting output (default: no file/print to stdout)\n--allow-undefined - Do not throw a parser error if there is a function in the Stan program that is declared but not defined in the functions block.\n--include_paths=&lt;dir1,...dirN&gt; - Takes a comma-separated list of directories that may contain a file in an #include directive.\n--use-opencl - If set, will use additional Stan OpenCL features enabled in the Stan-to-C++ compiler.\n--auto-format - Pretty prints the program to the console. See more on auto formatting.\n--canonicalize - Make changes to the program before pretty-printing by specifying options in a comma separated list. Options are ‘deprecations’, ‘parentheses’, ‘braces’, ‘includes’.\n--max-line-length=&lt;number&gt; - Set the column number at which formatting with --auto-format attempts to split lines. The default value is 78, which results in most lines being shorter than 80 characters.\n--print-canonical - Synonymous with --auto-format --canonicalize=[all   options].\n--print-cpp - If set, output the generated C++ Stan model class to stdout.\n--standalone-functions - If set, only generate the code for the functions defined in the file. This is the default behavior for .stanfunctions files.\n--O0 (Default) Do not apply optimizations to the Stan code.\n--O1 Apply level 1 compiler optimizations (only basic optimizations).\n--Oexperimental WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Allow the compiler to apply all optimizations to the Stan code.\n--O WARNING: This is currently an experimental feature whose components are not thoroughly tested and may not improve a programs performance! Same as --Oexperimental. Allow the compiler to apply all optimizations to the Stan code.\n--warn-uninitialized - Emit warnings about uninitialized variables to stderr. Currently an experimental feature.\n--warn-pedantic - Emit warnings in Pedantic mode which warns of potential issues in the meaning of your program.\n\nThe compiler also provides a number of debug options which are primarily of interest to stanc3 developers; use the --help option to see the full set.\n\n\n\nDuring model compilation, stanc can produce a variety of errors (issues that prevent the model from being compiled) and warnings (non-fatal issues that should still be considered).\n\n\nEven without the optional --warn-pedantic and --warn-uninitialized command line flags, both of which enable additional warnings, stanc can still produce warnings about your program. In particular, warnings will be produced in two situations\n\nA completely blank Stan program will produce the following warning message\nWarning: Empty file 'empty.stan' detected;\n    this is a valid stan model but likely unintended!\nThe use of any deprecated features will lead to warnings which will look as follows\nWarning in 'deprecated.stan', line 2, column 0: Comments beginning with # are\n deprecated and this syntax will be removed in Stan 2.32.0. Use // to\n begin line comments; this can be done automatically using stanc\n --auto-format\nA single Stan program can produce many warnings during compilation.\n\n\n\n\nErrors differ from warnings in their severity and format. In particular, errors are fatal and stop compilation, so at most one error is displayed per run of stanc.\nThere are five kinds of errors emitted by stanc3\n\nFile errors occur when the file passed to stanc is either missing or cannot be opened (i.e. has permissions issues). They look like\nError: file 'notfound.stan' not found or cannot be opened\nSyntactic errors occur whenever a program violates the Stan language’s syntax requirements. There are three kinds of errors within syntax errors; “lexing” errors mean that the input was unable to be read properly on the character level, “include” errors which occur when the #include directive fails, and “parsing” errors which result when the structure of the program is incorrect.\n\nThe lexing errors occur due to the use of invalid characters in a program. For example, a lexing error due to the use of $ in a variable name will look like the following.\nSyntax error in 'char.stan', line 2, column 6, lexing error:\n-------------------------------------------------\n  1:  data {\n  2:     int $ome_variable;\n             ^\n  3:  }\n-------------------------------------------------\nInvalid character found.\nWhen an include directive is used, it can lead to errors if the included file is not found, or if a file includes itself (including a recursive loop of includes, such as A -&gt; B -&gt; A).\nSyntax error in './incl.stan', line 1, column 0, included from\n'./incl.stan', line 1, column 0, included from\n'incl.stan', line 1, column 0, include error:\n-------------------------------------------------\n  1:  #include &lt;incl.stan&gt;\n      ^\n-------------------------------------------------\nFile incl.stan recursively included itself.\nIt is much more common to see parsing errors, which tend to have more in-depth explanations of the error found. For example, if a user forgets to put a size on a type like vector, as in the following, this raises a parsing (structural) error in the compiler.\nSyntax error in vec.stan', line 3, column 10 to column 11, parsing error:\n-------------------------------------------------\n  1:  data {\n  2:     int&lt;lower=0&gt; N;\n  3:     vector x;\n                ^\n  4:  }\n-------------------------------------------------\n\"[\" expression \"]\" expected for vector size.\n\nSemantic errors (also known as type errors) occur when a program is structured correctly but features an error in the type rules imposed by the language. An example of this is assigning a real value to a variable defined as an integer.\nSemantic error in 'type.stan', line 2, column 3 to column 15:\n-------------------------------------------------\n  1:  transformed data {\n  2:     int x = 1.5;\n         ^\n  3:  }\n-------------------------------------------------\nIll-typed arguments supplied to assignment operator =: lhs has\ntype int and rhs has type real\nThe compiler will raise an error for use of any removed features for at least one version following their removal. The deprecation warnings mentioned above eventually turn into this kind of error to prompt the user to update their model. After the version of removal, these errors will be converted to one of the other types listed here, depending on the feature.\nFinally, the compiler can raise an internal error. These are caused by bugs in the compiler, not your model, and we would appreciate it if you report them on the stanc3 repo with the error message provided. These errors usually say something like “This should never happen,” and we apologize if they do.\n\n\n\n\n\nPedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the Changes of variables section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.\n\n\n\n\n\nIn addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the Stan Program Style Guide. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.\n\n\n\n\n\nThe stanc3 compiler can optimize the code of Stan model during compilation. The optimized model code behaves the same as unoptimized code, but it may be faster, more memory efficient, or more numerically stable.\nThis section introduces the available optimization options and describes their effect.\nTo print out a representation of the optimized Stan program, use the stanc3 command-line flag --debug-optimized-mir-pretty. To print an analogous representation of the Stan program prior to optimization, use the flag --debug-transformed-mir-pretty.\n\n\nTo turn optimizations on, the user specifies the desired optimization level. The level specifies the set of optimizations to use. The chosen optimizations are used in a specific order, with some of them applied repeatedly.\nOptimization levels are specified by the numbers 0 and 1 and the ‘experimental’ tag:\n\nO0 No optimizations are applied.\nO1 Optimizations that are simple, do not dramatically change the program, and are unlikely to noticeably slow down compile times are applied.\nOexperimental All optimizations are applied. Some of these are not thorougly tested and may not always improve a programs performance.\n\nO0 is the default setting.\nThe levels include these optimizations:\n\nO0 includes no optimizations.\nO1 includes:\n\nDead code elimination\nCopy propagation\nConstant propagation\n\nOexperimental includes optimizations specified by O1 and also:\n\nAutomatic-differentiation level optimization\nOne step loop unrolling\nExpression propagation\nPartial evaluation\nLazy code motion\nFunction inlining\nStatic loop unrolling\n\n\nIn addition, Oexperimental will apply more repetitions of the optimizations, which may increase compile times.\n\n\n\n\n\nDead code is code that does not affect the behavior of the program. Code is not dead if it affects target, the value of any outside-observable variable like transformed parameters or generated quantities, or side effects such as print statements. Removing dead code can speed up a program by avoiding unnecessary computations.\nExample Stan program:\nmodel {\n  int i;\n  i = 5;\n  for (j in 1:10);\n  if (0) {\n    print(\"Dead code\");\n  } else {\n    print(\"Hi!\");\n  }\n}\nCompiler representation of program before dead code elimination (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  int i = 5;\n  for(j in 1:10) {\n    ;\n  }\n  if(0) {\n    FnPrint__(\"Dead code\");\n  } else {\n    FnPrint__(\"Hi!\");\n  }\n}\nCompiler representation of program after dead code elimination (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i;\n  FnPrint__(\"Hi!\");\n}\n\n\n\nConstant propagation replaces uses of a variable which is known to have a constant value C with that constant C. This removes the overhead of looking up the variable, and also makes many other optimizations possible (such as static loop unrolling and partial evaluation).\nExample Stan program:\ntransformed data {\n  int n = 100;\n  int a[n];\n  for (i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before constant propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, n] a;\n  for(i in 1:n) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after constant propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = 100;\n  data array[int, 100] a;\n  for(i in 1:100) {\n    a[i] = i;\n  }\n}\n\n\n\nCopy propagation is similar to expression propagation, but only propagates variables rather than arbitrary expressions. This can reduce the complexity of the code for other optimizations such as expression propagation.\nExample Stan program:\nmodel {\n  int i = 1;\n  int j = i;\n  int k = i + j;\n}\nCompiler representation of program before copy propagation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n    int i = 1;\n    int j = i;\n    int k = (i + j);\n}\nCompiler representation of program after copy propagation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  int i = 1;\n  int j = i;\n  int k = (i + i);\n}\n\n\n\nPartial evaluation searches for expressions that we can replace with a faster, simpler, more memory efficient, or more numerically stable expression with the same meaning.\nExample Stan program:\nmodel {\n  real a = 1 + 1;\n  real b = log(1 - a);\n  real c = a + b * 5;\n}\nCompiler representation of program before partial evaluation (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real a = (1 + 1);\n  real b = log((1 - a));\n  real c = (a + (b * 5));\n}\nCompiler representation of program after partial evaluation (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  real a = 2;\n  real b = log1m(a);\n  real c = fma(b, 5, a);\n}\n\n\n\n\n\n\nStan variables can have two auto-differentiation (AD) levels: AD or non-AD. AD variables carry gradient information with them, which allows Stan to calculate the log-density gradient, but they also have more overhead than non-AD variables. It is therefore inefficient for a variable to be AD unnecessarily. AD-level optimization sets every variable to be a floating point type unless its gradient is necessary.\nExample Stan program:\ndata {\n  real y;\n}\nmodel {\n  real x = y + 1;\n}\nCompiler representation of program before AD-level optimization (simplified from the output of --debug-transformed-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  real x = (y + 1);\n}\nCompiler representation of program after AD-level optimization (simplified from the output of --debug-optimized-mir-pretty):\ninput_vars {\n  real y;\n}\n\nlog_prob {\n  data real x = (y + 1);\n}\n\n\n\nOne step loop unrolling is similar to static loop unrolling. However, this optimization only ‘unrolls’ the first loop iteration, and can therefore work even when the total number of iterations is not predictable. This can speed up a program by providing more opportunities for further optimizations such as partial evaluation and lazy code motion.\nExample Stan program:\ndata {\n  int n;\n}\ntransformed data {\n  int x = 0;\n  for (i in 1:n) {\n    x += i;\n  }\n}\nCompiler representation of program before one step static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  data int x = 0;\n  for(i in 1:n) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after one step static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int n = FnReadData__(\"n\")[1];\n  int x = 0;\n  if((n &gt;= 1)) {\n    x = (x + 1);\n    for(i in (1 + 1):n) {\n      x = (x + i);\n    }\n  }\n}\n\n\n\nConstant propagation replaces the uses of a variable which is known to have a constant value E with that constant E. This often results in recalculating the expression, but provides more opportunities for further optimizations such as partial evaluation. Expression propagation is always followed by lazy code motion to avoid unnecessarily recomputing expressions.\nExample Stan program:\ndata {\n  int m;\n}\ntransformed data {\n  int n = m+1;\n  int a[n];\n  for (i in 1:n-1) {\n    a[i] = i;\n  }\n}\nCompiler representation of program before expression propagation (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, n] a;\n  for(i in 1:(n - 1)) {\n    a[i] = i;\n  }\n}\nCompiler representation of program after expression propagation (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int m = FnReadData__(\"m\")[1];\n  data int n = (m + 1);\n  data array[int, (m + 1)] a;\n  for(i in 1:((m + 1) - 1)) {\n    a[i] = i;\n  }\n}\n\n\n\nLazy code motion rearranges the statements and expressions in a program with the goals of:\n\nAvoiding computing expressions more than once, and\nComputing expressions as late as possible (to minimize the strain on the working memory set).\n\nTo accomplish these goals, lazy code motion will perform optimizations such as:\n\nMoving a repeatedly calculated expression to its own variable (also referred to as common-subexpression elimination)\nMoving an expression outside of a loop if it does not need to be in the loop (also referred to as loop-invariant code motion)\n\nLazy code motion can make some programs significantly more efficient by avoiding redundant or early computations.\nExample Stan program:\nmodel {\n  real x;\n  real y;\n  real z;\n\n  for (i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program before lazy code motion (simplified from the output of --debug-transformed-mir-pretty):\nlog_prob {\n  real x;\n  real y;\n  real z;\n  for(i in 1:10) {\n    x = sqrt(10);\n    y = sqrt(i);\n  }\n  z = sqrt(10);\n}\nCompiler representation of program after lazy code motion (simplified from the output of --debug-optimized-mir-pretty):\nlog_prob {\n  data real lcm_sym4__;\n  data real lcm_sym3__;\n  real x;\n  real y;\n  lcm_sym4__ = sqrt(10);\n  real z;\n  for(i in 1:10) {\n    x = lcm_sym4__;\n    y = sqrt(i);\n  }\n  z = lcm_sym4__;\n}\n\n\n\nFunction inlining replaces each function call to each user-defined function f with the body of f. It does this by copying the function body to the call site and doing appropriately renaming the argument variables. This optimization can speed up a program by avoiding the overhead of a function call and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return x + y;\n  }\n}\ntransformed data {\n  int a = 2;\n  int b = incr(a);\n}\nCompiler representation of program before function inlining (simplified from the output of --debug-transformed-mir-pretty):\nfunctions {\n  int incr(int x) {\n    int y = 1;\n    return (x + y);\n  }\n}\n\nprepare_data {\n  data int a = 2;\n  data int b = incr(a);\n}\nCompiler representation of program after function inlining (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int a;\n  a = 2;\n  data int b;\n  data int inline_sym1__;\n  data int inline_sym3__;\n  inline_sym3__ = 0;\n  for(inline_sym4__ in 1:1) {\n    int inline_sym2__;\n    inline_sym2__ = 1;\n    inline_sym3__ = 1;\n    inline_sym1__ = (a + inline_sym2__);\n    break;\n  }\n  b = inline_sym1__;\n}\nIn this code, the for loop and break is used to simulate the behavior of a return statement. The value to be returned is held in inline_sym1__. The flag variable inline_sym3__ indicates whether a return has occurred and is necessary to handle return statements nested inside loops within the function body.\n\n\n\nStatic loop unrolling takes a loop with a predictable number of iterations X and replaces it by writing out the loop body X times. The loop index in each repeat is replaced with the appropriate constant. This optimization can speed up a program by avoiding the overhead of a loop and providing more opportunities for further optimizations (such as partial evaluation).\nExample Stan program:\ntransformed data {\n  int x = 0;\n  for (i in 1:4) {\n    x += i;\n  }\n}\nCompiler representation of program before static loop unrolling (simplified from the output of --debug-transformed-mir-pretty):\nprepare_data {\n  data int x = 0;\n  for(i in 1:4) {\n    x = (x + i);\n  }\n}\nCompiler representation of program after static loop unrolling (simplified from the output of --debug-optimized-mir-pretty):\nprepare_data {\n  data int x;\n  x = 0;\n  x = (x + 1);\n  x = (x + 2);\n  x = (x + 3);\n  x = (x + 4);\n}",
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     "href": "stan-users-guide/using-stanc.html#pedantic-mode",
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-    "text": "Pedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.",
+    "text": "Pedantic mode is a compilation option built into Stanc3 that warns you about potential issues in your Stan program.\nFor example, consider the following program.\ndata {\n  int N;\n  array[N] real x;\n}\nparameters {\n  real sigma;\n}\nmodel {\n  real mu;\n  x ~ normal(mu, sigma);\n}\nWhen pedantic mode is turned on, the compiler will produce the following warnings.\nWarning:\n  The parameter sigma has no priors.\nWarning at 'ped-mode-ex1.stan', line 10, column 14 to column 16:\n  The variable mu may not have been assigned a value before its use.\nWarning at 'ped-mode-ex1.stan', line 10, column 18 to column 23:\n  A normal distribution is given parameter sigma as a scale parameter\n  (argument 2), but sigma was not constrained to be strictly positive.\nHere are the kinds of issues that pedantic mode will find (which are described in more detail in following sections):\n\nDistribution usages issues. Distribution arguments don’t match the distribution specification, or some specific distribution is used in an inadvisable way.\nUnused parameter. A parameter is defined but doesn’t contribute to target.\nLarge or small constant in a distribution. Very large or very small constants are used as distribution arguments.\nControl flow depends on a parameter. Branching control flow (like if/else) depends on a parameter value .\nParameter has multiple tildes. A parameter is on the left-hand side of multiple tildes.\nParameter has zero or multiple priors. A parameter has zero or more than one prior distribution.\nVariable is used before assignment. A variable is used before being assigned a value.\nStrict or nonsensical parameter bounds. A parameter is given questionable bounds.\nNonlinear transformations. When the left-hand side of a tilde statement (or first argument of a log probability function) contains a nonlinear transform which may require a Jacobian change of variables adjustment.\n\nSome important limitations of pedantic mode are listed at the end of this chapter.\n\n\nWhen an argument to a built-in distribution certainly does not match that distribution’s specification in the Stan Functions Reference, a warning is thrown. This primarily checks if any distribution argument’s bounds at declaration, compile-time value, or subtype at declaration (e.g. simplex) is incompatible with the domain of the distribution. x\nFor example, consider the following program.\nparameters {\n  real unb_p;\n  real&lt;lower=0&gt; pos_p;\n}\nmodel {\n  1 ~ poisson(unb_p);\n  1 ~ poisson(pos_p);\n}\nThe parameter of poisson should be strictly positive, but unb_p is not constrained to be positive.\nPedantic mode produces the following warning.\nWarning at 'ex-dist-args.stan', line 6, column 14 to column 19:\n  A poisson distribution is given parameter unb_p as a rate parameter\n  (argument 1), but unb_p was not constrained to be strictly positive.\n\n\n\nPedantic mode checks for some specific uses of distributions that may indicate a statistical mistake:\n\n\nAny use of uniform distribution generates a warning, except when the variate parameter’s declared upper and lower bounds exactly match the uniform distribution bounds. In general, assigning a parameter a uniform distribution can create non-differentiable boundary conditions and is not recommended.\nFor example, consider the following program.\nparameters {\n  real a;\n  real&lt;lower=0, upper=1&gt; b;\n}\nmodel {\n  a ~ uniform(0, 1);\n  b ~ uniform(0, 1);\n}\na is assigned a uniform distribution that doesn’t match its constraints.\nPedantic mode produces the following warning.\nWarning at 'uniform-warn.stan', line 6, column 2 to column 20:\n  Parameter a is given a uniform distribution. The uniform distribution is\n  not recommended, for two reasons: (a) Except when there are logical or\n  physical constraints, it is very unusual for you to be sure that a\n  parameter will fall inside a specified range, and (b) The infinite gradient\n  induced by a uniform density can cause difficulties for Stan's sampling\n  algorithm. As a consequence, we recommend soft constraints rather than hard\n  constraints; for example, instead of giving an elasticity parameter a\n  uniform(0, 1) distribution, try normal(0.5, 0.5).\n\n\n\nGamma distributions are sometimes used as an attempt to assign an improper prior to a parameter. Pedantic mode gives a warning when the Gamma arguments indicate that this may be the case.\n\n\n\nAny use of the lkj_corr distribution generates a warning that suggests using the Cholesky variant instead. See the LKJ correlation distribution section of the Stan Functions Reference for details.\n\n\n\n\nA warning is generated when a parameter is declared but does not have any effect on the program. This is determined by checking whether the value of the target variable depends in any way on each of the parameters.\nFor example, consider the following program.\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(1, 1);\n}\na participates in the density function but b does not.\nPedantic mode produces the following warning.\nWarning:\n  The parameter b was declared but was not used in the density calculation.\n\n\n\nWhen numbers with magnitude less than 0.1 or greater than 10 are used as arguments to a distribution, it indicates that some parameter is not scaled to unit value, so a warning is thrown. See the efficiency tuning section of the Stan User’s guide for a discussion of scaling parameters.\nFor example, consider the following program.\nparameters {\n  real x;\n  real y;\n}\nmodel {\n  x ~ normal(-100, 100);\n  y ~ normal(0, 1);\n}\nThe constants -100 and 100 suggest that x is not unit scaled.\nPedantic mode produces the following warning.\nWarning at 'constants-warn.stan', line 6, column 14 to column 17:\n  Argument -100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\nWarning at 'constants-warn.stan', line 6, column 19 to column 22:\n  Argument 100 suggests there may be parameters that are not unit scale;\n  consider rescaling with a multiplier (see manual section 22.12).\n\n\n\nControl flow statements, such as if, for and while should not depend on parameters or functions of parameters to determine their branching conditions. This is likely to introduce a discontinuity into the density function. Pedantic mode generates a warning when any branching condition may depend on a parameter value.\nFor example, consider the following program.\nparameters {\n  real a;\n}\nmodel {\n  // x depends on parameter a\n  real x = a * a;\n\n  int m;\n\n  // the if-then-else depends on x which depends on a\n  if(x &gt; 0) {\n    //now m depends on x which depends on a\n    m = 1;\n  } else {\n    m = 2;\n  }\n\n  // for loop depends on m -&gt; x -&gt; a\n  for (i in 0:m) {\n    a ~ normal(i, 1);\n  }\n}\nThe if and for statements are control flow that depend (indirectly) on the value of the parameter m.\nPedantic mode produces the following warning.\nWarning at 'param-dep-cf-warn.stan', line 11, column 2 to line 16, column 3:\n  A control flow statement depends on parameter(s): a.\nWarning at 'param-dep-cf-warn.stan', line 19, column 2 to line 21, column 3:\n  A control flow statement depends on parameter(s): a.\n\n\n\nA warning is generated when a parameter is found on the left-hand side of more than one ~ statements (or an equivalent target += conditional density statement). This pattern is not inherently an issue, but it is unusual and may indicate a mistake.\nPedantic mode only searches for repeated statements, it will not for example generate a warning when a ~ statement is executed repeatedly inside of a loop.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n}\nmodel {\n  a ~ normal(0, 1);\n  a ~ normal(x, 1);\n\n  b ~ normal(1, 1);\n}\nPedantic mode produces the following warning.\nWarning at 'multi-tildes.stan', line 9, column 2 to column 19:\n  The parameter a is on the left-hand side of more than one tildes\n  statement.\n\n\n\nA warning is generated when a parameter appears to have greater than or less than one prior distribution factor.\nThis analysis depends on a factor graph representation of a Stan program. A factor F that depends on a parameter P is called a prior factor for P if there is no path in the factor graph from F to any data variable except through P.\nOne limitation of this approach is that the compiler cannot distinguish between modeled data variables and other convenient uses of data variables such as data sizes or hyperparameters. This warning assumes that all data variables (except for int variables) are modeled data, which may cause extra warnings.\nFor example, consider the following program.\ndata {\n  real x;\n}\nparameters {\n  real a;\n  real b;\n  real c;\n  real d;\n}\nmodel\n{\n  a ~ normal(0, 1); // this is a prior\n  x ~ normal(a, 1); // this is not a prior, since data is involved\n\n  b ~ normal(x, 1); // this is also not a prior, since data is involved\n\n  // this is not a prior for c, since data is involved through b\n  // but it is a prior for b, since the data is only involved through b\n  c ~ normal(b, 1);\n\n  //these are multiple priors:\n  d ~ normal(0, 1);\n  1 ~ normal(d, 1);\n}\nOne prior is found for a and for b, while c only has a factor that touches a data variable and d has multiple priors.\nPedantic mode produces the following warning.\nWarning:\n  The parameter c has no priors.\nWarning:\n  The parameter d has 2 priors.\n\n\n\nA warning is generated when any variable is used before it has been assigned a value.\nFor example, consider the following program.\ntransformed data {\n  real x;\n  if (1 &gt; 2) {\n    x = 1;\n  } else {\n    print(\"oops\");\n  }\n  print(x);\n}\nSince x is only assigned in one of the branches of the if statement, it might get to print(x) without having been assigned to.\nPedantic mode produces the following warning.\nWarning at 'uninit-warn.stan', line 7, column 8 to column 9:\n  The variable x may not have been assigned a value before its use.\n\n\n\nExcept when there are logical or physical constraints, it is very unusual for you to be sure that a parameter will fall inside a specified range. A warning is generated for all parameters declared with the bounds &lt;lower=.., upper=..&gt; except for &lt;lower=0, upper=1&gt; or &lt;lower=-1, upper=1&gt;.\nIn addition, a warning is generated when a parameter bound is found to have lower &gt;= upper.\nFor example, consider the following program.\nparameters {\n  real&lt;lower=0, upper=1&gt; a;\n  real&lt;lower=-1, upper=1&gt; b;\n  real&lt;lower=-2, upper=1012&gt; c;\n}\nmodel {\n  c ~ normal(b, a);\n}\nPedantic mode produces the following warning.\nWarning:\n  Your Stan program has a parameter c with a lower and upper bound in its\n  declaration. These hard constraints are not recommended, for two reasons:\n  (a) Except when there are logical or physical constraints, it is very\n  unusual for you to be sure that a parameter will fall inside a specified\n  range, and (b) The infinite gradient induced by a hard constraint can cause\n  difficulties for Stan's sampling algorithm. As a consequence, we recommend\n  soft constraints rather than hard constraints; for example, instead of\n  constraining an elasticity parameter to fall between 0, and 1, leave it\n  unconstrained and give it a normal(0.5, 0.5) prior distribution.\n\n\n\nWhen a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the Changes of variables section.\nThis portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.\nFor example, consider the following program.\nparameters {\n  real y;\n}\nmodel {\n  log(y) ~ normal(0,1);\n}\nPedantic mode produces the following warning.\nWarning:\n    Left-hand side of sampling statement (~) may contain a non-linear\n    transform of a parameter or local variable. If it does, you need\n    to include a target += statement with the log absolute determinant\n    of the Jacobian of the transform.\n\n\n\n\nConstant values are sometimes uncomputable\nPedantic mode attempts to evaluate expressions down to literal values so that they can be used to generate warnings. For example, in the code normal(x, 1 - 2), the expression 1 - 2 will be evaluated to -1, which is not a valid variance argument so a warning is generated. However, this strategy is limited; it is often impossible to fully evaluate expressions in finite time.\nContainer types\nCurrently, indexed variables are not handled intelligently, so they are treated as monolithic variables. Each analysis treats indexed variables conservatively (erring toward generating fewer warnings).\nData variables\nThe declaration information for data variables is currently not considered, so using data as incompatible arguments to distributions may not generate the appropriate warnings.\nControl flow dependent on parameters in nested functions\nIf a parameter is passed as an argument to a user-defined function within another user-defined function, and then some control flow depends on that argument, the appropriate warning will not be thrown.",
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     "title": "Using the Stan Compiler",
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-    "text": "In addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.",
+    "text": "In addition to compiling Stan programs, stanc3 features several flags which can be used to format Stan programs and update them to the most recent Stan syntax by removing any deprecation features which can be automatically replaced.\nThese flags work for both .stan model files and .stanfunctions function files. They can be combined with --o to redirect the formatted output to a new file.\n\n\nInvoking stanc --auto-format &lt;model_file&gt; will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the Stan Program Style Guide. This means spacing, indentation, and line length are all regularized. Some deprecated features, like the use of # for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.\nBy default, this will try to split lines at or before column 78. This number can be changed using --max-line-length.\n\n\n\nIn addition to automatic formatting, stanc can also “canonicalize” programs by updating deprecated syntax, removing unnecessary parenthesis, and adding braces around bodies of if statements and for and while loops.\nThis can be done by using stanc --auto-format --canonicalize=... where ... is a comma-separated list of options. Currently these options are:\n\ndeprecations\nRemoves deprecated syntax such as replacing deprecated functions with their drop-in replacements.\nparentheses\nRemoves unnecessary extra parentheses, such as converting y = ((x-1)) to y = x - 1\nbraces\nPlaces braces around all blocks. For example, the following statement\nif (cond)\n  //result\nwill be formatted as\nif (cond) {\n  //result\n}\nand similarly for both kinds of loops containing a single statement.\nincludes\nThis will pretty-print code from other files included with #include as part of the program. This was the default behavior prior to Stan 2.29. When not enabled, the pretty-printer output will include the same #include directives as the input program.\n\nInvoking stanc --print-canonical &lt;model_file&gt; is synonymous with running stanc --auto-format --canonicalize=deprecations,braces,parentheses,includes\n\n\n\nThe formatting and canonicalizing features of stanc3 are still under development. The following are some known issues one should be aware of before using either:\n\nOddly placed comments\nIf your Stan program features comments in unexpected places, such as inside an expression, they may be moved in the process of formatting. Moved comments are prefixed with the string ^^^: to indicate they originally appeared higher in the program.\nWe hope to improve this functionality in future versions. For now, this can usually be avoided by manually moving the comment outside of an expression, either by placing it on its own line or following a separator such as a comma or keyword.\nFailure to recreate strange #include structure\nPrinting without include inlining (--canonicalize=includes) can fail when includes were used in atypical locations, such as in the middle of statements. We recommend either printing with inlining enabled or reconsidering the use of includes in this way.",
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     "title": "Posterior and Prior Predictive Checks",
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-    "text": "Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.\nPrior predictive checks evaluate the prior the same way. Specifically, they evaluate what data sets would be consistent with the prior. They will not be calibrated with actual data, but extreme values help diagnose priors that are either too strong, too weak, poorly shaped, or poorly located.\nPrior and posterior predictive checks are two cases of the general concept of predictive checks, just conditioning on different things (no data and the observed data, respectively). For hierarchical models, there are intermediate versions, as discussed in the section on hierarchical models and mixed replication.\n\n\nThe posterior predictive distribution is the distribution over new observations given previous observations. It’s predictive in the sense that it’s predicting behavior on new data that is not part of the training set. It’s posterior in that everything is conditioned on observed data \\(y\\).\nThe posterior predictive distribution for replications \\(y^{\\textrm{rep}}\\) of the original data set \\(y\\) given model parameters \\(\\theta\\) is defined by \\[\np(y^{\\textrm{rep}} \\mid y)\n= \\int p(y^{\\textrm{rep}} \\mid \\theta)\n       \\cdot p(\\theta \\mid y) \\, \\textrm{d}\\theta.\n\\]\nAs with other posterior predictive quantities, generating a replicated data set \\(y^{\\textrm{rep}}\\) from the posterior predictive distribution is straightforward using the generated quantities block. Consider a simple regression model with parameters \\(\\theta = (\\alpha, \\beta, \\sigma).\\)\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n  vector[N] y;\n}\nparameters {\n  real alpha;\n  real beta;\n  real&lt;lower=0&gt; sigma;\n}\nmodel {\n  alpha ~ normal(0, 2);\n  beta ~ normal(0, 1);\n  sigma ~ normal(0, 1);\n  y ~ normal(alpha + beta * x, sigma);\n}\nTo generate a replicated data set y_rep for this simple model, the following generated quantities block suffices.\ngenerated quantities {\n  array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n}\nThe vectorized form of the normal random number generator is used with the original predictors x and the model parameters alpha, beta, and sigma. The replicated data variable y_rep is declared to be the same size as the original data y, but instead of a vector type, it is declared to be an array of reals to match the return type of the function normal_rng. Because the vector and real array types have the same dimensions and layout, they can be plotted against one another and otherwise compared during downstream processing.\nThe posterior predictive sampling for posterior predictive checks is different from usual posterior predictive sampling discussed in the chapter on posterior predictions in that the original predictors \\(x\\) are used. That is, the posterior predictions are for the original data.\n\n\n\nA standard posterior predictive check would plot a histogram of each replicated data set along with the original data set and compare them by eye. For this purpose, only a few replications are needed. These should be taken by thinning a larger set of replications down to the size needed to ensure rough independence of the replications.\nHere’s a complete example where the model is a simple Poisson with a weakly informative exponential prior with a mean of 10 and standard deviation of 10.\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0&gt; y;\n}\ntransformed data {\n  real&lt;lower=0&gt; mean_y = mean(to_vector(y));\n  real&lt;lower=0&gt; sd_y = sd(to_vector(y));\n}\nparameters {\n  real&lt;lower=0&gt; lambda;\n}\nmodel {\n  y ~ poisson(lambda);\n  lambda ~ exponential(0.2);\n}\ngenerated quantities {\n  array[N] int&lt;lower=0&gt; y_rep = poisson_rng(rep_array(lambda, N));\n  real&lt;lower=0&gt; mean_y_rep = mean(to_vector(y_rep));\n  real&lt;lower=0&gt; sd_y_rep = sd(to_vector(y_rep));\n  int&lt;lower=0, upper=1&gt; mean_gte = (mean_y_rep &gt;= mean_y);\n  int&lt;lower=0, upper=1&gt; sd_gte = (sd_y_rep &gt;= sd_y);\n}\nThe generated quantities block creates a variable y_rep for the replicated data, variables mean_y_rep and sd_y_rep for the statistics of the replicated data, and indicator variables mean_gte and sd_gte for whether the replicated statistic is greater than or equal to the statistic applied to the original data.\nNow consider generating data \\(y \\sim \\textrm{Poisson}(5)\\). The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\n\n\nWith a Poisson data-generating process and Poisson model, the posterior replications look similar to the original data. If it were easy to pick the original data out of the lineup, there would be a problem.\nNow consider generating over-dispersed data \\(y \\sim \\textrm{negative-binomial2}(5, 1).\\) This has the same mean as \\(\\textrm{Poisson}(5)\\), namely \\(5\\), but a standard deviation of \\(\\sqrt{5 + 5^2 /1} \\approx 5.5.\\) There is no way to fit this data with the Poisson model, because a variable distributed as \\(\\textrm{Poisson}(\\lambda)\\) has mean \\(\\lambda\\) and standard deviation \\(\\sqrt{\\lambda},\\) which is \\(\\sqrt{5}\\) for \\(\\textrm{Poisson}(5).\\) Here’s the resulting small multiples plot, again with original data in the upper left.\n\n\n\nPosterior predictive checks for negative binomial data generating process and Poisson model.\n\n\nThis time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.\n\n\n\nIf a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.\n\n\n\n\nPrior predictive checks generate data according to the prior in order to asses whether a prior is appropriate (Gabry et al. 2019). A posterior predictive check generates replicated data according to the posterior predictive distribution. In contrast, the prior predictive check generates data according to the prior predictive distribution, \\[\ny^{\\textrm{sim}} \\sim p(y).\n\\] The prior predictive distribution is just like the posterior predictive distribution with no observed data, so that a prior predictive check is nothing more than the limiting case of a posterior predictive check with no data.\nThis is easy to carry out mechanically by simulating parameters \\[\n\\theta^{\\textrm{sim}} \\sim p(\\theta)\n\\] according to the priors, then simulating data \\[\ny^{\\textrm{sim}} \\sim p(y \\mid \\theta^{\\textrm{sim}})\n\\] according to the sampling distribution given the simulated parameters. The result is a simulation from the joint distribution, \\[\n(y^{\\textrm{sim}}, \\theta^{\\textrm{sim}}) \\sim p(y, \\theta)\n\\] and thus \\[\ny^{\\textrm{sim}} \\sim p(y)\n\\] is a simulation from the prior predictive distribution.\n\n\nA prior predictive check is coded just like a posterior predictive check. If a posterior predictive check has already been coded and it’s possible to set the data to be empty, then no additional coding is necessary. The disadvantage to coding prior predictive checks as posterior predictive checks with no data is that Markov chain Monte Carlo will be used to sample the parameters, which is less efficient than taking independent draws using random number generation.\nPrior predictive checks can be coded entirely within the generated quantities block using random number generation. The resulting draws will be independent. Predictors must be read in from the actual data set—they do not have a generative model from which to be simulated. For a Poisson regression, prior predictive sampling can be encoded as the following complete Stan program.\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n}\ngenerated quantities {\n  real alpha = normal_rng(0, 1);\n  real beta = normal_rng(0, 1);\n  array[N] real y_sim = poisson_log_rng(alpha + beta * x);\n}\nRunning this program using Stan’s fixed-parameter sampler yields draws from the prior. These may be plotted to consider their appropriateness.\n\n\n\n\nSuppose we have a model for a football (aka soccer) league where there are \\(J\\) teams. Each team has a scoring rate \\(\\lambda_j\\) and in each game will be assumed to score \\(\\textrm{poisson}(\\lambda_j)\\) points. Yes, this model completely ignores defense. Suppose the modeler does not want to “put their thumb on the scale” and would rather “let the data speak for themselves” and so uses a prior with very wide tails, because it seems uninformative, such as the widely deployed \\[\n\\lambda_j \\sim \\textrm{gamma}(\\epsilon_1, \\epsilon_2).\n\\] This is not just a manufactured example; The BUGS Book recommends setting \\(\\epsilon = (0.5, 0.00001)\\), which corresponds to a Jeffreys prior for a Poisson rate parameter prior (Lunn et al. 2012, 85).\nSuppose the league plays a round-robin tournament wherein every team plays every other team. The following Stan model generates random team abilities and the results of such a round-robin tournament, which may be used to perform prior predictive checks.\ndata {\n  int&lt;lower=0&gt; J;\n  array[2] real&lt;lower=0&gt; epsilon;\n}\ngenerated quantities {\n  array[J] real&lt;lower=0&gt; lambda;\n  array[J, J] int y;\n  for (j in 1:J) lambda[j] = gamma_rng(epsilon[1], epsilon[2]);\n  for (i in 1:J) {\n    for (j in 1:J) {\n      y[i, j] = poisson_rng(lambda[i]) - poisson_rng(lambda[j]);\n    }\n  }\n}\nIn this simulation, teams play each other twice and play themselves once. This could be made more realistic by controlling the combinatorics to only generate a single result for each pair of teams, of which there are \\(\\binom{J}{2} = \\frac{J \\cdot (J - 1)}{2}.\\)\nUsing the \\(\\textrm{gamma}(0.5, 0.00001)\\) reference prior on team abilities, the following are the first 20 simulated point differences for the match between the first two teams, \\(y^{(1:20)}_{1, 2}\\).\n2597 -26000   5725  22496   1270   1072   4502  -2809   -302   4987\n7513   7527  -3268 -12374   3828   -158 -29889   2986  -1392     66\nThat’s some pretty highly scoring football games being simulated; all but one has a score differential greater than 100! In other words, this \\(\\textrm{gamma}(0.5, 0.00001)\\) prior is putting around 95% of its weight on score differentials above 100. Given that two teams combined rarely score 10 points, this prior is way out of line with prior knowledge about football matches; it is not only consistent with outcomes that have never occurred in the history of the sport, it puts most of the prior probability mass there.\nThe posterior predictive distribution can be strongly affected by the prior when there is not much observed data and substantial prior mass is concentrated around infeasible values (A. Gelman 2006).\nJust as with posterior predictive distributions, any statistics of the generated data may be evaluated. Here, the focus was on score difference between a single pair of teams, but it could’ve been on maximums, minimums, averages, variances, etc.\nIn this textbook example, the prior is univariate and directly related to the expected number of points scored, and could thus be directly inspected for consistency with prior knowledge about scoring rates in football. There will not be the same kind of direct connection when the prior and sampling distributions are multivariate. In these more challenging situations, prior predictive checks are an easy way to get a handle on the implications of a prior in terms of what it says the data is going to look like; for a more complex application involving spatially heterogeneous air pollution concentration, see (Gabry et al. 2019).\nPrior predictive checks can also be compared with the data, but one should not expect them to be calibrated in the same way as posterior predictive checks. That would require guessing the posterior and encoding it in the prior. The goal is make sure the prior is not so wide that it will pull probability mass away from feasible values.\n\n\n\nAndrew Gelman, Meng, and Stern (1996) discuss the case of mixed replication for hierarchical models in which the hyperparameters remain fixed, but varying effects are replicated. This is neither a purely prior nor purely posterior predictive check, but falls somewhere in between.\nFor example, consider a simple varying intercept logistic regression, with intercepts \\(\\alpha_k\\) for \\(k \\in 1:K\\). Each data item \\(y_n \\in \\{ 0, 1 \\}\\) is assumed to correspond to group \\(kk_n \\in 1:K.\\) The sampling distribution is thus \\[\ny_n \\sim \\textrm{bernoulli}(\\textrm{logit}^{-1}(\\alpha_{kk[n]})).\n\\] The varying intercepts have a hierarchical normal prior, \\[\n\\alpha_k \\sim \\textrm{normal}(\\mu, \\sigma).\n\\] The hyperparameters are themselves given weakly informative priors, \\[\\begin{eqnarray*}\n\\mu & \\sim & \\textrm{normal}(0, 2)\n\\\\[4pt]\n\\sigma & \\sim & \\textrm{lognormal}(0, 1).\n\\end{eqnarray*}\\]\nLike in a posterior predictive check, the hyperparameters \\(\\mu\\) and \\(\\sigma\\) are drawn from the posterior, \\[\n\\mu^{(m)}, \\sigma^{(m)} \\sim p(\\mu, \\sigma \\mid y)\n\\] Like in a prior predictive check, replicated values of \\(\\alpha\\) are drawn from the hyperparameters, \\[\n\\alpha^{\\textrm{rep}(m)}_k \\sim \\textrm{normal}(\\alpha_k \\mid\n\\mu^{(m)}, \\sigma^{(m)}).\n\\] The data items are then each replicated using the replicated intercepts, \\[\ny^{\\textrm{rep}(m)}_n \\sim\n\\textrm{bernoulli}\n  (\\textrm{logit}^{-1}(\\alpha^{\\textrm{rep}(m)}_{kk[n]})).\n\\] Thus the \\(y^{\\textrm{rep}(m)}\\) can be seen as a kind of posterior predictive replication of observations from new groups that were not among the original \\(K\\) groups.\nIn Stan, mixed predictive replications \\(y^{\\textrm{rep}(m)}\\) can be programmed directly.\ndata {\n  int&lt;lower=0&gt; K;\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=1, upper=K&gt; kk;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real mu;\n  real&lt;lower=0&gt; sigma;\n  vector&lt;offset=mu, multiplier=sigma&gt;[K] alpha;\n}\nmodel {\n  mu ~ normal(0, 2);               // hyperprior\n  sigma ~ lognormal(0, 1);\n  alpha ~ normal(mu, sigma);       // hierarchical prior\n  y ~ bernoulli_logit(alpha[kk]);  // sampling distribution\n}\ngenerated quantities {\n  // alpha replicated;  mu and sigma not replicated\n  array[K] real alpha_rep\n    = normal_rng(rep_vector(mu, K), sigma);\n  array[N] int&lt;lower=0, upper=1&gt; y_rep\n    = bernoulli_logit_rng(alpha_rep[kk]);\n}\n\n\n\nFollowing Andrew Gelman, Meng, and Stern (1996), prior, posterior, and mixed replications may all be defined as posteriors from joint models over parameters and observed and replicated data.\n\n\nFor example, posterior predictive replication may be formulated using sampling notation as follows. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] The heavily overloaded sampling notation is meant to indicate that both \\(y\\) and \\(y^{\\textrm{rep}}\\) are drawn from the same distribution, or more formally using capital letters to distinguish random variables, that the conditional densities \\(p_{Y^{\\textrm{rep}} \\mid\n\\Theta}\\) and \\(p_{Y \\mid \\Theta}\\) are the same.\nThe joint density is \\[\np(\\theta, y, y^{\\textrm{rep}})\n= p(\\theta) \\cdot p(y \\mid \\theta) \\cdot p(y^{\\textrm{rep}} \\mid \\theta).\n\\] This again is assuming that the two distributions for \\(y\\) and \\(y^{\\textrm{rep}}\\) are identical.\nThe variable \\(y\\) is observed, with the predictive simulation \\(y^{\\textrm{rep}}\\) and parameter vector \\(\\theta\\) not observed. The posterior is \\(p(y^{\\textrm{rep}}, \\theta \\mid y)\\). Given draws from the posterior, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are retained.\n\n\n\nThe prior predictive model simply drops the data component of the posterior predictive model. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] This corresponds to the joint density \\[\np(\\theta, y^{\\textrm{rep}}) = p(\\theta) \\cdot p(y^{\\textrm{rep}} \\mid\n\\theta).\n\\]\nIt is typically straightforward to draw \\(\\theta\\) from the prior and \\(y^{\\textrm{rep}}\\) from the sampling distribution given \\(\\theta\\) efficiently. In cases where it is not, the model may be coded and executed just as the posterior predictive model, only with no data.\n\n\n\nThe mixed replication corresponds to the model \\[\\begin{eqnarray*}\n\\phi & \\sim & p(\\phi)\n\\\\[2pt]\n\\alpha & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\alpha)\n\\\\[2pt]\n\\alpha^{\\textrm{rep}} & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\phi)\n\\end{eqnarray*}\\] The notation here is meant to indicate that \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\) have identical distributions, as do \\(y\\) and \\(y^{\\textrm{rep}}\\).\nThis corresponds to a joint model \\[\np(\\phi, \\alpha, \\alpha^{\\textrm{rep}}, y, y^{\\textrm{rep}})\n=\np(\\phi)\n\\cdot p(\\alpha \\mid \\phi)\n\\cdot p(y \\mid \\alpha)\n\\cdot p(\\alpha^{\\textrm{rep}} \\mid \\phi)\n\\cdot p(y^{\\textrm{rep}} \\mid \\alpha^{\\textrm{rep}}),\n\\] where \\(y\\) is the only observed variable, \\(\\alpha\\) contains the lower-level parameters and \\(\\phi\\) the hyperparameters. Note that \\(\\phi\\) is not replicated and instead appears in the distribution for both \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\).\nThe posterior is \\(p(\\phi, \\alpha, \\alpha^{\\textrm{rep}},\ny^{\\textrm{rep}} \\mid y)\\). From posterior draws, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are kept.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\nPosterior predictive checks for negative binomial data generating process and Poisson model.\nHistogram of means of replicated data sets; vertical red line at mean of original data.\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.",
+    "text": "Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.\nPrior predictive checks evaluate the prior the same way. Specifically, they evaluate what data sets would be consistent with the prior. They will not be calibrated with actual data, but extreme values help diagnose priors that are either too strong, too weak, poorly shaped, or poorly located.\nPrior and posterior predictive checks are two cases of the general concept of predictive checks, just conditioning on different things (no data and the observed data, respectively). For hierarchical models, there are intermediate versions, as discussed in the section on hierarchical models and mixed replication.\n\n\nThe posterior predictive distribution is the distribution over new observations given previous observations. It’s predictive in the sense that it’s predicting behavior on new data that is not part of the training set. It’s posterior in that everything is conditioned on observed data \\(y\\).\nThe posterior predictive distribution for replications \\(y^{\\textrm{rep}}\\) of the original data set \\(y\\) given model parameters \\(\\theta\\) is defined by \\[\np(y^{\\textrm{rep}} \\mid y)\n= \\int p(y^{\\textrm{rep}} \\mid \\theta)\n       \\cdot p(\\theta \\mid y) \\, \\textrm{d}\\theta.\n\\]\nAs with other posterior predictive quantities, generating a replicated data set \\(y^{\\textrm{rep}}\\) from the posterior predictive distribution is straightforward using the generated quantities block. Consider a simple regression model with parameters \\(\\theta = (\\alpha, \\beta, \\sigma).\\)\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n  vector[N] y;\n}\nparameters {\n  real alpha;\n  real beta;\n  real&lt;lower=0&gt; sigma;\n}\nmodel {\n  alpha ~ normal(0, 2);\n  beta ~ normal(0, 1);\n  sigma ~ normal(0, 1);\n  y ~ normal(alpha + beta * x, sigma);\n}\nTo generate a replicated data set y_rep for this simple model, the following generated quantities block suffices.\ngenerated quantities {\n  array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n}\nThe vectorized form of the normal random number generator is used with the original predictors x and the model parameters alpha, beta, and sigma. The replicated data variable y_rep is declared to be the same size as the original data y, but instead of a vector type, it is declared to be an array of reals to match the return type of the function normal_rng. Because the vector and real array types have the same dimensions and layout, they can be plotted against one another and otherwise compared during downstream processing.\nThe posterior predictive sampling for posterior predictive checks is different from usual posterior predictive sampling discussed in the chapter on posterior predictions in that the original predictors \\(x\\) are used. That is, the posterior predictions are for the original data.\n\n\n\nA standard posterior predictive check would plot a histogram of each replicated data set along with the original data set and compare them by eye. For this purpose, only a few replications are needed. These should be taken by thinning a larger set of replications down to the size needed to ensure rough independence of the replications.\nHere’s a complete example where the model is a simple Poisson with a weakly informative exponential prior with a mean of 10 and standard deviation of 10.\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0&gt; y;\n}\ntransformed data {\n  real&lt;lower=0&gt; mean_y = mean(to_vector(y));\n  real&lt;lower=0&gt; sd_y = sd(to_vector(y));\n}\nparameters {\n  real&lt;lower=0&gt; lambda;\n}\nmodel {\n  y ~ poisson(lambda);\n  lambda ~ exponential(0.2);\n}\ngenerated quantities {\n  array[N] int&lt;lower=0&gt; y_rep = poisson_rng(rep_array(lambda, N));\n  real&lt;lower=0&gt; mean_y_rep = mean(to_vector(y_rep));\n  real&lt;lower=0&gt; sd_y_rep = sd(to_vector(y_rep));\n  int&lt;lower=0, upper=1&gt; mean_gte = (mean_y_rep &gt;= mean_y);\n  int&lt;lower=0, upper=1&gt; sd_gte = (sd_y_rep &gt;= sd_y);\n}\nThe generated quantities block creates a variable y_rep for the replicated data, variables mean_y_rep and sd_y_rep for the statistics of the replicated data, and indicator variables mean_gte and sd_gte for whether the replicated statistic is greater than or equal to the statistic applied to the original data.\nNow consider generating data \\(y \\sim \\textrm{Poisson}(5)\\). The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\n\n\nWith a Poisson data-generating process and Poisson model, the posterior replications look similar to the original data. If it were easy to pick the original data out of the lineup, there would be a problem.\nNow consider generating over-dispersed data \\(y \\sim \\textrm{negative-binomial2}(5, 1).\\) This has the same mean as \\(\\textrm{Poisson}(5)\\), namely \\(5\\), but a standard deviation of \\(\\sqrt{5 + 5^2 /1} \\approx 5.5.\\) There is no way to fit this data with the Poisson model, because a variable distributed as \\(\\textrm{Poisson}(\\lambda)\\) has mean \\(\\lambda\\) and standard deviation \\(\\sqrt{\\lambda},\\) which is \\(\\sqrt{5}\\) for \\(\\textrm{Poisson}(5).\\) Here’s the resulting small multiples plot, again with original data in the upper left.\n\n\n\nPosterior predictive checks for negative binomial data generating process and Poisson model.\n\n\nThis time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.\n\n\n\nIf a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.\n\n\n\n\nPrior predictive checks generate data according to the prior in order to asses whether a prior is appropriate (Gabry et al. 2019). A posterior predictive check generates replicated data according to the posterior predictive distribution. In contrast, the prior predictive check generates data according to the prior predictive distribution, \\[\ny^{\\textrm{sim}} \\sim p(y).\n\\] The prior predictive distribution is just like the posterior predictive distribution with no observed data, so that a prior predictive check is nothing more than the limiting case of a posterior predictive check with no data.\nThis is easy to carry out mechanically by simulating parameters \\[\n\\theta^{\\textrm{sim}} \\sim p(\\theta)\n\\] according to the priors, then simulating data \\[\ny^{\\textrm{sim}} \\sim p(y \\mid \\theta^{\\textrm{sim}})\n\\] according to the sampling distribution given the simulated parameters. The result is a simulation from the joint distribution, \\[\n(y^{\\textrm{sim}}, \\theta^{\\textrm{sim}}) \\sim p(y, \\theta)\n\\] and thus \\[\ny^{\\textrm{sim}} \\sim p(y)\n\\] is a simulation from the prior predictive distribution.\n\n\nA prior predictive check is coded just like a posterior predictive check. If a posterior predictive check has already been coded and it’s possible to set the data to be empty, then no additional coding is necessary. The disadvantage to coding prior predictive checks as posterior predictive checks with no data is that Markov chain Monte Carlo will be used to sample the parameters, which is less efficient than taking independent draws using random number generation.\nPrior predictive checks can be coded entirely within the generated quantities block using random number generation. The resulting draws will be independent. Predictors must be read in from the actual data set—they do not have a generative model from which to be simulated. For a Poisson regression, prior predictive sampling can be encoded as the following complete Stan program.\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] x;\n}\ngenerated quantities {\n  real alpha = normal_rng(0, 1);\n  real beta = normal_rng(0, 1);\n  array[N] real y_sim = poisson_log_rng(alpha + beta * x);\n}\nRunning this program using Stan’s fixed-parameter sampler yields draws from the prior. These may be plotted to consider their appropriateness.\n\n\n\n\nSuppose we have a model for a football (aka soccer) league where there are \\(J\\) teams. Each team has a scoring rate \\(\\lambda_j\\) and in each game will be assumed to score \\(\\textrm{poisson}(\\lambda_j)\\) points. Yes, this model completely ignores defense. Suppose the modeler does not want to “put their thumb on the scale” and would rather “let the data speak for themselves” and so uses a prior with very wide tails, because it seems uninformative, such as the widely deployed \\[\n\\lambda_j \\sim \\textrm{gamma}(\\epsilon_1, \\epsilon_2).\n\\] This is not just a manufactured example; The BUGS Book recommends setting \\(\\epsilon = (0.5, 0.00001)\\), which corresponds to a Jeffreys prior for a Poisson rate parameter prior (Lunn et al. 2012, 85).\nSuppose the league plays a round-robin tournament wherein every team plays every other team. The following Stan model generates random team abilities and the results of such a round-robin tournament, which may be used to perform prior predictive checks.\ndata {\n  int&lt;lower=0&gt; J;\n  array[2] real&lt;lower=0&gt; epsilon;\n}\ngenerated quantities {\n  array[J] real&lt;lower=0&gt; lambda;\n  array[J, J] int y;\n  for (j in 1:J) lambda[j] = gamma_rng(epsilon[1], epsilon[2]);\n  for (i in 1:J) {\n    for (j in 1:J) {\n      y[i, j] = poisson_rng(lambda[i]) - poisson_rng(lambda[j]);\n    }\n  }\n}\nIn this simulation, teams play each other twice and play themselves once. This could be made more realistic by controlling the combinatorics to only generate a single result for each pair of teams, of which there are \\(\\binom{J}{2} = \\frac{J \\cdot (J - 1)}{2}.\\)\nUsing the \\(\\textrm{gamma}(0.5, 0.00001)\\) reference prior on team abilities, the following are the first 20 simulated point differences for the match between the first two teams, \\(y^{(1:20)}_{1, 2}\\).\n2597 -26000   5725  22496   1270   1072   4502  -2809   -302   4987\n7513   7527  -3268 -12374   3828   -158 -29889   2986  -1392     66\nThat’s some pretty highly scoring football games being simulated; all but one has a score differential greater than 100! In other words, this \\(\\textrm{gamma}(0.5, 0.00001)\\) prior is putting around 95% of its weight on score differentials above 100. Given that two teams combined rarely score 10 points, this prior is way out of line with prior knowledge about football matches; it is not only consistent with outcomes that have never occurred in the history of the sport, it puts most of the prior probability mass there.\nThe posterior predictive distribution can be strongly affected by the prior when there is not much observed data and substantial prior mass is concentrated around infeasible values (A. Gelman 2006).\nJust as with posterior predictive distributions, any statistics of the generated data may be evaluated. Here, the focus was on score difference between a single pair of teams, but it could’ve been on maximums, minimums, averages, variances, etc.\nIn this textbook example, the prior is univariate and directly related to the expected number of points scored, and could thus be directly inspected for consistency with prior knowledge about scoring rates in football. There will not be the same kind of direct connection when the prior and sampling distributions are multivariate. In these more challenging situations, prior predictive checks are an easy way to get a handle on the implications of a prior in terms of what it says the data is going to look like; for a more complex application involving spatially heterogeneous air pollution concentration, see (Gabry et al. 2019).\nPrior predictive checks can also be compared with the data, but one should not expect them to be calibrated in the same way as posterior predictive checks. That would require guessing the posterior and encoding it in the prior. The goal is make sure the prior is not so wide that it will pull probability mass away from feasible values.\n\n\n\nAndrew Gelman, Meng, and Stern (1996) discuss the case of mixed replication for hierarchical models in which the hyperparameters remain fixed, but varying effects are replicated. This is neither a purely prior nor purely posterior predictive check, but falls somewhere in between.\nFor example, consider a simple varying intercept logistic regression, with intercepts \\(\\alpha_k\\) for \\(k \\in 1:K\\). Each data item \\(y_n \\in \\{ 0, 1 \\}\\) is assumed to correspond to group \\(kk_n \\in 1:K.\\) The sampling distribution is thus \\[\ny_n \\sim \\textrm{bernoulli}(\\textrm{logit}^{-1}(\\alpha_{kk[n]})).\n\\] The varying intercepts have a hierarchical normal prior, \\[\n\\alpha_k \\sim \\textrm{normal}(\\mu, \\sigma).\n\\] The hyperparameters are themselves given weakly informative priors, \\[\\begin{eqnarray*}\n\\mu & \\sim & \\textrm{normal}(0, 2)\n\\\\[4pt]\n\\sigma & \\sim & \\textrm{lognormal}(0, 1).\n\\end{eqnarray*}\\]\nLike in a posterior predictive check, the hyperparameters \\(\\mu\\) and \\(\\sigma\\) are drawn from the posterior, \\[\n\\mu^{(m)}, \\sigma^{(m)} \\sim p(\\mu, \\sigma \\mid y)\n\\] Like in a prior predictive check, replicated values of \\(\\alpha\\) are drawn from the hyperparameters, \\[\n\\alpha^{\\textrm{rep}(m)}_k \\sim \\textrm{normal}(\\alpha_k \\mid\n\\mu^{(m)}, \\sigma^{(m)}).\n\\] The data items are then each replicated using the replicated intercepts, \\[\ny^{\\textrm{rep}(m)}_n \\sim\n\\textrm{bernoulli}\n  (\\textrm{logit}^{-1}(\\alpha^{\\textrm{rep}(m)}_{kk[n]})).\n\\] Thus the \\(y^{\\textrm{rep}(m)}\\) can be seen as a kind of posterior predictive replication of observations from new groups that were not among the original \\(K\\) groups.\nIn Stan, mixed predictive replications \\(y^{\\textrm{rep}(m)}\\) can be programmed directly.\ndata {\n  int&lt;lower=0&gt; K;\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=1, upper=K&gt; kk;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real mu;\n  real&lt;lower=0&gt; sigma;\n  vector&lt;offset=mu, multiplier=sigma&gt;[K] alpha;\n}\nmodel {\n  mu ~ normal(0, 2);               // hyperprior\n  sigma ~ lognormal(0, 1);\n  alpha ~ normal(mu, sigma);       // hierarchical prior\n  y ~ bernoulli_logit(alpha[kk]);  // sampling distribution\n}\ngenerated quantities {\n  // alpha replicated;  mu and sigma not replicated\n  array[K] real alpha_rep\n    = normal_rng(rep_vector(mu, K), sigma);\n  array[N] int&lt;lower=0, upper=1&gt; y_rep\n    = bernoulli_logit_rng(alpha_rep[kk]);\n}\n\n\n\nFollowing Andrew Gelman, Meng, and Stern (1996), prior, posterior, and mixed replications may all be defined as posteriors from joint models over parameters and observed and replicated data.\n\n\nFor example, posterior predictive replication may be formulated using sampling notation as follows. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] The heavily overloaded sampling notation is meant to indicate that both \\(y\\) and \\(y^{\\textrm{rep}}\\) are drawn from the same distribution, or more formally using capital letters to distinguish random variables, that the conditional densities \\(p_{Y^{\\textrm{rep}} \\mid\n\\Theta}\\) and \\(p_{Y \\mid \\Theta}\\) are the same.\nThe joint density is \\[\np(\\theta, y, y^{\\textrm{rep}})\n= p(\\theta) \\cdot p(y \\mid \\theta) \\cdot p(y^{\\textrm{rep}} \\mid \\theta).\n\\] This again is assuming that the two distributions for \\(y\\) and \\(y^{\\textrm{rep}}\\) are identical.\nThe variable \\(y\\) is observed, with the predictive simulation \\(y^{\\textrm{rep}}\\) and parameter vector \\(\\theta\\) not observed. The posterior is \\(p(y^{\\textrm{rep}}, \\theta \\mid y)\\). Given draws from the posterior, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are retained.\n\n\n\nThe prior predictive model simply drops the data component of the posterior predictive model. \\[\\begin{eqnarray*}\n\\theta & \\sim & p(\\theta)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\theta)\n\\end{eqnarray*}\\] This corresponds to the joint density \\[\np(\\theta, y^{\\textrm{rep}}) = p(\\theta) \\cdot p(y^{\\textrm{rep}} \\mid\n\\theta).\n\\]\nIt is typically straightforward to draw \\(\\theta\\) from the prior and \\(y^{\\textrm{rep}}\\) from the sampling distribution given \\(\\theta\\) efficiently. In cases where it is not, the model may be coded and executed just as the posterior predictive model, only with no data.\n\n\n\nThe mixed replication corresponds to the model \\[\\begin{eqnarray*}\n\\phi & \\sim & p(\\phi)\n\\\\[2pt]\n\\alpha & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny & \\sim & p(y \\mid \\alpha)\n\\\\[2pt]\n\\alpha^{\\textrm{rep}} & \\sim & p(\\alpha \\mid \\phi)\n\\\\[2pt]\ny^{\\textrm{rep}} & \\sim & p(y \\mid \\phi)\n\\end{eqnarray*}\\] The notation here is meant to indicate that \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\) have identical distributions, as do \\(y\\) and \\(y^{\\textrm{rep}}\\).\nThis corresponds to a joint model \\[\np(\\phi, \\alpha, \\alpha^{\\textrm{rep}}, y, y^{\\textrm{rep}})\n=\np(\\phi)\n\\cdot p(\\alpha \\mid \\phi)\n\\cdot p(y \\mid \\alpha)\n\\cdot p(\\alpha^{\\textrm{rep}} \\mid \\phi)\n\\cdot p(y^{\\textrm{rep}} \\mid \\alpha^{\\textrm{rep}}),\n\\] where \\(y\\) is the only observed variable, \\(\\alpha\\) contains the lower-level parameters and \\(\\phi\\) the hyperparameters. Note that \\(\\phi\\) is not replicated and instead appears in the distribution for both \\(\\alpha\\) and \\(\\alpha^{\\textrm{rep}}\\).\nThe posterior is \\(p(\\phi, \\alpha, \\alpha^{\\textrm{rep}},\ny^{\\textrm{rep}} \\mid y)\\). From posterior draws, the posterior predictive simulations \\(y^{\\textrm{rep}}\\) are kept.\n\n\n\nPosterior predictive checks for Poisson data generating process and Poisson model.\nPosterior predictive checks for negative binomial data generating process and Poisson model.\nHistogram of means of replicated data sets; vertical red line at mean of original data.\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.",
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     "href": "stan-users-guide/posterior-predictive-checks.html#posterior-p-values",
     "title": "Posterior and Prior Predictive Checks",
     "section": "",
-    "text": "If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.",
+    "text": "If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic \\(s(\\cdot)\\) in a replicated data set exceeds that in the original data, \\[\n\\Pr\\!\\left[ s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right]\n=\n\\int\n\\textrm{I}\\left( s(y^{\\textrm{rep}}) \\geq s(y) \\mid y \\right)\n\\cdot p\\left( y^{\\textrm{rep}} \\mid y \\right)\n\\, \\textrm{d}{y^{\\textrm{rep}}}.\n\\] It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified (Bayarri and Berger 2000).\nNevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.\nTo calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.\ngenerated quantities {\n  int&lt;lower=0, upper=1&gt; mean_gt;\n  int&lt;lower=0, upper=1&gt; sd_gt;\n  {\n    array[N] real y_rep = normal_rng(alpha + beta * x, sigma);\n    mean_gt = mean(y_rep) &gt; mean(y);\n    sd_gt = sd(y_rep) &gt; sd(y);\n  }\n}\nThe indicator variable mean_gt will have value 1 if the mean of the simulated data y_rep is greater than or equal to the mean of he original data y. Because the values of y_rep are not needed for the posterior predictive checks, the program saves output space by using a local variable for y_rep. The statistics mean(u) and sd(y) could also be computed in the transformed data block and saved.\nFor the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.\n\n\n\nHistogram of means of replicated data sets; vertical red line at mean of original data.\n\n\nThe p-value for the mean is just the percentage of replicated data sets whose statistic is greater than or equal that of the original data. Using a Poisson model for negative binomial data still fits the mean well, with a posterior \\(p\\)-value of 0.49. In Stan terms, it is extracted as the posterior mean of the indicator variable mean_gt.\nThe standard deviation statistic tells a different story.\n\n\n\nScatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.\n\n\nHere, the original data has much higher standard deviation than any of the replicated data sets. The resulting \\(p\\)-value estimated by Stan after a large number of iterations is exactly zero (the absolute error bounds are fine, but a lot of iterations are required to get good relative error bounds on small \\(p\\)-values by sampling). In other words, there were no posterior draws in which the replicated data set had a standard deviation greater than or equal to that of the original data set. Clearly, the model is not capturing the dispersion of the original data. The point of this exercise isn’t just to figure out that there’s a problem with a model, but to isolate where it is. Seeing that the data is over-dispersed compared to the Poisson model would be reason to fit a more general model like the negative binomial or a latent varying effects (aka random effects) model that can account for the over-dispersion.\n\n\nAny statistic may be used for the data, but these can be guided by the quantities of interest in the model itself. Popular choices in addition to mean and standard deviation are quantiles, such as the median, 5% or 95% quantiles, or even the maximum or minimum value to test extremes.\nDespite the range of choices, test statistics should ideally be ancillary, in the sense that they should be testing something other than the fit of a parameter. For example, a simple normal model of a data set will typically fit the mean and variance of the data quite well as long as the prior doesn’t dominate the posterior. In contrast, a Poisson model of the same data cannot capture both the mean and the variance of a data set if they are different, so they bear checking in the Poisson case. As we saw with the Poisson case, the posterior mean for the single rate parameter was located near the data mean, not the data variance. Other distributions such as the lognormal and gamma distribution, have means and variances that are functions of two or more parameters.",
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     "href": "reference-manual/types.html",
     "title": "Data Types and Declarations",
     "section": "",
-    "text": "This chapter covers the data types for expressions in Stan. Every variable used in a Stan program must have a declared data type. Only values of that type will be assignable to the variable (except for temporary states of transformed data and transformed parameter values). This follows the convention of programming languages like C++, not the conventions of scripting languages like Python or statistical languages such as R or BUGS.\nThe motivation for strong, static typing is threefold.\n\nStrong typing forces the programmer’s intent to be declared with the variable, making programs easier to comprehend and hence easier to debug and maintain.\nStrong typing allows programming errors relative to the declared intent to be caught sooner (at compile time) rather than later (at run time). The Stan compiler (called through an interface such as CmdStan, RStan, or PyStan) will flag any type errors and indicate the offending expressions quickly when the program is compiled.\nConstrained types will catch runtime data, initialization, and intermediate value errors as soon as they occur rather than allowing them to propagate and potentially pollute final results.\n\nStrong typing disallows assigning the same variable to objects of different types at different points in the program or in different invocations of the program.\n\n\nArguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.\n\n\n\n\nUnfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.\n\n\n\n\nThe complex data type is a scalar, but unlike real and int types, it contains two components, a real and imaginary component, both of which are of type real. That is, the real and imaginary components of a complex number are 64-bit, IEEE 754-complaint floating point numbers.\n\n\nImaginary literals are written in mathematical notation using a numeral followed by the suffix i. For example, the following example constructs a complex number \\(2 - 1.3i\\) and assigns it to the variable z.\ncomplex z = 2 - 1.3i;\nreal re = get_real(z);  // re has value 2.0\nreal im = get_imag(z);  // im has value -1.3\nThe getter functions then extract the real and imaginary components of z and assign them to re and im respectively.\nThe function to_complex constructs a complex number from its real and imaginary components. The functional form needs to be used whenever the components are not literal numerals, as in the following example.\nvector[K] re;\nvector[K] im;\n// ...\nfor (k in 1:K) {\n  complex z = to_complex(re[k], im[k]);\n  // ...\n}\n\n\n\nExpressions of type real may be assigned to variables of type complex. For example, the following is a valid sequence of Stan statements.\nreal x = 5.0;\ncomplex z = x;  // get_real(z) == 5.0, get_imag(z) == 0\nThe real number assigned to a complex number determine’s the complex number’s real component, with the imaginary component set to zero.\nAssignability is transitive, so that expressions of type int may also be assigned to variables of type complex, as in the following example.\nint n = 2;\ncomplex z = n;\nFunction arguments also support promotion of integer or real typed expressions to type complex.\n\n\n\n\nAll variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be read as data in the dump format.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.\n\n\n\n\nStan provides three types of container objects: arrays, vectors, and matrices. Vectors and matrices are more limited kinds of data structures than arrays. Vectors are intrinsically one-dimensional collections of real or complex values, whereas matrices are intrinsically two dimensional. Vectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical. A \\(3 \\times 4\\) matrix is a different kind of object in Stan than a \\(3\n\\times 4\\) array.\nThe intention of using matrix types is to call out their usage in the code. There are three situations in Stan where only vectors and matrices may be used,\n\nmatrix arithmetic operations (e.g., matrix multiplication)\nlinear algebra functions (e.g., eigenvalues and determinants), and\nmultivariate function parameters and outcomes (e.g., multivariate normal distribution arguments).\n\nVectors and matrices cannot be typed to return integer values. They are restricted to real and complex values.\nFor constructing vectors and matrices in Stan, see Vector, Matrix, and Array Expressions.\n\n\nVectors and matrices, as well as arrays, are indexed starting from one (1) in Stan. This follows the convention in statistics and linear algebra as well as their implementations in the statistical software packages R, MATLAB, BUGS, and JAGS. General computer programming languages, on the other hand, such as C++ and Python, index arrays starting from zero.\n\n\n\nVectors in Stan are column vectors; see below for information on row vectors. Vectors are declared with a size (i.e., a dimensionality). For example, a 3-dimensional real vector is declared with the keyword vector, as follows.\nvector[3] u;\nVectors may also be declared with constraints, as in the following declaration of a 3-vector of non-negative values.\nvector&lt;lower=0&gt;[3] u;\nSimilarly, they may be declared with a offset and/or multiplier, as in the following example\nvector&lt;offset=42, multiplier=3&gt;[3] u;\n\n\n\nLike real vectors, complex vectors are column vectors and are declared with a size. For example, a 3-dimensional complex vector is declared with the keyword complex_vector, as follows.\ncomplex_vector[3] v;\nComplex vector declarations do not support any constraints.\n\n\n\nA unit simplex is a vector with non-negative values whose entries sum to 1. For instance, \\([0.2,0.3,0.4,0.1]^{\\top}\\) is a unit 4-simplex. Unit simplexes are most often used as parameters in categorical or multinomial distributions, and they are also the sampled variate in a Dirichlet distribution. Simplexes are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-simplex by\nsimplex[5] theta;\nUnit simplexes are implemented as vectors and may be assigned to other vectors and vice-versa. Simplex variables, like other constrained variables, are validated to ensure they contain simplex values; for simplexes, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\nIn high dimensional problems, simplexes may require smaller step sizes in the inference algorithms in order to remain stable; this can be achieved through higher target acceptance rates for samplers and longer warmup periods, tighter tolerances for optimization with more iterations, and in either case, with less dispersed parameter initialization or custom initialization if there are informative priors for some parameters.\n\n\n\nA unit vector is a vector with a norm of one. For instance, \\([0.5,\n0.5, 0.5, 0.5]^{\\top}\\) is a unit 4-vector. Unit vectors are sometimes used in directional statistics. Unit vectors are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-vector by\nunit_vector[5] theta;\nUnit vectors are implemented as vectors and may be assigned to other vectors and vice-versa. Unit vector variables, like other constrained variables, are validated to ensure that they are indeed unit length; for unit vectors, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\n\n\n\nAn ordered vector type in Stan represents a vector whose entries are sorted in ascending order. For instance, \\((-1.3,2.7,2.71)^{\\top}\\) is an ordered 3-vector. Ordered vectors are most often employed as cut points in ordered logistic regression models (see section).\nThe variable c is declared as an ordered 5-vector by\nordered[5] c;\nAfter their declaration, ordered vectors, like unit simplexes, may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nThere is also a positive, ordered vector type which operates similarly to ordered vectors, but all entries are constrained to be positive. For instance, \\((2,3.7,4,12.9)\\) is a positive, ordered 4-vector.\nThe variable d is declared as a positive, ordered 5-vector by\npositive_ordered[5] d;\nLike ordered vectors, after their declaration, positive ordered vectors may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nRow vectors are declared with the keyword row_vector. Like (column) vectors, they are declared with a size. For example, a 1093-dimensional row vector u would be declared as\nrow_vector[1093] u;\nConstraints are declared as for vectors, as in the following example of a 10-vector with values between -1 and 1.\nrow_vector&lt;lower=-1, upper=1&gt;[10] u;\nOffset and multiplier are also similar as for the following 3-row-vector with offset -42 and multiplier 3.\nrow_vector&lt;offset=-42, multiplier=3&gt;[3] u;\nRow vectors may not be assigned to column vectors, nor may column vectors be assigned to row vectors. If assignments are required, they may be accommodated through the transposition operator.\n\n\n\nComplex row vectors are declared with the keyword complex_row_vector and given a size in basic declarations. For example, a 12-dimensional complex row vector v would be declared as\ncomplex_row_vector[12] v;\nComplex row vectors do not allow constraints.\n\n\n\nMatrices are declared with the keyword matrix along with a number of rows and number of columns. For example,\nmatrix[3, 3] A;\nmatrix[M, N] B;\ndeclares A to be a \\(3 \\times 3\\) matrix and B to be a \\(M\n\\times N\\) matrix. For the second declaration to be well formed, the variables M and N must be declared as integers in either the data or transformed data block and before the matrix declaration.\nMatrices may also be declared with constraints, as in this (\\(3 \\times 4\\)) matrix of non-positive values.\nmatrix&lt;upper=0&gt;[3, 4] B;\nSimilarly, matrices can be declared to have a set offset and/or multiplier, as in this matrix with multiplier 5.\nmatrix&lt;multiplier=5&gt;[3, 4] B;\n\n\nRows of a matrix can be assigned by indexing the left-hand side of an assignment statement. For example, this is possible.\nmatrix[M, N] a;\nrow_vector[N] b;\n// ...\na[1] = b;\nThis copies the values from row vector b to a[1], which is the first row of the matrix a. If the number of columns in a is not the same as the size of b, a run-time error is raised; the number of columns of a is N, which is also the number of columns of b.\nAssignment works by copying values in Stan. That means any subsequent assignment to a[1] does not affect b, nor does an assignment to b affect a.\n\n\n\n\nComplex matrices are declared with the keyword complex_matrix and a number of rows and columns. For example,\ncomplex_matrix[3, 3] C;\nComplex matrices do not allow constraints.\n\n\n\nMatrix variables may be constrained to represent covariance matrices. A matrix is a covariance matrix if it is symmetric and positive definite. Like correlation matrices, covariance matrices only need a single dimension in their declaration. For instance,\ncov_matrix[K] Omega;\ndeclares Omega to be a \\(K \\times K\\) covariance matrix, where \\(K\\) is the value of the data variable K.\n\n\n\nMatrix variables may be constrained to represent correlation matrices. A matrix is a correlation matrix if it is symmetric and positive definite, has entries between \\(-1\\) and \\(1\\), and has a unit diagonal. Because correlation matrices are square, only one dimension needs to be declared. For example,\ncorr_matrix[3] Sigma;\ndeclares Sigma to be a \\(3 \\times 3\\) correlation matrix.\nCorrelation matrices may be assigned to other matrices, including unconstrained matrices, if their dimensions match, and vice-versa.\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a covariance matrix. This is often more convenient or more efficient than representing covariance matrices directly.\nA Cholesky factor \\(L\\) is an \\(M \\times N\\) lower-triangular matrix (if \\(m &lt; n\\) then \\(L[m, n] =0\\)) with a strictly positive diagonal (\\(L[k, k]\n&gt; 0\\)) and \\(M \\geq N\\). If \\(L\\) is a Cholesky factor, then \\(\\Sigma = L\n\\, L^{\\top}\\) is a covariance matrix (i.e., it is positive definite). The mapping between positive definite matrices and their Cholesky factors is bijective—every covariance matrix has a unique Cholesky factorization.\nThe typical case of a square Cholesky factor may be declared with a single dimension,\ncholesky_factor_cov[4] L;\n\n\nIn general, two dimensions may be declared, with the above being equal to cholesky_factor_cov[4, 4]. The type cholesky_factor_cov[M, N] may be used for the general \\(M \\times N\\) case to produce positive semi-definite matrices of rank \\(M\\).\n\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a correlation matrix.\nA Cholesky factor for a correlation matrix \\(L\\) is a \\(K \\times K\\) lower-triangular matrix with positive diagonal entries and rows that are of length 1 (i.e., \\(\\sum_{n=1}^K L_{m,n}^2 = 1\\)). If \\(L\\) is a Cholesky factor for a correlation matrix, then \\(L\\,L^{\\top}\\) is a correlation matrix (i.e., symmetric positive definite with a unit diagonal).\nTo declare the variable L to be a K by K Cholesky factor of a correlation matrix, the following code may be used.\ncholesky_factor_corr[K] L;\n\n\n\nConstrained variables of all types may be assigned to other variables of the same unconstrained type and vice-versa. Matching is interpreted strictly as having the same basic type and number of array dimensions. Constraints are not considered, but basic data types are. For instance, a variable declared to be real&lt;lower=0, upper=1&gt; could be assigned to a variable declared as real and vice-versa. Similarly, a variable declared as matrix[3, 3] may be assigned to a variable declared as cov_matrix[3] or cholesky_factor_cov[3], and vice-versa.\nChecks are carried out at the end of each relevant block of statements to ensure constraints are enforced. This includes run-time size checks. The Stan compiler isn’t able to catch the fact that an attempt may be made to assign a matrix of one dimensionality to a matrix of mismatching dimensionality.\n\n\n\nReal-valued vectors, row vectors and matrices may be assigned to complex-valued vectors, row vectors and matrices, respectively. For example, the following is legal.\nvector[N] v = ...;\ncomplex_vector[N] u = 2 * v;\nRow vectors and matrices work the same way.\n\n\n\nVariables may be declared with sizes given by expressions. Such expressions are constrained to only contain data or transformed data variables. This ensures that all sizes are determined once the data is read in and transformed data variables defined by their statements. For example, the following is legal.\ndata {\n  int&lt;lower=0&gt; N_observed, N_missing;\n  // ...\ntransformed parameters {\n  vector[N_observed + N_missing] y;\n  // ...\n\n\n\nIf v is a column vector or row vector, then v[2] is the second element in the vector. If m is a matrix, then m[2, 3] is the value in the second row and third column.\nProviding a matrix with a single index returns the specified row. For instance, if m is a matrix, then m[2] is the second row. This allows Stan blocks such as\nmatrix[M, N] m;\nrow_vector[N] v;\nreal x;\n// ...\nv = m[2];\nx = v[3];   // x == m[2][3] == m[2, 3]\nThe type of m[2] is row_vector because it is the second row of m. Thus it is possible to write m[2][3] instead of m[2, 3] to access the third element in the second row. When given a choice, the form m[2, 3] is preferred.\nComplex versions work the same way,\ncomplex_matrix[M, N] m = ...;\ncomplex_row_vector[N] u = m[3];\ncomplex_vector[M] v = m[ , 2];\n\n\nThe form m[2, 3] is more efficient because it does not require the creation and use of an intermediate expression template for m[2]. In later versions, explicit calls to m[2][3] may be optimized to be as efficient as m[2, 3] by the Stan compiler.\n\n\n\n\nAn integer expression is used to pick out the sizes of vectors, matrices, and arrays. For instance, we can declare a vector of size M + N using\nvector[M + N] y;\nAny integer-denoting expression may be used for the size declaration, providing all variables involved are either data, transformed data, or local variables. That is, expressions used for size declarations may not include parameters or transformed parameters or generated quantities.\n\n\n\n\nStan supports arrays of arbitrary dimension. The values in an array can be any type, so that arrays may contain values that are simple reals or integers, vectors, matrices, or other arrays. Arrays are the only way to store sequences of integers, and some functions in Stan, such as discrete distributions, require integer arguments.\nA two-dimensional array is just an array of arrays, both conceptually and in terms of current implementation. When an index is supplied to an array, it returns the value at that index. When more than one index is supplied, this indexing operation is chained. For example, if a is a two-dimensional array, then a[m, n] is just a convenient shorthand for a[m][n].\nVectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical.\nFor constructing arrays in Stan, see Vector, Matrix, and Array Expressions.\n\n\nArrays are declared with the keyword array followed by the dimensions enclosed in square brackets, the element type, and the name of the variable.\nThe variable n is declared as an array of five integers as follows.\narray[5] int n;\nA two-dimensional array of complex values with three rows and four columns is declared as follows.\narray[3, 4] complex a;\nA three-dimensional array z of positive reals with five rows, four columns, and two shelves can be declared as follows.\narray[5, 4, 2] real&lt;lower=0&gt; z;\nArrays may also be declared to contain vectors. For example,\narray[3] vector[7] mu;\ndeclares mu to be an array of size 3 containing vectors with 7 elements. Arrays may also contain matrices. The example\narray[15, 12] complex_matrix[7, 2] mu;\ndeclares a 15 by 12 array of \\(7 \\times 2\\) complex matrices. Any of the constrained types may also be used in arrays, as in the declaration\narray[2, 3, 4] cholesky_factor_cov[5, 6] mu;\nof a \\(2 \\times 3 \\times 4\\) array of \\(5 \\times 6\\) Cholesky factors of covariance matrices.\n\n\n\nIf x is a 1-dimensional array of length 5, then x[1] is the first element in the array and x[5] is the last. For a \\(3\n\\times 4\\) array y of two dimensions, y[1, 1] is the first element and y[3, 4] the last element. For a three-dimensional array z, the first element is z[1, 1, 1], and so on.\nSubarrays of arrays may be accessed by providing fewer than the full number of indexes. For example, suppose y is a two-dimensional array with three rows and four columns. Then y[3] is one-dimensional array of length four. This means that y[3][1] may be used instead of y[3, 1] to access the value of the first column of the third row of y. The form y[3, 1] is the preferred form (see note in this chapter).\n\n\n\nSubarrays may be manipulated and assigned just like any other variables. Similar to the behavior of matrices, Stan allows blocks such as\narray[9, 10, 11] real w;\narray[10, 11] real x;\narray[11] real y;\nreal z;\n// ...\nx = w[5];\ny = x[4];  // y == w[5][4] == w[5, 4]\nz = y[3];  // z == w[5][4][3] == w[5, 4, 3]\nComplex-valued arrays work the same way.\n\n\n\nArrays of vectors and matrices are accessed in the same way as arrays of doubles. Consider the following vector and scalar declarations.\narray[3, 4] vector[5] a;\narray[4] vector[5] b;\nvector[5] c;\nreal x;\nWith these declarations, the following assignments are legal.\nb = a[1];       // result is array of vectors\nc = a[1, 3];    // result is vector\nc = b[3];       //   same result as above\nx = a[1, 3, 5]; // result is scalar\nx = b[3, 5];    //   same result as above\nx = c[5];       //   same result as above\nRow vectors and other derived vector types (simplex and ordered) behave the same way in terms of indexing.\nConsider the following matrix, vector and scalar declarations.\narray[3, 4] matrix[6, 5] d;\narray[4] matrix[6, 5] e;\nmatrix[6, 5] f;\nrow_vector[5] g;\nreal x;\nWith these declarations, the following definitions are legal.\ne = d[1];           // result is array of matrices\nf = d[1, 3];        // result is matrix\nf = e[3];           //   same result as above\ng = d[1, 3, 2];     // result is row vector\ng = e[3, 2];        //   same result as above\ng = f[2];           //   same result as above\nx = d[1, 3, 5, 2];  // result is scalar\nx = e[3, 5, 2];     //   same result as above\nx = f[5, 2];        //   same result as above\nx = g[2];           //   same result as above\nAs shown, the result f[2] of supplying a single index to a matrix is the indexed row, here row 2 of matrix f.\n\n\n\nSubarrays of arrays may be assigned by indexing on the left-hand side of an assignment statement. For example, the following is legal.\narray[I, J, K] real x;\narray[J, K] real y;\narray[K] real z;\n// ...\nx[1] = y;\nx[1, 1] = z;\nThe sizes must match. Here, x[1] is a J by K array, as is y.\nPartial array assignment also works for arrays of matrices, vectors, and row vectors.\n\n\n\nArrays, row vectors, column vectors and matrices are not interchangeable in Stan. Thus a variable of any one of these fundamental types is not assignable to any of the others, nor may it be used as an argument where the other is required (use as arguments follows the assignment rules).\n\n\nFor example, vectors cannot be assigned to arrays or vice-versa.\narray[4] real a;\nvector[4] b;\nrow_vector[4] c;\n// ...\na = b; // illegal assignment of vector to array\nb = a; // illegal assignment of array to vector\na = c; // illegal assignment of row vector to array\nc = a; // illegal assignment of array to row vector\n\n\n\nIt is not even legal to assign row vectors to column vectors or vice versa.\nvector[4] b;\nrow_vector[4] c;\n// ...\nb = c; // illegal assignment of row vector to column vector\nc = b; // illegal assignment of column vector to row vector\n\n\n\nThe same holds for matrices, where 2-dimensional arrays may not be assigned to matrices or vice-versa.\narray[3, 4] real a;\nmatrix[3, 4] b;\n// ...\na = b;  // illegal assignment of matrix to array\nb = a;  // illegal assignment of array to matrix\n\n\n\nA \\(1 \\times N\\) matrix cannot be assigned a row vector or vice versa.\nmatrix[1, 4] a;\nrow_vector[4] b;\n// ...\na = b;  // illegal assignment of row vector to matrix\nb = a;  // illegal assignment of matrix to row vector\nSimilarly, an \\(M \\times 1\\) matrix may not be assigned to a column vector.\nmatrix[4, 1] a;\nvector[4] b;\n// ...\na = b;  // illegal assignment of column vector to matrix\nb = a;  // illegal assignment of matrix to column vector\n\n\n\n\nAn integer expression is used to pick out the sizes of arrays. The same restrictions as for vector and matrix sizes apply, namely that the size is declared with an integer-denoting expression that does not contain any parameters, transformed parameters, or generated quantities.\n\n\n\nIf any of an array’s dimensions is size zero, the entire array will be of size zero. That is, if we declare\narray[3, 0] real a;\nthen the resulting size of a is zero and querying any of its dimensions at run time will result in the value zero. Declared as above, a[1] will be a size-zero one-dimensional array. For comparison, declaring\narray[0, 3] real b;\nalso produces an array with an overall size of zero, but in this case, there is no way to index legally into b, because b[0] is undefined. The array will behave at run time as if it’s a \\(0 \\times\n0\\) array. For example, the result of to_matrix(b) will be a \\(0 \\times 0\\) matrix, not a \\(0 \\times 3\\) matrix.\n\n\n\n\nStan supports tuples of arbitrary size. The values in a tuple can be of arbitrary type, but the component types must be declared along with the declaration of the tuple. Tuples can be manipulated as a whole, or their elements may be accessed and set individually.\n\n\nTuples are declared with the keyword tuple followed by a parenthesized sequence of types, which determine the types of the respective tuple entries. For example, a tuple with three elements may be declared as\ntuple(int, vector[3], complex) abc;\nTuples must have at least two entries, so the following declarations are illegal.\ntuple() nil;  // ILLEGAL\ntuple(int) n;  // ILLEGAL\nTuples can be assigned as a whole if their elements can be assigned individually. For example, a can be assigned to b in the following example because int can be promoted to complex.\ntuple(int, real) a;\n...\ntuple(complex, real) b = a;\nTuple types may have elements which are declared as tuples, such as the following example.\ntuple(int, tuple(real, complex)) x;\nIn this case, it would probably be simpler to use a 3-tuple type, tuple(int, real, complex).\nTuples can be declared with constraints anywhere that ordinary variables can (i.e., as top-level block variables). That means any context in which it is legal to have a declaration\nreal&lt;lower=0&gt; sigma;\nreal&lt;lower=0, upper=1&gt; theta;\nit is legal to have a tuple with constraints such as\ntuple(real&lt;lower=0&gt;, real&lt;lower=0, upper=1&gt;) sigma_theta;\n\n\n\nTuple elements may be accessed directly. For example, with our declaration of abc from the last section, Stan uses abc.1 for the first element, abc.2 for the second, and abc.3 for the third. These numbers must be integer literals (i.e., they cannot be variables), and must be within the size of the number of elements of tuples. The types of elements are as declared, so that abc.1 is of type int, abc.2 of type vector[3] and abc.3 of type complex.\n\n\n\nTuple elements can be assigned individually, allowing, e.g.,\ntuple(int, real) ab;\nab.1 = 123;\nab.2 = 12.9;\nAs with other assignments, promotions will happen if necessary (of int to real and of real to complex, along with the corresponding container type promotions).\n\n\n\nFor convenience of using values stored in tuples, Stan supports “unpacking” (or “destructuring”) of tuples in an assignment statement.\nGiven a tuple t of type tuple(T1, ..., Tn) and a sequence of assignable expressions of types v1, …, vn, where each vi has a type which is assignable from type Ti, individual elements of the tuple may be assigned to the corresponding variables in the sequence by the statement\n(v1, /*...*/, vn) = t;\nNote that the above parenthesis are required, unlike in some other languages with similar features (e.g., Python).\nThese unpacking assignments can be nested if the tuple on the right hand side contains nested tuples.\nFor example, if T is a tuple of type tuple(int, (real, real), complex), then the program\nint i;\nreal x, y;\ncomplex z;\n\n(i, (x, y), z) = T;\nAssigns the result of T.1 to i, the result of T.2.1 to x, the result of T.2.2 to y, and the result of T.3 to z.\nThe left hand side must match in size the tuple on the right. Additionally, the same variable may not appear more than once in the left hand side of an unpacking assignment.\n\n\n\n\nThe type information associated with a variable only contains the underlying type and dimensionality of the variable.\n\n\nThe size associated with a given variable is not part of its data type. For example, declaring a variable using\narray[3] real a;\ndeclares the variable a to be an array. The fact that it was declared to have size 3 is part of its declaration, but not part of its underlying type.\n\n\nSizes are determined dynamically (at run time) and thus cannot be type-checked statically when the program is compiled. As a result, any conformance error on size will raise a run-time error. For example, trying to assign an array of size 5 to an array of size 6 will cause a run-time error. Similarly, multiplying an \\(N\n\\times M\\) by a \\(J \\times K\\) matrix will raise a run-time error if \\(M\n\\neq J\\).\n\n\n\n\nLike sizes, constraints are not treated as part of a variable’s type in Stan when it comes to the compile-time check of operations it may participate in. Anywhere Stan accepts a matrix as an argument, it will syntactically accept a correlation matrix or covariance matrix or Cholesky factor. Thus a covariance matrix may be assigned to a matrix and vice-versa.\nSimilarly, a bounded real may be assigned to an unconstrained real and vice-versa.\n\n\nFor arguments to functions, constraints are sometimes, but not always checked when the function is called. Exclusions include C++ standard library functions. All probability functions and cumulative distribution functions check that their arguments are appropriate at run time as the function is called.\n\n\n\nFor data variables, constraints are checked after the variable is read from a data file or other source. For transformed data variables, the check is done after the statements in the transformed data block have executed. Thus it is legal for intermediate values of variables to not satisfy declared constraints.\nFor parameters, constraints are enforced by the transform applied and do not need to be checked. For transformed parameters, the check is done after the statements in the transformed parameter block have executed.\nFor all blocks defining variables (transformed data, transformed parameters, generated quantities), real values are initialized to NaN and integer values are initialized to the smallest legal integer (i.e., a large absolute value negative number).\nFor generated quantities, constraints are enforced after the statements in the generated quantities block have executed.\n\n\n\n\nIn order to refer to data types, it is convenient to have a way to refer to them. The type naming notation outlined in this section is not part of the Stan programming language, but rather a convention adopted in this document to enable a concise description of a type.\nBecause size information is not part of a data type, data types will be written without size information. For instance, array[] real is the type of one-dimensional array of reals and matrix is the type of matrices. The three-dimensional integer array type is written as array[,,] int, indicating the number slots available for indexing. Similarly, array[,] vector is the type of a two-dimensional array of vectors.\n\n\n\n\nVariables in Stan are declared by giving a type and a name. For example\nint N;\nvector[N] y;\narray[5] matrix[3, 4] A;\ndeclares a variable N that is an integer, a variable y that is a vector of length N (the previously declared variable), and a variable A, which is a length-5 array where each element is a 3 by 4 matrix.\nThere are several different places a variable is declared in Stan. They are block variables, like those inside data, which can have constraints and must include sizes for their types, like in the above examples. Local variables, like those defined inside loops or local blocks cannot be constrained, but still include sizes. Finally, variables declared as function parameters are not constrained types and exclude sizes.\n// valid block variables, but not locals or function parameters\nvector&lt;lower=0&gt;[N] u;\n\n// valid as a block or local variable, but not a function parameter\narray[3] int is;\n\n// function parameters exclude sizes and cannot be constrained\nvoid pretty_print_tri_lower(matrix x) { ... }\nIn the following table, the leftmost column is a list of the unconstrained and undimensioned basic types; these are used as function return types and argument types. The middle column is of unconstrained types with dimensions; these are used as local variable types. The variables M and N indicate number of columns and rows, respectively. The variable K is used for square matrices, i.e., K denotes both the number of rows and columns. The rightmost column lists the corresponding constrained types. An expression of any right-hand column type may be assigned to its corresponding left-hand column basic type. At runtime, dimensions are checked for consistency for all variables; containers of any sizes may be assigned to function arguments. The constrained matrix types cov_matrix[K], corr_matrix[K], cholesky_factor_cov[K], and cholesky_factor_corr[K] are only assignable to matrices of dimensions matrix[K, K] types.\n\n\n\n\n\n\n\n\nFunction Argument (unsized)\n   Local\n(unconstrained)\n              Block\n          (constrained)\n\n\n\n\nint\nint\nint\n\n\n\n\nint&lt;lower=L&gt;\n\n\n\n\nint&lt;upper=U&gt;\n\n\n\n\nint&lt;lower=L, upper=U&gt;\n\n\n\n\nint&lt;offset=O&gt;\n\n\n\n\nint&lt;multiplier=M&gt;\n\n\n\n\nint&lt;offset=O, multiplier=M&gt;\n\n\nreal\nreal\nreal\n\n\n\n\nreal&lt;lower=L&gt;\n\n\n\n\nreal&lt;upper=U&gt;\n\n\n\n\nreal&lt;lower=L, upper=U&gt;\n\n\n\n\nreal&lt;offset=O&gt;\n\n\n\n\nreal&lt;multiplier=M&gt;\n\n\n\n\nreal&lt;offset=O, multiplier=M&gt;\n\n\ncomplex\ncomplex\ncomplex\n\n\nvector\nvector[N]\nvector[N]\n\n\n\n\nvector[N]&lt;lower=L&gt;\n\n\n\n\nvector[N]&lt;upper=U&gt;\n\n\n\n\nvector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nvector[N]&lt;offset=O&gt;\n\n\n\n\nvector[N]&lt;multiplier=M&gt;\n\n\n\n\nvector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\nordered[N]\n\n\n\n\npositive_ordered[N]\n\n\n\n\nsimplex[N]\n\n\n\n\nunit_vector[N]\n\n\nrow_vector\nrow_vector[N]\nrow_vector[N]\n\n\n\n\nrow_vector[N]&lt;lower=L&gt;\n\n\n\n\nrow_vector[N]&lt;upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O&gt;\n\n\n\n\nrow_vector[N]&lt;multiplier=M&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O, multiplier=M&gt;\n\n\nmatrix\nmatrix[M, N]\nmatrix[M, N]\n\n\n\n\nmatrix[M, N]&lt;lower=L&gt;\n\n\n\n\nmatrix[M, N]&lt;upper=U&gt;\n\n\n\n\nmatrix[M, N]&lt;lower=L, uppers=U&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O&gt;\n\n\n\n\nmatrix[M, N]&lt;multiplier=M&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O, multiplier=M&gt;\n\n\n\nmatrix[K, K]\ncorr_matrix[K]\n\n\n\nmatrix[K, K]\ncov_matrix[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_corr[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_cov[K]\n\n\ncomplex_vector\ncomplex_vector[M]\ncomplex_vector[M]\n\n\ncomplex_row_vector\ncomplex_row_vector[N]\ncomplex_row_vector[N]\n\n\ncomplex_matrix\ncomplex_matrix[M, N]\ncomplex_matrix[M,N]\n\n\narray[] vector\narray[M] vector[N]\narray[M] vector[N]\n\n\n\n\narray[M] vector[N]&lt;lower=L&gt;\n\n\n\n\narray[M] vector[N]&lt;upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O&gt;\n\n\n\n\narray[M] vector[N]&lt;multiplier=M&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\narray[M] ordered[N]\n\n\n\n\narray[M] positive_ordered[N]\n\n\n\n\narray[M] simplex[N]\n\n\n\n\narray[M] unit_vector[N]\n\n\n\n\nAdditional array types follow the same basic template as the final example in the table and can contain any of the previous types. The unsized version of arrays with more than one dimension is specified by using commas, e.g. array[ , ] is a 2-D array.\nFor more on how function arguments and return types are declared, consult the User’s Guide chapter on functions.\n\n\n\nStan allows assignable variables to be declared and defined in a single statement. Assignable variables are\n\nlocal variables, and\nvariables declared in the transformed data, transformed parameters, or generated quantities blocks.\n\nFor example, the statement\nint N = 5;\ndeclares the variable N to be an integer scalar type and at the same time defines it to be the value of the expression 5.\n\n\nThe type of the expression on the right-hand side of the assignment must be assignable to the type of the variable being declared. For example, it is legal to have\nreal sum = 0;\neven though 0 is of type int and sum is of type real, because integer-typed scalar expressions can be assigned to real-valued scalar variables. In all other cases, the type of the expression on the right-hand side of the assignment must be identical to the type of the variable being declared.\nVariables of any type may have values assigned to them. For example,\nmatrix[3, 2] a = b;\ndeclares a \\(3 \\times 2\\) matrix variable a and assigns a copy of the value of b to the variable a. The variable b must be of type matrix for the statement to be well formed. For the code to execute successfully, b must be the same shape as a, but this cannot be validated until run time. Because a copy is assigned, subsequent changes to a do not affect b and subsequent changes to b do not affect a.\n\n\n\nThe right-hand side may be any expression which has a type which is assignable to the variable being declared. For example,\nmatrix[3, 2] a = 0.5 * (b + c);\nassigns the matrix variable a to half of the sum of b and c. The only requirement on b and c is that the expression b + c be of type matrix. For example, b could be of type matrix and c of type real, because adding a matrix to a scalar produces a matrix, and the multiplying by a scalar produces another matrix.\nSimilarly,\ncomplex z = 2 + 3i;\nassigns the the complex number \\(2 + 3i\\) to the complex scalar z. The right-hand side expression can be a call to a user defined function, allowing general algorithms to be applied that might not be otherwise expressible as simple expressions (e.g., iterative or recursive algorithms).\n\n\n\nAny variable that is in scope and any function that is available in the block in which the compound declaration and definition appears may be used in the expression on the right-hand side of the compound declaration and definition statement.\n\n\n\n\nStan will interpret multiple comma-separated variable names following a single type as declaring multiple new variables. This is available for all variable declarations in all blocks.\n\n\nThe code:\nreal x, y;\nis equivalent to\nreal x;\nreal y;\nAs a result, all declarations on the same line must be of the same type.\n\n\n\nThe ability to declare multiple variables can be combined with assignments whenever a declare-define is valid, as documented in the section introducing compound declarations and definitions :\nreal x = 3, y = 5.6;\nConstrained data types can also be declared together, so long as the constraint for each variable is the same:\nreal&lt;lower=0&gt; x, y;",
+    "text": "This chapter covers the data types for expressions in Stan. Every variable used in a Stan program must have a declared data type. Only values of that type will be assignable to the variable (except for temporary states of transformed data and transformed parameter values). This follows the convention of programming languages like C++, not the conventions of scripting languages like Python or statistical languages such as R or BUGS.\nThe motivation for strong, static typing is threefold.\n\nStrong typing forces the programmer’s intent to be declared with the variable, making programs easier to comprehend and hence easier to debug and maintain.\nStrong typing allows programming errors relative to the declared intent to be caught sooner (at compile time) rather than later (at run time). The Stan compiler (called through an interface such as CmdStan, RStan, or PyStan) will flag any type errors and indicate the offending expressions quickly when the program is compiled.\nConstrained types will catch runtime data, initialization, and intermediate value errors as soon as they occur rather than allowing them to propagate and potentially pollute final results.\n\nStrong typing disallows assigning the same variable to objects of different types at different points in the program or in different invocations of the program.\n\n\nArguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the Removed Features chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.\n\n\n\n\nUnfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.\n\n\n\n\nThe complex data type is a scalar, but unlike real and int types, it contains two components, a real and imaginary component, both of which are of type real. That is, the real and imaginary components of a complex number are 64-bit, IEEE 754-complaint floating point numbers.\n\n\nImaginary literals are written in mathematical notation using a numeral followed by the suffix i. For example, the following example constructs a complex number \\(2 - 1.3i\\) and assigns it to the variable z.\ncomplex z = 2 - 1.3i;\nreal re = get_real(z);  // re has value 2.0\nreal im = get_imag(z);  // im has value -1.3\nThe getter functions then extract the real and imaginary components of z and assign them to re and im respectively.\nThe function to_complex constructs a complex number from its real and imaginary components. The functional form needs to be used whenever the components are not literal numerals, as in the following example.\nvector[K] re;\nvector[K] im;\n// ...\nfor (k in 1:K) {\n  complex z = to_complex(re[k], im[k]);\n  // ...\n}\n\n\n\nExpressions of type real may be assigned to variables of type complex. For example, the following is a valid sequence of Stan statements.\nreal x = 5.0;\ncomplex z = x;  // get_real(z) == 5.0, get_imag(z) == 0\nThe real number assigned to a complex number determine’s the complex number’s real component, with the imaginary component set to zero.\nAssignability is transitive, so that expressions of type int may also be assigned to variables of type complex, as in the following example.\nint n = 2;\ncomplex z = n;\nFunction arguments also support promotion of integer or real typed expressions to type complex.\n\n\n\n\nAll variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be supplied as data.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.\n\n\n\n\nStan provides three types of container objects: arrays, vectors, and matrices. Vectors and matrices are more limited kinds of data structures than arrays. Vectors are intrinsically one-dimensional collections of real or complex values, whereas matrices are intrinsically two dimensional. Vectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical. A \\(3 \\times 4\\) matrix is a different kind of object in Stan than a \\(3\n\\times 4\\) array.\nThe intention of using matrix types is to call out their usage in the code. There are three situations in Stan where only vectors and matrices may be used,\n\nmatrix arithmetic operations (e.g., matrix multiplication)\nlinear algebra functions (e.g., eigenvalues and determinants), and\nmultivariate function parameters and outcomes (e.g., multivariate normal distribution arguments).\n\nVectors and matrices cannot be typed to return integer values. They are restricted to real and complex values.\nFor constructing vectors and matrices in Stan, see Vector, Matrix, and Array Expressions.\n\n\nVectors and matrices, as well as arrays, are indexed starting from one (1) in Stan. This follows the convention in statistics and linear algebra as well as their implementations in the statistical software packages R, MATLAB, BUGS, and JAGS. General computer programming languages, on the other hand, such as C++ and Python, index arrays starting from zero.\n\n\n\nVectors in Stan are column vectors; see below for information on row vectors. Vectors are declared with a size (i.e., a dimensionality). For example, a 3-dimensional real vector is declared with the keyword vector, as follows.\nvector[3] u;\nVectors may also be declared with constraints, as in the following declaration of a 3-vector of non-negative values.\nvector&lt;lower=0&gt;[3] u;\nSimilarly, they may be declared with a offset and/or multiplier, as in the following example\nvector&lt;offset=42, multiplier=3&gt;[3] u;\n\n\n\nLike real vectors, complex vectors are column vectors and are declared with a size. For example, a 3-dimensional complex vector is declared with the keyword complex_vector, as follows.\ncomplex_vector[3] v;\nComplex vector declarations do not support any constraints.\n\n\n\nA unit simplex is a vector with non-negative values whose entries sum to 1. For instance, \\([0.2,0.3,0.4,0.1]^{\\top}\\) is a unit 4-simplex. Unit simplexes are most often used as parameters in categorical or multinomial distributions, and they are also the sampled variate in a Dirichlet distribution. Simplexes are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-simplex by\nsimplex[5] theta;\nUnit simplexes are implemented as vectors and may be assigned to other vectors and vice-versa. Simplex variables, like other constrained variables, are validated to ensure they contain simplex values; for simplexes, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\nIn high dimensional problems, simplexes may require smaller step sizes in the inference algorithms in order to remain stable; this can be achieved through higher target acceptance rates for samplers and longer warmup periods, tighter tolerances for optimization with more iterations, and in either case, with less dispersed parameter initialization or custom initialization if there are informative priors for some parameters.\n\n\n\nA unit vector is a vector with a norm of one. For instance, \\([0.5,\n0.5, 0.5, 0.5]^{\\top}\\) is a unit 4-vector. Unit vectors are sometimes used in directional statistics. Unit vectors are declared with their full dimensionality. For instance, theta is declared to be a unit \\(5\\)-vector by\nunit_vector[5] theta;\nUnit vectors are implemented as vectors and may be assigned to other vectors and vice-versa. Unit vector variables, like other constrained variables, are validated to ensure that they are indeed unit length; for unit vectors, this is only done up to a statically specified accuracy threshold \\(\\epsilon\\) to account for errors arising from floating-point imprecision.\n\n\n\nAn ordered vector type in Stan represents a vector whose entries are sorted in ascending order. For instance, \\((-1.3,2.7,2.71)^{\\top}\\) is an ordered 3-vector. Ordered vectors are most often employed as cut points in ordered logistic regression models (see section).\nThe variable c is declared as an ordered 5-vector by\nordered[5] c;\nAfter their declaration, ordered vectors, like unit simplexes, may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nThere is also a positive, ordered vector type which operates similarly to ordered vectors, but all entries are constrained to be positive. For instance, \\((2,3.7,4,12.9)\\) is a positive, ordered 4-vector.\nThe variable d is declared as a positive, ordered 5-vector by\npositive_ordered[5] d;\nLike ordered vectors, after their declaration, positive ordered vectors may be assigned to other vectors and other vectors may be assigned to them. Constraints will be checked after executing the block in which the variables were declared.\n\n\n\nRow vectors are declared with the keyword row_vector. Like (column) vectors, they are declared with a size. For example, a 1093-dimensional row vector u would be declared as\nrow_vector[1093] u;\nConstraints are declared as for vectors, as in the following example of a 10-vector with values between -1 and 1.\nrow_vector&lt;lower=-1, upper=1&gt;[10] u;\nOffset and multiplier are also similar as for the following 3-row-vector with offset -42 and multiplier 3.\nrow_vector&lt;offset=-42, multiplier=3&gt;[3] u;\nRow vectors may not be assigned to column vectors, nor may column vectors be assigned to row vectors. If assignments are required, they may be accommodated through the transposition operator.\n\n\n\nComplex row vectors are declared with the keyword complex_row_vector and given a size in basic declarations. For example, a 12-dimensional complex row vector v would be declared as\ncomplex_row_vector[12] v;\nComplex row vectors do not allow constraints.\n\n\n\nMatrices are declared with the keyword matrix along with a number of rows and number of columns. For example,\nmatrix[3, 3] A;\nmatrix[M, N] B;\ndeclares A to be a \\(3 \\times 3\\) matrix and B to be a \\(M\n\\times N\\) matrix. For the second declaration to be well formed, the variables M and N must be declared as integers in either the data or transformed data block and before the matrix declaration.\nMatrices may also be declared with constraints, as in this (\\(3 \\times 4\\)) matrix of non-positive values.\nmatrix&lt;upper=0&gt;[3, 4] B;\nSimilarly, matrices can be declared to have a set offset and/or multiplier, as in this matrix with multiplier 5.\nmatrix&lt;multiplier=5&gt;[3, 4] B;\n\n\nRows of a matrix can be assigned by indexing the left-hand side of an assignment statement. For example, this is possible.\nmatrix[M, N] a;\nrow_vector[N] b;\n// ...\na[1] = b;\nThis copies the values from row vector b to a[1], which is the first row of the matrix a. If the number of columns in a is not the same as the size of b, a run-time error is raised; the number of columns of a is N, which is also the number of columns of b.\nAssignment works by copying values in Stan. That means any subsequent assignment to a[1] does not affect b, nor does an assignment to b affect a.\n\n\n\n\nComplex matrices are declared with the keyword complex_matrix and a number of rows and columns. For example,\ncomplex_matrix[3, 3] C;\nComplex matrices do not allow constraints.\n\n\n\nMatrix variables may be constrained to represent covariance matrices. A matrix is a covariance matrix if it is symmetric and positive definite. Like correlation matrices, covariance matrices only need a single dimension in their declaration. For instance,\ncov_matrix[K] Omega;\ndeclares Omega to be a \\(K \\times K\\) covariance matrix, where \\(K\\) is the value of the data variable K.\n\n\n\nMatrix variables may be constrained to represent correlation matrices. A matrix is a correlation matrix if it is symmetric and positive definite, has entries between \\(-1\\) and \\(1\\), and has a unit diagonal. Because correlation matrices are square, only one dimension needs to be declared. For example,\ncorr_matrix[3] Sigma;\ndeclares Sigma to be a \\(3 \\times 3\\) correlation matrix.\nCorrelation matrices may be assigned to other matrices, including unconstrained matrices, if their dimensions match, and vice-versa.\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a covariance matrix. This is often more convenient or more efficient than representing covariance matrices directly.\nA Cholesky factor \\(L\\) is an \\(M \\times N\\) lower-triangular matrix (if \\(m &lt; n\\) then \\(L[m, n] =0\\)) with a strictly positive diagonal (\\(L[k, k]\n&gt; 0\\)) and \\(M \\geq N\\). If \\(L\\) is a Cholesky factor, then \\(\\Sigma = L\n\\, L^{\\top}\\) is a covariance matrix (i.e., it is positive definite). The mapping between positive definite matrices and their Cholesky factors is bijective—every covariance matrix has a unique Cholesky factorization.\nThe typical case of a square Cholesky factor may be declared with a single dimension,\ncholesky_factor_cov[4] L;\n\n\nIn general, two dimensions may be declared, with the above being equal to cholesky_factor_cov[4, 4]. The type cholesky_factor_cov[M, N] may be used for the general \\(M \\times N\\) case to produce positive semi-definite matrices of rank \\(M\\).\n\n\n\n\nMatrix variables may be constrained to represent the Cholesky factors of a correlation matrix.\nA Cholesky factor for a correlation matrix \\(L\\) is a \\(K \\times K\\) lower-triangular matrix with positive diagonal entries and rows that are of length 1 (i.e., \\(\\sum_{n=1}^K L_{m,n}^2 = 1\\)). If \\(L\\) is a Cholesky factor for a correlation matrix, then \\(L\\,L^{\\top}\\) is a correlation matrix (i.e., symmetric positive definite with a unit diagonal).\nTo declare the variable L to be a K by K Cholesky factor of a correlation matrix, the following code may be used.\ncholesky_factor_corr[K] L;\n\n\n\nConstrained variables of all types may be assigned to other variables of the same unconstrained type and vice-versa. Matching is interpreted strictly as having the same basic type and number of array dimensions. Constraints are not considered, but basic data types are. For instance, a variable declared to be real&lt;lower=0, upper=1&gt; could be assigned to a variable declared as real and vice-versa. Similarly, a variable declared as matrix[3, 3] may be assigned to a variable declared as cov_matrix[3] or cholesky_factor_cov[3], and vice-versa.\nChecks are carried out at the end of each relevant block of statements to ensure constraints are enforced. This includes run-time size checks. The Stan compiler isn’t able to catch the fact that an attempt may be made to assign a matrix of one dimensionality to a matrix of mismatching dimensionality.\n\n\n\nReal-valued vectors, row vectors and matrices may be assigned to complex-valued vectors, row vectors and matrices, respectively. For example, the following is legal.\nvector[N] v = ...;\ncomplex_vector[N] u = 2 * v;\nRow vectors and matrices work the same way.\n\n\n\nVariables may be declared with sizes given by expressions. Such expressions are constrained to only contain data or transformed data variables. This ensures that all sizes are determined once the data is read in and transformed data variables defined by their statements. For example, the following is legal.\ndata {\n  int&lt;lower=0&gt; N_observed, N_missing;\n  // ...\ntransformed parameters {\n  vector[N_observed + N_missing] y;\n  // ...\n\n\n\nIf v is a column vector or row vector, then v[2] is the second element in the vector. If m is a matrix, then m[2, 3] is the value in the second row and third column.\nProviding a matrix with a single index returns the specified row. For instance, if m is a matrix, then m[2] is the second row. This allows Stan blocks such as\nmatrix[M, N] m;\nrow_vector[N] v;\nreal x;\n// ...\nv = m[2];\nx = v[3];   // x == m[2][3] == m[2, 3]\nThe type of m[2] is row_vector because it is the second row of m. Thus it is possible to write m[2][3] instead of m[2, 3] to access the third element in the second row. When given a choice, the form m[2, 3] is preferred.\nComplex versions work the same way,\ncomplex_matrix[M, N] m = ...;\ncomplex_row_vector[N] u = m[3];\ncomplex_vector[M] v = m[ , 2];\n\n\nThe form m[2, 3] is more efficient because it does not require the creation and use of an intermediate expression template for m[2]. In later versions, explicit calls to m[2][3] may be optimized to be as efficient as m[2, 3] by the Stan compiler.\n\n\n\n\nAn integer expression is used to pick out the sizes of vectors, matrices, and arrays. For instance, we can declare a vector of size M + N using\nvector[M + N] y;\nAny integer-denoting expression may be used for the size declaration, providing all variables involved are either data, transformed data, or local variables. That is, expressions used for size declarations may not include parameters or transformed parameters or generated quantities.\n\n\n\n\nStan supports arrays of arbitrary dimension. The values in an array can be any type, so that arrays may contain values that are simple reals or integers, vectors, matrices, or other arrays. Arrays are the only way to store sequences of integers, and some functions in Stan, such as discrete distributions, require integer arguments.\nA two-dimensional array is just an array of arrays, both conceptually and in terms of current implementation. When an index is supplied to an array, it returns the value at that index. When more than one index is supplied, this indexing operation is chained. For example, if a is a two-dimensional array, then a[m, n] is just a convenient shorthand for a[m][n].\nVectors, matrices, and arrays are not assignable to one another, even if their dimensions are identical.\nFor constructing arrays in Stan, see Vector, Matrix, and Array Expressions.\n\n\nArrays are declared with the keyword array followed by the dimensions enclosed in square brackets, the element type, and the name of the variable.\nThe variable n is declared as an array of five integers as follows.\narray[5] int n;\nA two-dimensional array of complex values with three rows and four columns is declared as follows.\narray[3, 4] complex a;\nA three-dimensional array z of positive reals with five rows, four columns, and two shelves can be declared as follows.\narray[5, 4, 2] real&lt;lower=0&gt; z;\nArrays may also be declared to contain vectors. For example,\narray[3] vector[7] mu;\ndeclares mu to be an array of size 3 containing vectors with 7 elements. Arrays may also contain matrices. The example\narray[15, 12] complex_matrix[7, 2] mu;\ndeclares a 15 by 12 array of \\(7 \\times 2\\) complex matrices. Any of the constrained types may also be used in arrays, as in the declaration\narray[2, 3, 4] cholesky_factor_cov[5, 6] mu;\nof a \\(2 \\times 3 \\times 4\\) array of \\(5 \\times 6\\) Cholesky factors of covariance matrices.\n\n\n\nIf x is a 1-dimensional array of length 5, then x[1] is the first element in the array and x[5] is the last. For a \\(3\n\\times 4\\) array y of two dimensions, y[1, 1] is the first element and y[3, 4] the last element. For a three-dimensional array z, the first element is z[1, 1, 1], and so on.\nSubarrays of arrays may be accessed by providing fewer than the full number of indexes. For example, suppose y is a two-dimensional array with three rows and four columns. Then y[3] is one-dimensional array of length four. This means that y[3][1] may be used instead of y[3, 1] to access the value of the first column of the third row of y. The form y[3, 1] is the preferred form (see note in this chapter).\n\n\n\nSubarrays may be manipulated and assigned just like any other variables. Similar to the behavior of matrices, Stan allows blocks such as\narray[9, 10, 11] real w;\narray[10, 11] real x;\narray[11] real y;\nreal z;\n// ...\nx = w[5];\ny = x[4];  // y == w[5][4] == w[5, 4]\nz = y[3];  // z == w[5][4][3] == w[5, 4, 3]\nComplex-valued arrays work the same way.\n\n\n\nArrays of vectors and matrices are accessed in the same way as arrays of doubles. Consider the following vector and scalar declarations.\narray[3, 4] vector[5] a;\narray[4] vector[5] b;\nvector[5] c;\nreal x;\nWith these declarations, the following assignments are legal.\nb = a[1];       // result is array of vectors\nc = a[1, 3];    // result is vector\nc = b[3];       //   same result as above\nx = a[1, 3, 5]; // result is scalar\nx = b[3, 5];    //   same result as above\nx = c[5];       //   same result as above\nRow vectors and other derived vector types (simplex and ordered) behave the same way in terms of indexing.\nConsider the following matrix, vector and scalar declarations.\narray[3, 4] matrix[6, 5] d;\narray[4] matrix[6, 5] e;\nmatrix[6, 5] f;\nrow_vector[5] g;\nreal x;\nWith these declarations, the following definitions are legal.\ne = d[1];           // result is array of matrices\nf = d[1, 3];        // result is matrix\nf = e[3];           //   same result as above\ng = d[1, 3, 2];     // result is row vector\ng = e[3, 2];        //   same result as above\ng = f[2];           //   same result as above\nx = d[1, 3, 5, 2];  // result is scalar\nx = e[3, 5, 2];     //   same result as above\nx = f[5, 2];        //   same result as above\nx = g[2];           //   same result as above\nAs shown, the result f[2] of supplying a single index to a matrix is the indexed row, here row 2 of matrix f.\n\n\n\nSubarrays of arrays may be assigned by indexing on the left-hand side of an assignment statement. For example, the following is legal.\narray[I, J, K] real x;\narray[J, K] real y;\narray[K] real z;\n// ...\nx[1] = y;\nx[1, 1] = z;\nThe sizes must match. Here, x[1] is a J by K array, as is y.\nPartial array assignment also works for arrays of matrices, vectors, and row vectors.\n\n\n\nArrays, row vectors, column vectors and matrices are not interchangeable in Stan. Thus a variable of any one of these fundamental types is not assignable to any of the others, nor may it be used as an argument where the other is required (use as arguments follows the assignment rules).\n\n\nFor example, vectors cannot be assigned to arrays or vice-versa.\narray[4] real a;\nvector[4] b;\nrow_vector[4] c;\n// ...\na = b; // illegal assignment of vector to array\nb = a; // illegal assignment of array to vector\na = c; // illegal assignment of row vector to array\nc = a; // illegal assignment of array to row vector\n\n\n\nIt is not even legal to assign row vectors to column vectors or vice versa.\nvector[4] b;\nrow_vector[4] c;\n// ...\nb = c; // illegal assignment of row vector to column vector\nc = b; // illegal assignment of column vector to row vector\n\n\n\nThe same holds for matrices, where 2-dimensional arrays may not be assigned to matrices or vice-versa.\narray[3, 4] real a;\nmatrix[3, 4] b;\n// ...\na = b;  // illegal assignment of matrix to array\nb = a;  // illegal assignment of array to matrix\n\n\n\nA \\(1 \\times N\\) matrix cannot be assigned a row vector or vice versa.\nmatrix[1, 4] a;\nrow_vector[4] b;\n// ...\na = b;  // illegal assignment of row vector to matrix\nb = a;  // illegal assignment of matrix to row vector\nSimilarly, an \\(M \\times 1\\) matrix may not be assigned to a column vector.\nmatrix[4, 1] a;\nvector[4] b;\n// ...\na = b;  // illegal assignment of column vector to matrix\nb = a;  // illegal assignment of matrix to column vector\n\n\n\n\nAn integer expression is used to pick out the sizes of arrays. The same restrictions as for vector and matrix sizes apply, namely that the size is declared with an integer-denoting expression that does not contain any parameters, transformed parameters, or generated quantities.\n\n\n\nIf any of an array’s dimensions is size zero, the entire array will be of size zero. That is, if we declare\narray[3, 0] real a;\nthen the resulting size of a is zero and querying any of its dimensions at run time will result in the value zero. Declared as above, a[1] will be a size-zero one-dimensional array. For comparison, declaring\narray[0, 3] real b;\nalso produces an array with an overall size of zero, but in this case, there is no way to index legally into b, because b[0] is undefined. The array will behave at run time as if it’s a \\(0 \\times\n0\\) array. For example, the result of to_matrix(b) will be a \\(0 \\times 0\\) matrix, not a \\(0 \\times 3\\) matrix.\n\n\n\n\nStan supports tuples of arbitrary size. The values in a tuple can be of arbitrary type, but the component types must be declared along with the declaration of the tuple. Tuples can be manipulated as a whole, or their elements may be accessed and set individually.\n\n\nTuples are declared with the keyword tuple followed by a parenthesized sequence of types, which determine the types of the respective tuple entries. For example, a tuple with three elements may be declared as\ntuple(int, vector[3], complex) abc;\nTuples must have at least two entries, so the following declarations are illegal.\ntuple() nil;  // ILLEGAL\ntuple(int) n;  // ILLEGAL\nTuples can be assigned as a whole if their elements can be assigned individually. For example, a can be assigned to b in the following example because int can be promoted to complex.\ntuple(int, real) a;\n...\ntuple(complex, real) b = a;\nTuple types may have elements which are declared as tuples, such as the following example.\ntuple(int, tuple(real, complex)) x;\nIn this case, it would probably be simpler to use a 3-tuple type, tuple(int, real, complex).\nTuples can be declared with constraints anywhere that ordinary variables can (i.e., as top-level block variables). That means any context in which it is legal to have a declaration\nreal&lt;lower=0&gt; sigma;\nreal&lt;lower=0, upper=1&gt; theta;\nit is legal to have a tuple with constraints such as\ntuple(real&lt;lower=0&gt;, real&lt;lower=0, upper=1&gt;) sigma_theta;\n\n\n\nTuple elements may be accessed directly. For example, with our declaration of abc from the last section, Stan uses abc.1 for the first element, abc.2 for the second, and abc.3 for the third. These numbers must be integer literals (i.e., they cannot be variables), and must be within the size of the number of elements of tuples. The types of elements are as declared, so that abc.1 is of type int, abc.2 of type vector[3] and abc.3 of type complex.\n\n\n\nTuple elements can be assigned individually, allowing, e.g.,\ntuple(int, real) ab;\nab.1 = 123;\nab.2 = 12.9;\nAs with other assignments, promotions will happen if necessary (of int to real and of real to complex, along with the corresponding container type promotions).\n\n\n\nFor convenience of using values stored in tuples, Stan supports “unpacking” (or “destructuring”) of tuples in an assignment statement.\nGiven a tuple t of type tuple(T1, ..., Tn) and a sequence of assignable expressions of types v1, …, vn, where each vi has a type which is assignable from type Ti, individual elements of the tuple may be assigned to the corresponding variables in the sequence by the statement\n(v1, /*...*/, vn) = t;\nNote that the above parenthesis are required, unlike in some other languages with similar features (e.g., Python).\nThese unpacking assignments can be nested if the tuple on the right hand side contains nested tuples.\nFor example, if T is a tuple of type tuple(int, (real, real), complex), then the program\nint i;\nreal x, y;\ncomplex z;\n\n(i, (x, y), z) = T;\nAssigns the result of T.1 to i, the result of T.2.1 to x, the result of T.2.2 to y, and the result of T.3 to z.\nThe left hand side must match in size the tuple on the right. Additionally, the same variable may not appear more than once in the left hand side of an unpacking assignment.\n\n\n\n\nThe type information associated with a variable only contains the underlying type and dimensionality of the variable.\n\n\nThe size associated with a given variable is not part of its data type. For example, declaring a variable using\narray[3] real a;\ndeclares the variable a to be an array. The fact that it was declared to have size 3 is part of its declaration, but not part of its underlying type.\n\n\nSizes are determined dynamically (at run time) and thus cannot be type-checked statically when the program is compiled. As a result, any conformance error on size will raise a run-time error. For example, trying to assign an array of size 5 to an array of size 6 will cause a run-time error. Similarly, multiplying an \\(N\n\\times M\\) by a \\(J \\times K\\) matrix will raise a run-time error if \\(M\n\\neq J\\).\n\n\n\n\nLike sizes, constraints are not treated as part of a variable’s type in Stan when it comes to the compile-time check of operations it may participate in. Anywhere Stan accepts a matrix as an argument, it will syntactically accept a correlation matrix or covariance matrix or Cholesky factor. Thus a covariance matrix may be assigned to a matrix and vice-versa.\nSimilarly, a bounded real may be assigned to an unconstrained real and vice-versa.\n\n\nFor arguments to functions, constraints are sometimes, but not always checked when the function is called. Exclusions include C++ standard library functions. All probability functions and cumulative distribution functions check that their arguments are appropriate at run time as the function is called.\n\n\n\nFor data variables, constraints are checked after the variable is read from a data file or other source. For transformed data variables, the check is done after the statements in the transformed data block have executed. Thus it is legal for intermediate values of variables to not satisfy declared constraints.\nFor parameters, constraints are enforced by the transform applied and do not need to be checked. For transformed parameters, the check is done after the statements in the transformed parameter block have executed.\nFor all blocks defining variables (transformed data, transformed parameters, generated quantities), real values are initialized to NaN and integer values are initialized to the smallest legal integer (i.e., a large absolute value negative number).\nFor generated quantities, constraints are enforced after the statements in the generated quantities block have executed.\n\n\n\n\nIn order to refer to data types, it is convenient to have a way to refer to them. The type naming notation outlined in this section is not part of the Stan programming language, but rather a convention adopted in this document to enable a concise description of a type.\nBecause size information is not part of a data type, data types will be written without size information. For instance, array[] real is the type of one-dimensional array of reals and matrix is the type of matrices. The three-dimensional integer array type is written as array[,,] int, indicating the number slots available for indexing. Similarly, array[,] vector is the type of a two-dimensional array of vectors.\n\n\n\n\nVariables in Stan are declared by giving a type and a name. For example\nint N;\nvector[N] y;\narray[5] matrix[3, 4] A;\ndeclares a variable N that is an integer, a variable y that is a vector of length N (the previously declared variable), and a variable A, which is a length-5 array where each element is a 3 by 4 matrix.\nThere are several different places a variable is declared in Stan. They are block variables, like those inside data, which can have constraints and must include sizes for their types, like in the above examples. Local variables, like those defined inside loops or local blocks cannot be constrained, but still include sizes. Finally, variables declared as function parameters are not constrained types and exclude sizes.\n// valid block variables, but not locals or function parameters\nvector&lt;lower=0&gt;[N] u;\n\n// valid as a block or local variable, but not a function parameter\narray[3] int is;\n\n// function parameters exclude sizes and cannot be constrained\nvoid pretty_print_tri_lower(matrix x) { ... }\nIn the following table, the leftmost column is a list of the unconstrained and undimensioned basic types; these are used as function return types and argument types. The middle column is of unconstrained types with dimensions; these are used as local variable types. The variables M and N indicate number of columns and rows, respectively. The variable K is used for square matrices, i.e., K denotes both the number of rows and columns. The rightmost column lists the corresponding constrained types. An expression of any right-hand column type may be assigned to its corresponding left-hand column basic type. At runtime, dimensions are checked for consistency for all variables; containers of any sizes may be assigned to function arguments. The constrained matrix types cov_matrix[K], corr_matrix[K], cholesky_factor_cov[K], and cholesky_factor_corr[K] are only assignable to matrices of dimensions matrix[K, K] types.\n\n\n\n\n\n\n\n\nFunction Argument (unsized)\n   Local\n(unconstrained)\n              Block\n          (constrained)\n\n\n\n\nint\nint\nint\n\n\n\n\nint&lt;lower=L&gt;\n\n\n\n\nint&lt;upper=U&gt;\n\n\n\n\nint&lt;lower=L, upper=U&gt;\n\n\n\n\nint&lt;offset=O&gt;\n\n\n\n\nint&lt;multiplier=M&gt;\n\n\n\n\nint&lt;offset=O, multiplier=M&gt;\n\n\nreal\nreal\nreal\n\n\n\n\nreal&lt;lower=L&gt;\n\n\n\n\nreal&lt;upper=U&gt;\n\n\n\n\nreal&lt;lower=L, upper=U&gt;\n\n\n\n\nreal&lt;offset=O&gt;\n\n\n\n\nreal&lt;multiplier=M&gt;\n\n\n\n\nreal&lt;offset=O, multiplier=M&gt;\n\n\ncomplex\ncomplex\ncomplex\n\n\nvector\nvector[N]\nvector[N]\n\n\n\n\nvector[N]&lt;lower=L&gt;\n\n\n\n\nvector[N]&lt;upper=U&gt;\n\n\n\n\nvector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nvector[N]&lt;offset=O&gt;\n\n\n\n\nvector[N]&lt;multiplier=M&gt;\n\n\n\n\nvector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\nordered[N]\n\n\n\n\npositive_ordered[N]\n\n\n\n\nsimplex[N]\n\n\n\n\nunit_vector[N]\n\n\nrow_vector\nrow_vector[N]\nrow_vector[N]\n\n\n\n\nrow_vector[N]&lt;lower=L&gt;\n\n\n\n\nrow_vector[N]&lt;upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O&gt;\n\n\n\n\nrow_vector[N]&lt;multiplier=M&gt;\n\n\n\n\nrow_vector[N]&lt;offset=O, multiplier=M&gt;\n\n\nmatrix\nmatrix[M, N]\nmatrix[M, N]\n\n\n\n\nmatrix[M, N]&lt;lower=L&gt;\n\n\n\n\nmatrix[M, N]&lt;upper=U&gt;\n\n\n\n\nmatrix[M, N]&lt;lower=L, uppers=U&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O&gt;\n\n\n\n\nmatrix[M, N]&lt;multiplier=M&gt;\n\n\n\n\nmatrix[M, N]&lt;offset=O, multiplier=M&gt;\n\n\n\nmatrix[K, K]\ncorr_matrix[K]\n\n\n\nmatrix[K, K]\ncov_matrix[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_corr[K]\n\n\n\nmatrix[K, K]\ncholesky_factor_cov[K]\n\n\ncomplex_vector\ncomplex_vector[M]\ncomplex_vector[M]\n\n\ncomplex_row_vector\ncomplex_row_vector[N]\ncomplex_row_vector[N]\n\n\ncomplex_matrix\ncomplex_matrix[M, N]\ncomplex_matrix[M,N]\n\n\narray[] vector\narray[M] vector[N]\narray[M] vector[N]\n\n\n\n\narray[M] vector[N]&lt;lower=L&gt;\n\n\n\n\narray[M] vector[N]&lt;upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;lower=L, upper=U&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O&gt;\n\n\n\n\narray[M] vector[N]&lt;multiplier=M&gt;\n\n\n\n\narray[M] vector[N]&lt;offset=O, multiplier=M&gt;\n\n\n\n\narray[M] ordered[N]\n\n\n\n\narray[M] positive_ordered[N]\n\n\n\n\narray[M] simplex[N]\n\n\n\n\narray[M] unit_vector[N]\n\n\n\n\nAdditional array types follow the same basic template as the final example in the table and can contain any of the previous types. The unsized version of arrays with more than one dimension is specified by using commas, e.g. array[ , ] is a 2-D array.\nFor more on how function arguments and return types are declared, consult the User’s Guide chapter on functions.\n\n\n\nStan allows assignable variables to be declared and defined in a single statement. Assignable variables are\n\nlocal variables, and\nvariables declared in the transformed data, transformed parameters, or generated quantities blocks.\n\nFor example, the statement\nint N = 5;\ndeclares the variable N to be an integer scalar type and at the same time defines it to be the value of the expression 5.\n\n\nThe type of the expression on the right-hand side of the assignment must be assignable to the type of the variable being declared. For example, it is legal to have\nreal sum = 0;\neven though 0 is of type int and sum is of type real, because integer-typed scalar expressions can be assigned to real-valued scalar variables. In all other cases, the type of the expression on the right-hand side of the assignment must be identical to the type of the variable being declared.\nVariables of any type may have values assigned to them. For example,\nmatrix[3, 2] a = b;\ndeclares a \\(3 \\times 2\\) matrix variable a and assigns a copy of the value of b to the variable a. The variable b must be of type matrix for the statement to be well formed. For the code to execute successfully, b must be the same shape as a, but this cannot be validated until run time. Because a copy is assigned, subsequent changes to a do not affect b and subsequent changes to b do not affect a.\n\n\n\nThe right-hand side may be any expression which has a type which is assignable to the variable being declared. For example,\nmatrix[3, 2] a = 0.5 * (b + c);\nassigns the matrix variable a to half of the sum of b and c. The only requirement on b and c is that the expression b + c be of type matrix. For example, b could be of type matrix and c of type real, because adding a matrix to a scalar produces a matrix, and the multiplying by a scalar produces another matrix.\nSimilarly,\ncomplex z = 2 + 3i;\nassigns the the complex number \\(2 + 3i\\) to the complex scalar z. The right-hand side expression can be a call to a user defined function, allowing general algorithms to be applied that might not be otherwise expressible as simple expressions (e.g., iterative or recursive algorithms).\n\n\n\nAny variable that is in scope and any function that is available in the block in which the compound declaration and definition appears may be used in the expression on the right-hand side of the compound declaration and definition statement.\n\n\n\n\nStan will interpret multiple comma-separated variable names following a single type as declaring multiple new variables. This is available for all variable declarations in all blocks.\n\n\nThe code:\nreal x, y;\nis equivalent to\nreal x;\nreal y;\nAs a result, all declarations on the same line must be of the same type.\n\n\n\nThe ability to declare multiple variables can be combined with assignments whenever a declare-define is valid, as documented in the section introducing compound declarations and definitions :\nreal x = 3, y = 5.6;\nConstrained data types can also be declared together, so long as the constraint for each variable is the same:\nreal&lt;lower=0&gt; x, y;",
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-    "text": "Arguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the [Removed Features] chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.",
+    "text": "Arguments for built-in and user-defined functions and local variables are required to be basic data types, meaning an unconstrained scalar, vector, or matrix type, or an array of such.\nPassing arguments to functions in Stan works just like assignment to basic types. Stan functions are only specified for the basic data types of their arguments, including array dimensionality, but not for sizes or constraints. Of course, functions often check constraints as part of their behavior.\n\n\nStan provides two primitive data types, real for continuous values and int for integer values. These are both considered scalar types.\n\n\n\nStan provides a complex number data type complex, where a complex number contains both a real and an imaginary component, both of which are of type real. Complex types are considered scalar types.\n\n\n\nStan provides three real-valued matrix data types, vector for column vectors, row_vector for row vectors, and matrix for matrices.\nStan also provides three complex-valued matrix data types, complex_vector for column vectors, complex_row_vector for row vectors, and complex_matrix for matrices.\n\n\n\nAny type (including the constrained types discussed in the next section) can be made into an array type by declaring array arguments. For example,\narray[10] real x;\narray[6, 7] matrix[3, 3] m;\narray[12, 8, 15] complex z;\ndeclares x to be a one-dimensional array of size 10 containing real values, declares m to be a two-dimensional array of size \\(6 \\times 7\\) containing values that are \\(3 \\times 3\\) matrices, and declares z to be a \\(12 \\times 8 \\times 15\\) array of complex numbers.\nPrior to 2.26 Stan models used a different syntax which has since been removed. See the Removed Features chapter for more details.\n\n\n\nFor any sequence of types, Stan provides a tuple data type. For example,\ntuple(real, array[5] int) xi;\ndeclares xi to be a tuple holding two values, the first of which is of type type real and the second of which a 5-dimensional array of type int.\n\n\n\nDeclarations of variables other than local variables may be provided with constraints. These constraints are not part of the underlying data type for a variable, but determine error checking in the transformed data, transformed parameter, and generated quantities block, and the transform from unconstrained to constrained space in the parameters block.\nAll of the basic data types other than complex may be given lower and upper bounds using syntax such as\nint&lt;lower=1&gt; N;\nreal&lt;upper=0&gt; log_p;\nvector&lt;lower=-1, upper=1&gt;[3] rho;\nThere are also special data types for structured vectors and matrices. There are four constrained vector data types, simplex for unit simplexes, unit_vector for unit-length vectors, ordered for ordered vectors of scalars and positive_ordered for vectors of positive ordered scalars. There are specialized matrix data types corr_matrix and cov_matrix for correlation matrices (symmetric, positive definite, unit diagonal) and covariance matrices (symmetric, positive definite). The type cholesky_factor_cov is for Cholesky factors of covariance matrices (lower triangular, positive diagonal, product with own transpose is a covariance matrix). The type cholesky_factor_corr is for Cholesky factors of correlation matrices (lower triangular, positive diagonal, unit-length rows).\nConstraints provide error checking for variables defined in the data, transformed data, transformed parameters, and generated quantities blocks. Constraints are critical for variables declared in the parameters block, where they determine the transformation from constrained variables (those satisfying the declared constraint) to unconstrained variables (those ranging over all of \\(\\mathbb{R}^n\\)).\nIt is worth calling out the most important aspect of constrained data types:\n\nThe model must have support (non-zero density, equivalently finite log density) at parameter values that satisfy the declared constraints.\n\nIf this condition is violated with parameter values that satisfy declared constraints but do not have finite log density, then the samplers and optimizers may have any of a number of pathologies including just getting stuck, failure to initialize, excessive Metropolis rejection, or biased draws due to inability to explore the tails of the distribution.",
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-    "text": "Unfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their values wrap. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but rounds the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.",
+    "text": "Unfortunately, the lovely mathematical abstraction of integers and real numbers is only partially supported by finite-precision computer arithmetic.\n\n\nStan uses 32-bit (4-byte) integers for all of its integer representations. The maximum value that can be represented as an integer is \\(2^{31}-1\\); the minimum value is \\(-(2^{31})\\).\nWhen integers overflow, their value is determined by the underlying architecture. On most, their values wrap, but this cannot be guaranteed. Thus it is up to the Stan programmer to make sure the integer values in their programs stay in range. In particular, every intermediate expression must have an integer value that is in range.\nInteger arithmetic works in the expected way for addition, subtraction, and multiplication, but truncates the result of division (see the Stan Functions Reference integer-valued arithmetic operators section for more information).\n\n\n\nStan uses 64-bit (8-byte) floating point representations of real numbers. Stan roughly1 follows the IEEE 754 standard for floating-point computation. The range of a 64-bit number is roughly \\(\\pm 2^{1022}\\), which is slightly larger than \\(\\pm 10^{307}\\). It is a good idea to stay well away from such extreme values in Stan models as they are prone to cause overflow.\n64-bit floating point representations have roughly 15 decimal digits of accuracy. But when they are combined, the result often has less accuracy. In some cases, the difference in accuracy between two operands and their result is large.\nThere are three special real values used to represent (1) not-a-number value for error conditions, (2) positive infinity for overflow, and (3) negative infinity for overflow. The behavior of these special numbers follows standard IEEE 754 behavior.\n\n\nThe not-a-number value propagates. If an argument to a real-valued function is not-a-number, it either rejects (an exception in the underlying C++) or returns not-a-number itself. For boolean-valued comparison operators, if one of the arguments is not-a-number, the return value is always zero (i.e., false).\n\n\n\nPositive infinity is greater than all numbers other than itself and not-a-number; negative infinity is similarly smaller. Adding an infinite value to a finite value returns the infinite value. Dividing a finite number by an infinite value returns zero; dividing an infinite number by a finite number returns the infinite number of appropriate sign. Dividing a finite number by zero returns positive infinity. Dividing two infinite numbers produces a not-a-number value as does subtracting two infinite numbers. Some functions are sensitive to infinite values; for example, the exponential function returns zero if given negative infinity and positive infinity if given positive infinity. Often the gradients will break down when values are infinite, making these boundary conditions less useful than they may appear at first.\n\n\n\n\nStan automatically promotes integer values to real values if necessary, but does not automatically demote real values to integers. For very large integers, this will cause a rounding error to fewer significant digits in the floating point representation than in the integer representation.\nUnlike in C++, real values are never demoted to integers. Therefore, real values may only be assigned to real variables. Integer values may be assigned to either integer variables or real variables. Internally, the integer representation is cast to a floating-point representation. This operation is not without overhead and should thus be avoided where possible.",
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-    "text": "All variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be read as data in the dump format.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.",
+    "text": "All variables used in a Stan program must have an explicitly declared data type. The form of a declaration includes the type and the name of a variable. This section covers scalar types, namely integer, real, and complex. The next section covers vector and matrix types, and the following section array types.\n\n\nUnconstrained integers are declared using the int keyword. For example, the variable N is declared to be an integer as follows.\nint N;\n\n\n\nInteger data types may be constrained to allow values only in a specified interval by providing a lower bound, an upper bound, or both. For instance, to declare N to be a positive integer, use the following.\nint&lt;lower=1&gt; N;\nThis illustrates that the bounds are inclusive for integers.\nTo declare an integer variable cond to take only binary values, that is zero or one, a lower and upper bound must be provided, as in the following example.\nint&lt;lower=0, upper=1&gt; cond;\n\n\n\nUnconstrained real variables are declared using the keyword real. The following example declares theta to be an unconstrained continuous value.\nreal theta;\n\n\n\nUnconstrained complex numbers are declared using the keyword complex. The following example declares z to be an unconstrained complex variable.\ncomplex z;\n\n\n\nReal variables may be bounded using the same syntax as integers. In theory (that is, with arbitrary-precision arithmetic), the bounds on real values would be exclusive. Unfortunately, finite-precision arithmetic rounding errors will often lead to values on the boundaries, so they are allowed in Stan.\nThe variable sigma may be declared to be non-negative as follows.\nreal&lt;lower=0&gt; sigma;\nThe following declares the variable x to be less than or equal to \\(-1\\).\nreal&lt;upper=-1&gt; x;\nTo ensure rho takes on values between \\(-1\\) and \\(1\\), use the following declaration.\nreal&lt;lower=-1, upper=1&gt; rho;\n\n\nLower bounds that are negative infinity or upper bounds that are positive infinity are ignored. Stan provides constants positive_infinity() and negative_infinity() which may be used for this purpose, or they may be supplied as data.\n\n\n\n\nReal variables may be declared on a space that has been transformed using an affine transformation \\(x\\mapsto \\mu + \\sigma * x\\) with offset \\(\\mu\\) and (positive) multiplier \\(\\sigma\\), using a syntax similar to that for bounds. While these transforms do not change the asymptotic sampling behaviour of the resulting Stan program (in a sense, the model the program implements), they can be useful for making the sampling process more efficient by transforming the geometry of the problem to a more natural multiplier and to a more natural offset for the sampling process, for instance by facilitating a non-centered parameterisation. While these affine transformation declarations do not impose a hard constraint on variables, they behave like the bounds constraints in many ways and could perhaps be viewed as acting as a sort of soft constraint.\nThe variable x may be declared to have offset \\(1\\) as follows.\nreal&lt;offset=1&gt; x;\nSimilarly, it can be declared to have multiplier \\(2\\) as follows.\nreal&lt;multiplier=2&gt; x;\nFinally, we can combine both declarations to declare a variable with offset \\(1\\) and multiplier \\(2\\).\nreal&lt;offset=1, multiplier=2&gt; x;\nAs an example, we can give x a normal distribution with non-centered parameterization as follows.\nparameters {\n  real&lt;offset=mu, multiplier=sigma&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nRecall that the centered parameterization is achieved with the code\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\nor equivalently\nparameters {\n  real&lt;offset=0, multiplier=1&gt; x;\n}\nmodel {\n  x ~ normal(mu, sigma);\n}\n\n\n\nBounds (and offset and multiplier) for integer or real variables may be arbitrary expressions. The only requirement is that they only include variables that have been declared (though not necessarily defined) before the declaration. array[N] row_vector[D] x; If the bounds themselves are parameters, the behind-the-scenes variable transform accounts for them in the log Jacobian.\nFor example, it is acceptable to have the following declarations.\ndata {\n real lb;\n}\nparameters {\n   real&lt;lower=lb&gt; phi;\n}\nThis declares a real-valued parameter phi to take values greater than the value of the real-valued data variable lb. Constraints may be arbitrary expressions, but must be of type int for integer variables and of type real for real variables (including constraints on vectors, row vectors, and matrices). Variables used in constraints can be any variable that has been defined at the point the constraint is used. For instance,\ndata {\n   int&lt;lower=1&gt; N;\n   array[N] real y;\n}\nparameters {\n   real&lt;lower=min(y), upper=max(y)&gt; phi;\n}\nThis declares a positive integer data variable N, an array y of real-valued data of length N, and then a parameter ranging between the minimum and maximum value of y. As shown in the example code, the functions min() and max() may be applied to containers such as arrays.\nA more subtle case involves declarations of parameters or transformed parameters based on parameters declared previously. For example, the following program will work as intended.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;  // enforces a &lt; b\n}\ntransformed parameters {\n  real c;\n  real&lt;lower=c&gt; d;\n  c = a;\n  d = b;\n}\nThe parameters instance works because all parameters are defined externally before the block is executed. The transformed parameters case works even though c isn’t defined at the point it is used, because constraints on transformed parameters are only validated at the end of the block. Data variables work like parameter variables, whereas transformed data and generated quantity variables work like transformed parameter variables.\n\n\n\nA variable may be declared with a size that depends on a boolean constant. For example, consider the definition of alpha in the following program fragment.\ndata {\n  int&lt;lower=0, upper=1&gt; include_alpha;\n  // ...\n}\nparameters {\n  vector[include_alpha ? N : 0] alpha;\n  // ...\n}\nIf include_alpha is true, the model will include the vector alpha; if the flag is false, the model will not include alpha (technically, it will include alpha of size 0, which means it won’t contain any values and won’t be included in any output).\nThis technique is not just useful for containers. If the value of N is set to 1, then the vector alpha will contain a single element and thus alpha[1] behaves like an optional scalar, the existence of which is controlled by include_alpha.\nThis coding pattern allows a single Stan program to define different models based on the data provided as input. This strategy is used extensively in the implementation of the RStanArm package.",
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     "href": "reference-manual/syntax.html",
     "title": "Language Syntax",
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-    "text": "This chapter defines the basic syntax of the Stan modeling language using a Backus-Naur form (BNF) grammar plus extra-grammatical constraints on function typing and operator precedence and associativity.\n\n\n\n\nIn the following BNF grammars, tokens are represented in ALLCAPS. Grammar non-terminals are surrounded by &lt; and &gt;. A square brackets ([A]) indicates optionality of A. A postfixed Kleene star (A*) indicates zero or more occurrences of A. Parenthesis can be used to group symbols together in productions.\nFinally, this grammar uses the concept of “parameterized nonterminals” as used in the parsing library Menhir. A rule like &lt;list(x)&gt; ::= x (COMMA x)* declares a generic list rule, which can later be applied to others by the symbol &lt;list(&lt;expression&gt;)&gt;.\nThe following representation is constructed directly from the OCaml reference parser using a tool called Obelisk. The raw output is available here.\n\n\n\n\n&lt;program&gt; ::= [&lt;function_block&gt;] [&lt;data_block&gt;] [&lt;transformed_data_block&gt;]\n              [&lt;parameters_block&gt;] [&lt;transformed_parameters_block&gt;]\n              [&lt;model_block&gt;] [&lt;generated_quantities_block&gt;] EOF\n\n&lt;functions_only&gt; ::= &lt;function_def&gt;* EOF\n\n&lt;function_block&gt; ::= FUNCTIONBLOCK LBRACE &lt;function_def&gt;* RBRACE\n\n&lt;data_block&gt; ::= DATABLOCK LBRACE &lt;top_var_decl_no_assign&gt;* RBRACE\n\n&lt;transformed_data_block&gt; ::= TRANSFORMEDDATABLOCK LBRACE\n                             &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;parameters_block&gt; ::= PARAMETERSBLOCK LBRACE &lt;top_var_decl_no_assign&gt;*\n                       RBRACE\n\n&lt;transformed_parameters_block&gt; ::= TRANSFORMEDPARAMETERSBLOCK LBRACE\n                                   &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;model_block&gt; ::= MODELBLOCK LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;generated_quantities_block&gt; ::= GENERATEDQUANTITIESBLOCK LBRACE\n                                 &lt;top_vardecl_or_statement&gt;* RBRACE\n\n\n\n&lt;function_def&gt; ::= &lt;return_type&gt; &lt;decl_identifier&gt; LPAREN [&lt;arg_decl&gt; (COMMA\n                   &lt;arg_decl&gt;)*] RPAREN &lt;statement&gt;\n\n&lt;return_type&gt; ::= VOID\n                | &lt;unsized_type&gt;\n\n&lt;arg_decl&gt; ::= [DATABLOCK] &lt;unsized_type&gt; &lt;decl_identifier&gt;\n\n&lt;unsized_type&gt; ::= ARRAY &lt;unsized_dims&gt; &lt;basic_type&gt;\n                 | ARRAY &lt;unsized_dims&gt; &lt;unsized_tuple_type&gt;\n                 | &lt;basic_type&gt;\n                 | &lt;unsized_tuple_type&gt;\n\n&lt;unsized_tuple_type&gt; ::= TUPLE LPAREN &lt;unsized_type&gt; COMMA &lt;unsized_type&gt;\n                         (COMMA &lt;unsized_type&gt;)* RPAREN\n\n&lt;basic_type&gt; ::= INT\n               | REAL\n               | COMPLEX\n               | VECTOR\n               | ROWVECTOR\n               | MATRIX\n               | COMPLEXVECTOR\n               | COMPLEXROWVECTOR\n               | COMPLEXMATRIX\n\n&lt;unsized_dims&gt; ::= LBRACK COMMA* RBRACK\n\n\n\n\n&lt;identifier&gt; ::= IDENTIFIER\n               | TRUNCATE\n\n&lt;decl_identifier&gt; ::= &lt;identifier&gt;\n\n&lt;no_assign&gt; ::= UNREACHABLE\n\n&lt;optional_assignment(rhs)&gt; ::= [ASSIGN rhs]\n\n&lt;id_and_optional_assignment(rhs)&gt; ::= &lt;decl_identifier&gt;\n                                      &lt;optional_assignment(rhs)&gt;\n\n&lt;decl(type_rule, rhs)&gt; ::= type_rule &lt;decl_identifier&gt; &lt;dims&gt;\n                           &lt;optional_assignment(rhs)&gt; SEMICOLON\n                         | &lt;higher_type(type_rule)&gt;\n                           &lt;id_and_optional_assignment(rhs)&gt; (COMMA\n                           &lt;id_and_optional_assignment(rhs)&gt;)* SEMICOLON\n\n&lt;higher_type(type_rule)&gt; ::= &lt;array_type(type_rule)&gt;\n                           | &lt;tuple_type(type_rule)&gt;\n                           | type_rule\n\n&lt;array_type(type_rule)&gt; ::= &lt;arr_dims&gt; type_rule\n                          | &lt;arr_dims&gt; &lt;tuple_type(type_rule)&gt;\n\n&lt;tuple_type(type_rule)&gt; ::= TUPLE LPAREN &lt;higher_type(type_rule)&gt; COMMA\n                            &lt;higher_type(type_rule)&gt; (COMMA\n                            &lt;higher_type(type_rule)&gt;)* RPAREN\n\n&lt;var_decl&gt; ::= &lt;decl(&lt;sized_basic_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl_no_assign&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;no_assign&gt;)&gt;\n                           | SEMICOLON\n\n&lt;sized_basic_type&gt; ::= INT\n                     | REAL\n                     | COMPLEX\n                     | VECTOR LBRACK &lt;expression&gt; RBRACK\n                     | ROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | MATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt; RBRACK\n                     | COMPLEXVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXMATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt;\n                       RBRACK\n\n&lt;top_var_type&gt; ::= INT [LABRACK &lt;range&gt; RABRACK]\n                 | REAL &lt;type_constraint&gt;                  | TUPLE\n\n                 | COMPLEX &lt;type_constraint&gt;\n                 | VECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | ROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | MATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | COMPLEXVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | COMPLEXROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt;\n                   RBRACK\n                 | COMPLEXMATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | ORDERED LBRACK &lt;expression&gt; RBRACK\n                 | POSITIVEORDERED LBRACK &lt;expression&gt; RBRACK\n                 | SIMPLEX LBRACK &lt;expression&gt; RBRACK\n                 | UNITVECTOR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCORR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCOV LBRACK &lt;expression&gt; [COMMA &lt;expression&gt;]\n                   RBRACK\n                 | CORRMATRIX LBRACK &lt;expression&gt; RBRACK\n                 | COVMATRIX LBRACK &lt;expression&gt; RBRACK\n\n&lt;type_constraint&gt; ::= [LABRACK &lt;range&gt; RABRACK]\n                    | LABRACK &lt;offset_mult&gt; RABRACK\n\n&lt;range&gt; ::= LOWER ASSIGN &lt;constr_expression&gt; COMMA UPPER ASSIGN\n            &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt; COMMA LOWER ASSIGN\n            &lt;constr_expression&gt;\n          | LOWER ASSIGN &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt;\n\n&lt;offset_mult&gt; ::= OFFSET ASSIGN &lt;constr_expression&gt; COMMA MULTIPLIER ASSIGN\n                  &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt; COMMA OFFSET ASSIGN\n                  &lt;constr_expression&gt;\n                | OFFSET ASSIGN &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt;\n\n&lt;arr_dims&gt; ::= ARRAY LBRACK &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACK\n\n\n\n\n&lt;expression&gt; ::= &lt;expression&gt; QMARK &lt;expression&gt; COLON &lt;expression&gt;\n               | &lt;expression&gt; &lt;infixOp&gt; &lt;expression&gt;\n               | &lt;prefixOp&gt; &lt;expression&gt;\n               | &lt;expression&gt; &lt;postfixOp&gt;\n               | &lt;common_expression&gt;\n\n&lt;constr_expression&gt; ::= &lt;constr_expression&gt; &lt;arithmeticBinOp&gt;\n                        &lt;constr_expression&gt;\n                      | &lt;prefixOp&gt; &lt;constr_expression&gt;\n                      | &lt;constr_expression&gt; &lt;postfixOp&gt;\n                      | &lt;common_expression&gt;\n\n&lt;common_expression&gt; ::= &lt;identifier&gt;\n                      | INTNUMERAL\n                      | REALNUMERAL\n                      | DOTNUMERAL\n                      | IMAGNUMERAL\n                      | LBRACE &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACE\n                      | LBRACK [&lt;expression&gt; (COMMA &lt;expression&gt;)*] RBRACK\n                      | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                        &lt;expression&gt;)*] RPAREN\n                      | TARGET LPAREN RPAREN\n                      | &lt;identifier&gt; LPAREN &lt;expression&gt; BAR [&lt;expression&gt;\n                        (COMMA &lt;expression&gt;)*] RPAREN\n                      | LPAREN &lt;expression&gt; COMMA &lt;expression&gt; (COMMA\n                        &lt;expression&gt;)* RPAREN\n                      | &lt;common_expression&gt; DOTNUMERAL\n                      | &lt;common_expression&gt; LBRACK &lt;indexes&gt; RBRACK\n                      | LPAREN &lt;expression&gt; RPAREN\n\n&lt;prefixOp&gt; ::= BANG\n             | MINUS\n             | PLUS\n\n&lt;postfixOp&gt; ::= TRANSPOSE\n\n&lt;infixOp&gt; ::= &lt;arithmeticBinOp&gt;\n            | &lt;logicalBinOp&gt;\n\n&lt;arithmeticBinOp&gt; ::= PLUS\n                    | MINUS\n                    | TIMES\n                    | DIVIDE\n                    | IDIVIDE\n                    | MODULO\n                    | LDIVIDE\n                    | ELTTIMES\n                    | ELTDIVIDE\n                    | HAT\n                    | ELTPOW\n\n&lt;logicalBinOp&gt; ::= OR\n                 | AND\n                 | EQUALS\n                 | NEQUALS\n                 | LABRACK\n                 | LEQ\n                 | RABRACK\n                 | GEQ\n\n&lt;indexes&gt; ::= epsilon\n            | COLON\n            | &lt;expression&gt;\n            | &lt;expression&gt; COLON\n            | COLON &lt;expression&gt;\n            | &lt;expression&gt; COLON &lt;expression&gt;\n            | &lt;indexes&gt; COMMA &lt;indexes&gt;\n\n&lt;printables&gt; ::= &lt;expression&gt;\n               | &lt;string_literal&gt;\n               | &lt;printables&gt; COMMA &lt;printables&gt;\n\n\n\n&lt;statement&gt; ::= &lt;atomic_statement&gt;\n              | &lt;nested_statement&gt;\n\n&lt;atomic_statement&gt; ::= &lt;common_expression&gt; &lt;assignment_op&gt; &lt;expression&gt;\n                       SEMICOLON\n                     | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                       &lt;expression&gt;)*] RPAREN SEMICOLON\n                     | &lt;expression&gt; TILDE &lt;identifier&gt; LPAREN [&lt;expression&gt;\n                       (COMMA &lt;expression&gt;)*] RPAREN [&lt;truncation&gt;] SEMICOLON\n                     | TARGET PLUSASSIGN &lt;expression&gt; SEMICOLON\n                     | BREAK SEMICOLON\n                     | CONTINUE SEMICOLON\n                     | PRINT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | REJECT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | RETURN &lt;expression&gt; SEMICOLON\n                     | RETURN SEMICOLON\n                     | SEMICOLON\n\n&lt;assignment_op&gt; ::= ASSIGN\n                  | PLUSASSIGN\n                  | MINUSASSIGN\n                  | TIMESASSIGN\n                  | DIVIDEASSIGN\n                  | ELTTIMESASSIGN\n                  | ELTDIVIDEASSIGN\n\n&lt;string_literal&gt; ::= STRINGLITERAL\n\n&lt;truncation&gt; ::= TRUNCATE LBRACK [&lt;expression&gt;] COMMA [&lt;expression&gt;] RBRACK\n\n&lt;nested_statement&gt; ::= IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                       ELSE &lt;vardecl_or_statement&gt;\n                     | IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | WHILE LPAREN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; COLON\n                       &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | PROFILE LPAREN &lt;string_literal&gt; RPAREN LBRACE\n                       &lt;vardecl_or_statement&gt;* RBRACE\n                     | LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;vardecl_or_statement&gt; ::= &lt;statement&gt;\n                         | &lt;var_decl&gt;\n\n&lt;top_vardecl_or_statement&gt; ::= &lt;statement&gt;\n                             | &lt;top_var_decl&gt;\n\n\n\n\nMany of the tokens used in the BNF grammars follow obviously from their names: DATABLOCK is the literal string ‘data’, COMMA is a single ‘,’ character, etc. The literal representation of each operator is additionally provided in the operator precedence table.\nA few tokens are not so obvious, and are defined here in regular expressions:\nIDENTIFIER = [a-zA-Z] [a-zA-Z0-9_]*\n\nSTRINGLITERAL = \".*\"\n\nINTNUMERAL = [0-9]+ (_ [0-9]+)*\n\nEXPLITERAL = [eE] [+-]? INTNUMERAL\n\nREALNUMERAL = INTNUMERAL \\. INTNUMERAL? EXPLITERAL?\n            | \\. INTNUMERAL EXPLITERAL\n            | INTNUMERAL EXPLITERAL\n\nIMAGNUMERAL = (REALNUMERAL | INTNUMERAL) i\n\nDOTNUMERAL = \\. INTNUMERAL\n\n\n\n\n\nA well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type matrix[ , , ], a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
+    "text": "This chapter defines the basic syntax of the Stan modeling language using a Backus-Naur form (BNF) grammar plus extra-grammatical constraints on function typing and operator precedence and associativity.\n\n\n\n\nIn the following BNF grammars, tokens are represented in ALLCAPS. Grammar non-terminals are surrounded by &lt; and &gt;. A square brackets ([A]) indicates optionality of A. A postfixed Kleene star (A*) indicates zero or more occurrences of A. Parenthesis can be used to group symbols together in productions.\nFinally, this grammar uses the concept of “parameterized nonterminals” as used in the parsing library Menhir. A rule like &lt;list(x)&gt; ::= x (COMMA x)* declares a generic list rule, which can later be applied to others by the symbol &lt;list(&lt;expression&gt;)&gt;.\nThe following representation is constructed directly from the OCaml reference parser using a tool called Obelisk. The raw output is available here.\n\n\n\n\n&lt;program&gt; ::= [&lt;function_block&gt;] [&lt;data_block&gt;] [&lt;transformed_data_block&gt;]\n              [&lt;parameters_block&gt;] [&lt;transformed_parameters_block&gt;]\n              [&lt;model_block&gt;] [&lt;generated_quantities_block&gt;] EOF\n\n&lt;functions_only&gt; ::= &lt;function_def&gt;* EOF\n\n&lt;function_block&gt; ::= FUNCTIONBLOCK LBRACE &lt;function_def&gt;* RBRACE\n\n&lt;data_block&gt; ::= DATABLOCK LBRACE &lt;top_var_decl_no_assign&gt;* RBRACE\n\n&lt;transformed_data_block&gt; ::= TRANSFORMEDDATABLOCK LBRACE\n                             &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;parameters_block&gt; ::= PARAMETERSBLOCK LBRACE &lt;top_var_decl_no_assign&gt;*\n                       RBRACE\n\n&lt;transformed_parameters_block&gt; ::= TRANSFORMEDPARAMETERSBLOCK LBRACE\n                                   &lt;top_vardecl_or_statement&gt;* RBRACE\n\n&lt;model_block&gt; ::= MODELBLOCK LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;generated_quantities_block&gt; ::= GENERATEDQUANTITIESBLOCK LBRACE\n                                 &lt;top_vardecl_or_statement&gt;* RBRACE\n\n\n\n&lt;function_def&gt; ::= &lt;return_type&gt; &lt;decl_identifier&gt; LPAREN [&lt;arg_decl&gt; (COMMA\n                   &lt;arg_decl&gt;)*] RPAREN &lt;statement&gt;\n\n&lt;return_type&gt; ::= VOID\n                | &lt;unsized_type&gt;\n\n&lt;arg_decl&gt; ::= [DATABLOCK] &lt;unsized_type&gt; &lt;decl_identifier&gt;\n\n&lt;unsized_type&gt; ::= ARRAY &lt;unsized_dims&gt; &lt;basic_type&gt;\n                 | ARRAY &lt;unsized_dims&gt; &lt;unsized_tuple_type&gt;\n                 | &lt;basic_type&gt;\n                 | &lt;unsized_tuple_type&gt;\n\n&lt;unsized_tuple_type&gt; ::= TUPLE LPAREN &lt;unsized_type&gt; COMMA &lt;unsized_type&gt;\n                         (COMMA &lt;unsized_type&gt;)* RPAREN\n\n&lt;basic_type&gt; ::= INT\n               | REAL\n               | COMPLEX\n               | VECTOR\n               | ROWVECTOR\n               | MATRIX\n               | COMPLEXVECTOR\n               | COMPLEXROWVECTOR\n               | COMPLEXMATRIX\n\n&lt;unsized_dims&gt; ::= LBRACK COMMA* RBRACK\n\n\n\n\n&lt;identifier&gt; ::= IDENTIFIER\n               | TRUNCATE\n\n&lt;decl_identifier&gt; ::= &lt;identifier&gt;\n\n&lt;no_assign&gt; ::= UNREACHABLE\n\n&lt;optional_assignment(rhs)&gt; ::= [ASSIGN rhs]\n\n&lt;id_and_optional_assignment(rhs)&gt; ::= &lt;decl_identifier&gt;\n                                      &lt;optional_assignment(rhs)&gt;\n\n&lt;decl(type_rule, rhs)&gt; ::= type_rule &lt;decl_identifier&gt; &lt;dims&gt;\n                           &lt;optional_assignment(rhs)&gt; SEMICOLON\n                         | &lt;higher_type(type_rule)&gt;\n                           &lt;id_and_optional_assignment(rhs)&gt; (COMMA\n                           &lt;id_and_optional_assignment(rhs)&gt;)* SEMICOLON\n\n&lt;higher_type(type_rule)&gt; ::= &lt;array_type(type_rule)&gt;\n                           | &lt;tuple_type(type_rule)&gt;\n                           | type_rule\n\n&lt;array_type(type_rule)&gt; ::= &lt;arr_dims&gt; type_rule\n                          | &lt;arr_dims&gt; &lt;tuple_type(type_rule)&gt;\n\n&lt;tuple_type(type_rule)&gt; ::= TUPLE LPAREN &lt;higher_type(type_rule)&gt; COMMA\n                            &lt;higher_type(type_rule)&gt; (COMMA\n                            &lt;higher_type(type_rule)&gt;)* RPAREN\n\n&lt;var_decl&gt; ::= &lt;decl(&lt;sized_basic_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;expression&gt;)&gt;\n\n&lt;top_var_decl_no_assign&gt; ::= &lt;decl(&lt;top_var_type&gt;, &lt;no_assign&gt;)&gt;\n                           | SEMICOLON\n\n&lt;sized_basic_type&gt; ::= INT\n                     | REAL\n                     | COMPLEX\n                     | VECTOR LBRACK &lt;expression&gt; RBRACK\n                     | ROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | MATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt; RBRACK\n                     | COMPLEXVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXROWVECTOR LBRACK &lt;expression&gt; RBRACK\n                     | COMPLEXMATRIX LBRACK &lt;expression&gt; COMMA &lt;expression&gt;\n                       RBRACK\n\n&lt;top_var_type&gt; ::= INT [LABRACK &lt;range&gt; RABRACK]\n                 | REAL &lt;type_constraint&gt;                  | TUPLE\n\n                 | COMPLEX &lt;type_constraint&gt;\n                 | VECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | ROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | MATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | COMPLEXVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt; RBRACK\n                 | COMPLEXROWVECTOR &lt;type_constraint&gt; LBRACK &lt;expression&gt;\n                   RBRACK\n                 | COMPLEXMATRIX &lt;type_constraint&gt; LBRACK &lt;expression&gt; COMMA\n                   &lt;expression&gt; RBRACK\n                 | ORDERED LBRACK &lt;expression&gt; RBRACK\n                 | POSITIVEORDERED LBRACK &lt;expression&gt; RBRACK\n                 | SIMPLEX LBRACK &lt;expression&gt; RBRACK\n                 | UNITVECTOR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCORR LBRACK &lt;expression&gt; RBRACK\n                 | CHOLESKYFACTORCOV LBRACK &lt;expression&gt; [COMMA &lt;expression&gt;]\n                   RBRACK\n                 | CORRMATRIX LBRACK &lt;expression&gt; RBRACK\n                 | COVMATRIX LBRACK &lt;expression&gt; RBRACK\n\n&lt;type_constraint&gt; ::= [LABRACK &lt;range&gt; RABRACK]\n                    | LABRACK &lt;offset_mult&gt; RABRACK\n\n&lt;range&gt; ::= LOWER ASSIGN &lt;constr_expression&gt; COMMA UPPER ASSIGN\n            &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt; COMMA LOWER ASSIGN\n            &lt;constr_expression&gt;\n          | LOWER ASSIGN &lt;constr_expression&gt;\n          | UPPER ASSIGN &lt;constr_expression&gt;\n\n&lt;offset_mult&gt; ::= OFFSET ASSIGN &lt;constr_expression&gt; COMMA MULTIPLIER ASSIGN\n                  &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt; COMMA OFFSET ASSIGN\n                  &lt;constr_expression&gt;\n                | OFFSET ASSIGN &lt;constr_expression&gt;\n                | MULTIPLIER ASSIGN &lt;constr_expression&gt;\n\n&lt;arr_dims&gt; ::= ARRAY LBRACK &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACK\n\n\n\n\n&lt;expression&gt; ::= &lt;expression&gt; QMARK &lt;expression&gt; COLON &lt;expression&gt;\n               | &lt;expression&gt; &lt;infixOp&gt; &lt;expression&gt;\n               | &lt;prefixOp&gt; &lt;expression&gt;\n               | &lt;expression&gt; &lt;postfixOp&gt;\n               | &lt;common_expression&gt;\n\n&lt;constr_expression&gt; ::= &lt;constr_expression&gt; &lt;arithmeticBinOp&gt;\n                        &lt;constr_expression&gt;\n                      | &lt;prefixOp&gt; &lt;constr_expression&gt;\n                      | &lt;constr_expression&gt; &lt;postfixOp&gt;\n                      | &lt;common_expression&gt;\n\n&lt;common_expression&gt; ::= &lt;identifier&gt;\n                      | INTNUMERAL\n                      | REALNUMERAL\n                      | DOTNUMERAL\n                      | IMAGNUMERAL\n                      | LBRACE &lt;expression&gt; (COMMA &lt;expression&gt;)* RBRACE\n                      | LBRACK [&lt;expression&gt; (COMMA &lt;expression&gt;)*] RBRACK\n                      | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                        &lt;expression&gt;)*] RPAREN\n                      | TARGET LPAREN RPAREN\n                      | &lt;identifier&gt; LPAREN &lt;expression&gt; BAR [&lt;expression&gt;\n                        (COMMA &lt;expression&gt;)*] RPAREN\n                      | LPAREN &lt;expression&gt; COMMA &lt;expression&gt; (COMMA\n                        &lt;expression&gt;)* RPAREN\n                      | &lt;common_expression&gt; DOTNUMERAL\n                      | &lt;common_expression&gt; LBRACK &lt;indexes&gt; RBRACK\n                      | LPAREN &lt;expression&gt; RPAREN\n\n&lt;prefixOp&gt; ::= BANG\n             | MINUS\n             | PLUS\n\n&lt;postfixOp&gt; ::= TRANSPOSE\n\n&lt;infixOp&gt; ::= &lt;arithmeticBinOp&gt;\n            | &lt;logicalBinOp&gt;\n\n&lt;arithmeticBinOp&gt; ::= PLUS\n                    | MINUS\n                    | TIMES\n                    | DIVIDE\n                    | IDIVIDE\n                    | MODULO\n                    | LDIVIDE\n                    | ELTTIMES\n                    | ELTDIVIDE\n                    | HAT\n                    | ELTPOW\n\n&lt;logicalBinOp&gt; ::= OR\n                 | AND\n                 | EQUALS\n                 | NEQUALS\n                 | LABRACK\n                 | LEQ\n                 | RABRACK\n                 | GEQ\n\n&lt;indexes&gt; ::= epsilon\n            | COLON\n            | &lt;expression&gt;\n            | &lt;expression&gt; COLON\n            | COLON &lt;expression&gt;\n            | &lt;expression&gt; COLON &lt;expression&gt;\n            | &lt;indexes&gt; COMMA &lt;indexes&gt;\n\n&lt;printables&gt; ::= &lt;expression&gt;\n               | &lt;string_literal&gt;\n               | &lt;printables&gt; COMMA &lt;printables&gt;\n\n\n\n&lt;statement&gt; ::= &lt;atomic_statement&gt;\n              | &lt;nested_statement&gt;\n\n&lt;atomic_statement&gt; ::= &lt;common_expression&gt; &lt;assignment_op&gt; &lt;expression&gt;\n                       SEMICOLON\n                     | &lt;identifier&gt; LPAREN [&lt;expression&gt; (COMMA\n                       &lt;expression&gt;)*] RPAREN SEMICOLON\n                     | &lt;expression&gt; TILDE &lt;identifier&gt; LPAREN [&lt;expression&gt;\n                       (COMMA &lt;expression&gt;)*] RPAREN [&lt;truncation&gt;] SEMICOLON\n                     | TARGET PLUSASSIGN &lt;expression&gt; SEMICOLON\n                     | BREAK SEMICOLON\n                     | CONTINUE SEMICOLON\n                     | PRINT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | REJECT LPAREN &lt;printables&gt; RPAREN SEMICOLON\n                     | RETURN &lt;expression&gt; SEMICOLON\n                     | RETURN SEMICOLON\n                     | SEMICOLON\n\n&lt;assignment_op&gt; ::= ASSIGN\n                  | PLUSASSIGN\n                  | MINUSASSIGN\n                  | TIMESASSIGN\n                  | DIVIDEASSIGN\n                  | ELTTIMESASSIGN\n                  | ELTDIVIDEASSIGN\n\n&lt;string_literal&gt; ::= STRINGLITERAL\n\n&lt;truncation&gt; ::= TRUNCATE LBRACK [&lt;expression&gt;] COMMA [&lt;expression&gt;] RBRACK\n\n&lt;nested_statement&gt; ::= IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                       ELSE &lt;vardecl_or_statement&gt;\n                     | IF LPAREN &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | WHILE LPAREN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; COLON\n                       &lt;expression&gt; RPAREN &lt;vardecl_or_statement&gt;\n                     | FOR LPAREN &lt;identifier&gt; IN &lt;expression&gt; RPAREN\n                       &lt;vardecl_or_statement&gt;\n                     | PROFILE LPAREN &lt;string_literal&gt; RPAREN LBRACE\n                       &lt;vardecl_or_statement&gt;* RBRACE\n                     | LBRACE &lt;vardecl_or_statement&gt;* RBRACE\n\n&lt;vardecl_or_statement&gt; ::= &lt;statement&gt;\n                         | &lt;var_decl&gt;\n\n&lt;top_vardecl_or_statement&gt; ::= &lt;statement&gt;\n                             | &lt;top_var_decl&gt;\n\n\n\n\nMany of the tokens used in the BNF grammars follow obviously from their names: DATABLOCK is the literal string ‘data’, COMMA is a single ‘,’ character, etc. The literal representation of each operator is additionally provided in the operator precedence table.\nA few tokens are not so obvious, and are defined here in regular expressions:\nIDENTIFIER = [a-zA-Z] [a-zA-Z0-9_]*\n\nSTRINGLITERAL = \".*\"\n\nINTNUMERAL = [0-9]+ (_ [0-9]+)*\n\nEXPLITERAL = [eE] [+-]? INTNUMERAL\n\nREALNUMERAL = INTNUMERAL \\. INTNUMERAL? EXPLITERAL?\n            | \\. INTNUMERAL EXPLITERAL\n            | INTNUMERAL EXPLITERAL\n\nIMAGNUMERAL = (REALNUMERAL | INTNUMERAL) i\n\nDOTNUMERAL = \\. INTNUMERAL\n\n\n\n\n\nA well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type array[ , , ] matrix, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
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-    "text": "A well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type matrix[ , , ], a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
+    "text": "A well-formed Stan program must satisfy the type constraints imposed by functions and distributions. For example, the binomial distribution requires an integer total count parameter and integer variate and when truncated would require integer truncation points. If these constraints are violated, the program will be rejected during compilation with an error message indicating the location of the problem.\n\n\n\nIn the Stan grammar provided in this chapter, the expression 1 + 2 * 3 has two parses. As described in the operator precedence table, Stan disambiguates between the meaning \\(1\n+ (2 \\times 3)\\) and the meaning \\((1 + 2) \\times 3\\) based on operator precedences and associativities.\n\n\n\nIn a compound variable declaration and definition, the type of the right-hand side expression must be assignable to the variable being declared. The assignability constraint restricts compound declarations and definitions to local variables and variables declared in the transformed data, transformed parameters, and generated quantities blocks.\n\n\n\nThe types of expressions used for elements in array expressions ('{' expressions '}') must all be of the same type or a mixture of scalar (int, real and complex) types (in which case the result is promoted to be of the highest type on the int -&gt; real -&gt; complex hierarchy).\n\n\n\nInteger literals longer than one digit may not start with 0 and real literals cannot consist of only a period or only an exponent.\n\n\n\nBoth the conditional if-then-else statement and while-loop statement require the expression denoting the condition to be a primitive type, integer or real.\n\n\n\nThe for loop statement requires that we specify in addition to the loop identifier, either a range consisting of two expressions denoting an integer, separated by ‘:’, or a single expression denoting a container. The loop variable will be of type integer in the former case and of the contained type in the latter case. Furthermore, the loop variable must not be in scope (i.e., there is no masking of variables).\n\n\n\nThe arguments to a print statement cannot be void.\n\n\n\nThe break and continue statements may only be used within the body of a for-loop or while-loop.\n\n\n\nSome constructs in the Stan language are only allowed in certain blocks or in certain kinds of user-defined functions.\n\n\nFunctions ending in _rng may only be called in the transformed data and generated quantities block, and within the bodies of user-defined functions with names ending in _rng.\n\n\n\nUnnormalized distributions (with suffixes _lupmf or _lupdf) may only be called in the model block, user-defined probability functions, or within the bodies of user defined functions which end in _lp.\n\n\n\ntarget += statements can only be used inside of the model block or user-defined functions which end in _lp.\nUser defined functions which end in _lp and the target() function can only be used in the model block, transformed parameters block, and in the bodies of other user defined functions which end in _lp.\nSampling statements (using ~) can only be used in the model block or in the bodies of user-defined functions which end in _lp.\n\n\n\n\nA probability function literal must have one of the following suffixes: _lpdf, _lpmf, _lcdf, or _lccdf.\n\n\n\nStandalone expressions used as indexes must denote either an integer (int) or an integer array (array[] int). Expressions participating in range indexes (e.g., a and b in a : b) must denote integers (int).\nA second condition is that there not be more indexes provided than dimensions of the underlying expression (in general) or variable (on the left side of assignments) being indexed. A vector or row vector adds 1 to the array dimension and a matrix adds 2. That is, the type array[ , , ] matrix, a three-dimensional array of matrices, has five index positions: three for the array, one for the row of the matrix and one for the column.",
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     "title": "Licenses and Dependencies",
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-    "text": "Stan and its dependent libraries, are distributed under generous, freedom-respecting licenses approved by the Open Source Initiative.\nIn particular, the licenses for Stan and its dependent libraries have no “copyleft” provisions requiring applications of Stan to be open source if they are redistributed.\nThis chapter specifies the licenses for the libraries on which Stan’s math library, language, and algorithms depend. The last tool mentioned, Google Test, is only used for testing and is not needed to run Stan.\n\n\nStan is distributed under\n\nBSD 3-clause license (BSD New)\n\nThe copyright holder of each contribution is the developer or his or her assignee.1\n\n\n\nStan uses the Boost library for template metaprograms, traits programs, the parser, and various numerical libraries for special functions, probability functions, and random number generators. Boost is distributed under the\n\nBoost Software License version 1.0\n\nThe copyright for each Boost package is held by its developers or their assignees.\n\n\n\nStan uses the Eigen library for matrix arithmetic and linear algebra. Eigen is distributed under the\n\nMozilla Public License, version 2\n\nThe copyright of Eigen is owned jointly by its developers or their assignees.\n\n\n\nStan uses the SUNDIALS package for solving differential equations. SUNDIALS is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.\n\n\n\nStan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of Google Test is owned by Google, Inc.",
+    "text": "Stan and its dependent libraries, are distributed under generous, freedom-respecting licenses approved by the Open Source Initiative.\nIn particular, the licenses for Stan and its dependent libraries have no “copyleft” provisions requiring applications of Stan to be open source if they are redistributed.\nThis chapter specifies the licenses for the libraries on which Stan’s math library, language, and algorithms depend. The last tool mentioned, Google Test, is only used for testing and is not needed to run Stan.\n\n\nStan is distributed under\n\nBSD 3-clause license (BSD New)\n\nThe copyright holder of each contribution is the developer or his or her assignee.1\n\n\n\nStan uses the Boost library for template metaprograms, traits programs, the parser, and various numerical libraries for special functions, probability functions, and random number generators. Boost is distributed under the\n\nBoost Software License version 1.0\n\nThe copyright for each Boost package is held by its developers or their assignees.\n\n\n\nStan uses the Eigen library for matrix arithmetic and linear algebra. Eigen is distributed under the\n\nMozilla Public License, version 2\n\nThe copyright of Eigen is owned jointly by its developers or their assignees.\n\n\n\nStan uses the SUNDIALS package for solving differential equations. SUNDIALS is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of SUNDIALS is owned by Lawrence Livermore National Security Lab.\n\n\n\nStan uses the Threaded Building Blocks (TBB) library for parallel computations. TBB is distributed under the\n\nApache License, version 2\n\nThe copyright of TBB is owned by Intel Corporation.\n\n\n\nStan uses Google Test for unit testing; it is not required to compile or execute models. Google Test is distributed under the\n\nBSD 3-clause license (BSD New)\n\nThe copyright of Google Test is owned by Google, Inc.",
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-    "text": "An expression is the syntactic unit in a Stan program that denotes a value. Every expression in a well-formed Stan program has a type that is determined statically (at compile time), based only on the type of its variables and the types of the functions used in it. If an expressions type cannot be determined statically, the Stan compiler will report the location of the problem.\nThis chapter covers the syntax, typing, and usage of the various forms of expressions in Stan.\n\n\nThe simplest form of expression is a literal that denotes a primitive numerical value.\n\n\nInteger literals represent integers of type int. Integer literals are written in base 10 without any separators. Integer literals may contain a single negative sign. (The expression --1 is interpreted as the negation of the literal -1.)\nThe following list contains well-formed integer literals.\n0, 1, -1, 256, -127098, 24567898765\nInteger literals must have values that fall within the bounds for integer values (see the section on numerical data types).\nInteger literals may not contain decimal points (.). Thus the expressions 1. and 1.0 are of type real and may not be used where a value of type int is required.\n\n\n\nA number written with a period or with scientific notation is assigned to a the continuous numeric type real. Real literals are written in base 10 with a period (.) as a separator and optionally an exponent with optional sign. Examples of well-formed real literals include the following.\n0.0, 1.0, 3.14, -217.9387, 2.7e3, -2E-5, 1.23e+3.\nThe notation e or E followed by a positive or negative integer denotes a power of 10 to multiply. For instance, 2.7e3 and 2.7e+3 denote \\(2.7 \\times 10^3\\), whereas -2E-5 denotes \\(-2 \\times\n10^{-5}\\).\n\n\n\nA number followed by the character i denotes an imaginary number and is assigned to the numeric type complex. The number preceding i may be either a real or integer literal and determines the magnitude of the imaginary number. Examples of well-formed imaginary literals include the following.\n1i, 2i, -325.786i, 1e10i, 2.87e-10i.\nNote that the character i by itself is not a well-formed imaginary literal. The unit imaginary number must be written as 1i.\n\n\n\nStan does not include complex literals directly, but a real or integer literal can be added to an imaginary literal to derive an expression that behaves like a complex literal. Examples include the following.\n1 + 2i, -3.2e9 + 1e10i\nThese will be assigned the type complex, which is the result of adding a real or integer and a complex number. They will also function like literals in the sense that the C++ compiler is able to reduce them to a single complex constant at compile time.\n\n\n\n\nA variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, get_lp, increment_log_prob,\ntarget\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.\n\n\n\n\n\nExpressions for the Stan container objects, namely arrays, vectors, row vectors, matrices, and tuples, can all be constructed using expressions.\n\n\nSquare brackets may be wrapped around a sequence of comma separated primitive expressions to produce a row vector expression. For example, the expression [ 1, 10, 100 ] denotes a row vector of three elements with real values 1.0, 10.0, and 100.0. Applying the transpose operator to a row vector expression produces a vector expression. This syntax provides a way declare and define small vectors a single line, as follows.\nrow_vector[2] rv2 =  [ 1, 2 ];\nvector[3] v3 = [ 3, 4, 5 ]';\nThe vector expression values may be compound expressions or variable names, so it is legal to write [ 2 * 3, 1 + 4] or [ x, y ], providing that x and y are primitive variables.\n\n\n\nA matrix expression consists of square brackets wrapped around a sequence of comma separated row vector expressions. This syntax provides a way declare and define a matrix in a single line, as follows.\nmatrix[3, 2] m1 = [ [ 1, 2 ], [ 3, 4 ], [5, 6 ] ];\nAny expression denoting a row vector can be used in a matrix expression. For example, the following code is valid:\nvector[2] vX = [ 1, 10 ]';\nrow_vector[2] vY = [ 100, 1000 ];\nmatrix[3, 2] m2 = [ vX', vY, [ 1, 2 ]  ];\n\n\n\nComplex vector expressions work the same way as real vector expressions. For example, the following are all legal Stan expressions and assignments.\ncomplex_vector[3] = [1 + 2i, 3 - 1.7i, 0]';\ncomplex_row_vector[2] = [12, -2i];\ncomplex_matrix[2, 3] = [[1 + 2i, 3 - 1.7i, 0],\n                        [3.9 - 1.234i, 176i, 1 + 1i]];\n\n\nThe empty expression [ ] is ambiguous and therefore is not allowed and similarly expressions such as [ [ ] ] or [ [ ], [ ] ] are not allowed.\n\n\n\n\nCurly braces may be wrapped around a sequence of expressions to produce an array expression. For example, the expression { 1, 10, 100 } denotes an integer array of three elements with values 1, 10, and 100. This syntax is particularly convenient to define small arrays in a single line, as follows.\narray[3] int a = { 1, 10, 100 };\nThe values may be compound expressions, so it is legal to write { 2 * 3, 1 + 4 }. It is also possible to write two dimensional arrays directly, as in the following example.\narray[2, 3] int b = { { 1, 2, 3 }, { 4, 5, 6 } };\nThis way, b[1] is { 1, 2, 3 } and b[2] is { 4, 5, 6 }.\nWhitespace is always interchangeable in Stan, so the above can be laid out as follows to more clearly indicate the row and column structure of the resulting two dimensional array.\narray[2, 3] int b = { { 1, 2, 3 },\n                { 4, 5, 6 } };\n\n\n\nAny type of expression may be used within braces to form an array expression. In the simplest case, all of the elements will be of the same type and the result will be an array of elements of that type. For example, the elements of the array can be vectors, in which case the result is an array of vectors.\nvector[3] b;\nvector[3] c;\n// ...\narray[2] vector[3] d = { b, c };\nThe elements may also be a mixture of int and real typed expressions, in which case the result is an array of real values.\narray[2] real b = { 1, 1.9 };\n\n\n\nStan uses parentheses around a comma-separated sequence of expressions to construct a tuple. For example, we can construct a 2-tuple as follows.\ntuple(int, vector[3]) xy = (42, [1, 2.9, -1.3]');\nThe expression 42 is of type int and the expression [1, 2.9, -1.3] is of type row_vector so that [1, 2.9, -1.3]' is of type vector and of size 3. The whole tuple expression (42, [1, 2.9, -1.3]') thus has a sized type of tuple(int, vector[3]) and an unsized type (e.g., for a function argument) of tuple(int, vector).\nStan does not support the Python notation with trailing commas, such as (1, 2, 3, ) for a 3-tuple.\n\n\n\nThere are some restrictions on how array expressions may be used that arise from their types being calculated bottom up and the basic data type and assignment rules of Stan.\n\n\nAlthough it is tempting to try to define a ragged array expression, all Stan data types are rectangular (or boxes or other higher-dimensional generalizations). Thus the following nested array expression will cause an error when it tries to create a non-rectangular array.\n{ { 1, 2, 3 }, { 4, 5 } }  // compile time error: size mismatch\nThis may appear to be OK, because it is creating a two-dimensional integer array (array[,] int) out of two one-dimensional array integer arrays (array[] int). But it is not allowed because the two one-dimensional arrays are not the same size. If the elements are array expressions, this can be diagnosed at compile time. If one or both expressions is a variable, then that won’t be caught until runtime.\n{ { 1, 2, 3 }, m }  // runtime error if m not size 3\n\n\n\nBecause there is no way to infer the type of the result, the empty array expression ({ }) is not allowed. This does not sacrifice expressive power, because a declaration is sufficient to initialize a zero-element array.\narray[0] int a;   // a is fully defined as zero element array\n\n\n\nThere is no way to declare or construct a zero-tuple or one-tuple in Stan. Tuples must be at least two elements long. The expression () does not pick out a zero-tuple—it is ill formed. Similarly, the expression (1) is of type int rather than a tuple.\n\n\n\n\n\nAny expression wrapped in parentheses is also an expression. Like in C++, but unlike in R, only the round parentheses, ( and ), are allowed. The square brackets [ and ] are reserved for array indexing and the curly braces { and } for grouping statements.\nWith parentheses it is possible to explicitly group subexpressions with operators. Without parentheses, the expression 1 + 2 * 3 has a subexpression 2 * 3 and evaluates to 7. With parentheses, this grouping may be made explicit with the expression 1 + (2 * 3). More importantly, the expression (1 + 2) *   3 has 1 + 2 as a subexpression and evaluates to 9.\n\n\n\nFor integer and real-valued expressions, Stan supports the basic binary arithmetic operations of addition (+), subtraction (-), multiplication (*) and division (/) in the usual ways.\nFor integer expressions, Stan supports the modulus (%) binary arithmetic operation. Stan also supports the unary operation of negation for integer and real-valued expressions. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3.0 + 0.14\n-15\n2 * 3 + 1\n(x - y) / 2.0\n(n * (n + 1)) / 2\nx / n\nm % n\nThe negation, addition, subtraction, and multiplication operations are extended to matrices, vectors, and row vectors. The transpose operation, written using an apostrophe (') is also supported for vectors, row vectors, and matrices. Return types for matrix operations are the smallest types that can be statically guaranteed to contain the result. The full set of allowable input types and corresponding return types is detailed in the list of functions.\nFor example, if y and mu are variables of type vector and Sigma is a variable of type matrix, then (y - mu)' * Sigma * (y - mu) is a well-formed expression of type real. The type of the complete expression is inferred working outward from the subexpressions. The subexpression(s) y - mu are of type vector because the variables y and mu are of type vector. The transpose of this expression, the subexpression (y - mu)' is of type row_vector. Multiplication is left associative and transpose has higher precedence than multiplication, so the above expression is equivalent to the following fully specified form (((y - mu)') * Sigma) * (y - mu).\nThe type of subexpression (y - mu)' * Sigma is inferred to be row_vector, being the result of multiplying a row vector by a matrix. The whole expression’s type is thus the type of a row vector multiplied by a (column) vector, which produces a real value.\nStan provides elementwise matrix multiplication (e.g., a .* b) and division (e.g., a ./ b) operations. These provide a shorthand to replace loops, but are not intrinsically more efficient than a version programmed with an elementwise calculations and assignments in a loop. For example, given declarations,\nvector[N] a;\nvector[N] b;\nvector[N] c;\nthe assignment,\nc = a .* b;\nproduces the same result with roughly the same efficiency as the loop\nfor (n in 1:N) {\n  c[n] = a[n] * b[n];\n}\nStan supports exponentiation (^) of integer and real-valued expressions. The return type of exponentiation is always a real-value. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3 ^ 2\n3.0 ^ -2\n3.0 ^ 0.14\nx ^ n\nn ^ x\nn ^ m\nx ^ y\nExponentiation is right associative, so the expression 2 ^ 3 ^ 4 is equivalent to the fully specified form 2 ^ (3 ^ 4).\n\n\nThe precedence and associativity of operators, as well as built-in syntax such as array indexing and function application is given in tabular form in the following table.\n\n\nStan’s unary, binary, and ternary operators, with their precedences, associativities, place in an expression, and a description. The last two lines list the precedence of function application and array, matrix, and vector indexing. The operators are listed in order of precedence, from least tightly binding to most tightly binding. The full set of legal arguments and corresponding result types are provided in the function documentation for the operators (i.e., operator*(int, int):int indicates the application of the multiplication operator to two integers, which returns an integer). Parentheses may be used to group expressions explicitly rather than relying on precedence and associativity.\n\n\n\n\n\n\n\n\n\n\nOp.\nPrec.\nAssoc.\nPlacement\nDescription\n\n\n\n\n? ~ :\n10\nright\nternary infix\nconditional\n\n\n||\n9\nleft\nbinary infix\nlogical or\n\n\n&&\n8\nleft\nbinary infix\nlogical and\n\n\n==\n7\nleft\nbinary infix\nequality\n\n\n!=\n7\nleft\nbinary infix\ninequality\n\n\n&lt;\n6\nleft\nbinary infix\nless than\n\n\n&lt;=\n6\nleft\nbinary infix\nless than or equal\n\n\n&gt;\n6\nleft\nbinary infix\ngreater than\n\n\n&gt;=\n6\nleft\nbinary infix\ngreater than or equal\n\n\n+\n5\nleft\nbinary infix\naddition\n\n\n-\n5\nleft\nbinary infix\nsubtraction\n\n\n*\n4\nleft\nbinary infix\nmultiplication\n\n\n.*\n4\nleft\nbinary infix\nelementwise multiplication\n\n\n/\n4\nleft\nbinary infix\n(right) division\n\n\n./\n4\nleft\nbinary infix\nelementwise division\n\n\n%\n4\nleft\nbinary infix\nmodulus\n\n\n\\\n3\nleft\nbinary infix\nleft division\n\n\n%/%\n3\nleft\nbinary infix\ninteger division\n\n\n!\n2\nn/a\nunary prefix\nlogical negation\n\n\n-\n2\nn/a\nunary prefix\nnegation\n\n\n+\n2\nn/a\nunary prefix\npromotion (no-op in Stan)\n\n\n^\n1\nright\nbinary infix\nexponentiation\n\n\n.^\n1\nright\nbinary infix\nelementwise exponentiation\n\n\n'\n0\nn/a\nunary postfix\ntransposition\n\n\n()\n0\nn/a\nprefix, wrap\nfunction application\n\n\n[]\n0\nleft\nprefix, wrap\narray, matrix indexing\n\n\n\nOther expression-forming operations, such as function application and subscripting bind more tightly than any of the arithmetic operations.\nThe precedence and associativity determine how expressions are interpreted. Because addition is left associative, the expression a + b + c is interpreted as (a + b) + c. Similarly, a / b * c is interpreted as (a / b) * c.\nBecause multiplication has higher precedence than addition, the expression a * b + c is interpreted as (a * b) + c and the expression a + b * c is interpreted as a + (b * c). Similarly, 2 * x + 3 * - y is interpreted as (2 * x) + (3 * (-y)).\nTransposition and exponentiation bind more tightly than any other arithmetic or logical operation. For vectors, row vectors, and matrices, -u' is interpreted as -(u'), u * v' as u* (v'), and u' * v as (u') * v. For integer and reals, -n ^ 3 is interpreted as -(n ^ 3).\n\n\n\n\n\n\n\nThe ternary conditional operator is unique in that it takes three arguments and uses a mixed syntax. If a is an expression of type int and b and c are expressions that can be converted to one another (e.g., compared with ==), then\na ? b : c\nis an expression of the promoted type of b and c. The only promotion allowed in Stan is integer -&gt; real -&gt; complex; e.g. if one argument is of type int and the other of type real, the conditional expression as a whole is of type real. In other cases, the arguments have to be of the same underlying Stan type (i.e., constraints don’t count, only the shape) and the conditional expression is of that type.\n\n\nThe conditional operator is the most loosely binding operator, so its arguments rarely require parentheses for disambiguation. For example,\na &gt; 0 || b &lt; 0 ? c + d : e - f\nis equivalent to the explicitly grouped version\n(a &gt; 0 || b &lt; 0) ? (c + d) : (e - f)\nThe latter is easier to read even if the parentheses are not strictly necessary.\n\n\n\nThe conditional operator is right associative, so that\na ? b : c ? d : e\nparses as if explicitly grouped as\na ? b : (c ? d : e)\nAgain, the explicitly grouped version is easier to read.\n\n\n\n\nStan’s conditional operator works very much like its C++ analogue. The first argument must be an expression denoting an integer. Typically this is a variable or a relation operator, as in the variable a in the example above. Then there are two resulting arguments, the first being the result returned if the condition evaluates to true (i.e., non-zero) and the second if the condition evaluates to false (i.e., zero). In the example above, the value b is returned if the condition evaluates to a non-zero value and c is returned if the condition evaluates to zero.\n\n\nThe key property of the conditional operator that makes it so useful in high-performance computing is that it only evaluates the returned subexpression, not the alternative expression. In other words, it is not like a typical function that evaluates its argument expressions eagerly in order to pass their values to the function. As usual, the saving is mostly in the derivatives that do not get computed rather than the unnecessary function evaluation itself.\n\n\n\nIf one return expression is a data value (an expression involving only constants and variables defined in the data or transformed data block), and the other is not, then the ternary operator will promote the data value to a parameter value. This can cause needless work calculating derivatives in some cases and be less efficient than a full if-then conditional statement. For example,\ndata {\n  array[10] real x;\n  // ...\n}\nparameters {\n  array[10] real z;\n  // ...\n}\nmodel {\n  y ~ normal(cond ? x : z, sigma);\n  // ...\n}\nwould be more efficiently (if not more transparently) coded as\nif (cond) {\n  y ~ normal(x, sigma);\n} else {\n  y ~ normal(z, sigma);\n}\nThe conditional statement, like the conditional operator, only evaluates one of the result statements. In this case, the variable x will not be promoted to a parameter and thus not cause any needless work to be carried out when propagating the chain rule during derivative calculations.\n\n\n\n\n\nStan arrays, matrices, vectors, and row vectors are all accessed using the same array-like notation. For instance, if x is a variable of type array [] real (a one-dimensional array of reals) then x[1] is the value of the first element of the array.\nSubscripting has higher precedence than any of the arithmetic operations. For example, alpha * x[1] is equivalent to alpha * (x[1]).\nMultiple subscripts may be provided within a single pair of square brackets. If x is of type array[,] real, a two-dimensional array, then x[2, 501] is of type real.\n\n\nThe subscripting operator also returns subarrays of arrays. For example, if x is of type array[,,] real, then x[2] is of type array[,] real, and x[2, 3] is of type array[] real. As a result, the expressions x[2, 3] and x[2][3] have the same meaning.\n\n\n\nIf Sigma is a variable of type matrix, then Sigma[1] denotes the first row of Sigma and has the type row_vector.\n\n\n\nStan supports mixed indexing of arrays and their vector, row vector or matrix values. For example, if m is of type matrix[ , ], a two-dimensional array of matrices, then m[1] refers to the first row of the array, which is a one-dimensional array of matrices. More than one index may be used, so that m[1, 2] is of type matrix and denotes the matrix in the first row and second column of the array. Continuing to add indices, m[1, 2, 3] is of type row_vector and denotes the third row of the matrix denoted by m[1, 2]. Finally, m[1, 2, 3, 4] is of type real and denotes the value in the third row and fourth column of the matrix that is found at the first row and second column of the array m.\n\n\n\n\nIn addition to single integer indexes, as described in the language indexing section, Stan supports multiple indexing. Multiple indexes can be integer arrays of indexes, lower bounds, upper bounds, lower and upper bounds, or simply shorthand for all of the indexes. A complete table of index types is given in the following table.\n\n\nTypes of indexes and examples with one-dimensional containers of size N and an integer array ii of type array [] real size K.\n\n\n\nindex type\nexample\nvalue\n\n\n\n\ninteger\na[11]\nvalue of a at index 11\n\n\ninteger array\na[ii]\na[ii[1]], …, a[ii[K]]\n\n\nlower bound\na[3:]\na[3], …, a[N]\n\n\nupper bound\na[:5]\na[1], …, a[5]\n\n\nrange\na[2:7]\na[2], …, a[7]\n\n\nall\na[:]\na[1], …, a[N]\n\n\nall\na[]\na[1], …, a[N]\n\n\n\n\n\n\nThe fundamental semantic rule for dealing with multiple indexes is the following. If idxs is a multiple index, then it produces an indexable position in the result. To evaluate that index position in the result, the index is first passed to the multiple index, and the resulting index used.\na[idxs, ...][i, ...] = a[idxs[i], ...][...]\nOn the other hand, if idx is a single index, it reduces the dimensionality of the output, so that\na[idx, ...] = a[idx][...]\nThe only issue is what happens with matrices and vectors. Vectors work just like arrays. Matrices with multiple row indexes and multiple column indexes produce matrices. Matrices with multiple row indexes and a single column index become (column) vectors. Matrices with a single row index and multiple column indexes become row vectors. The types are summarized in the following table.\n\n\nSpecial rules for reducing matrices based on whether the argument is a single or multiple index. Examples are for a matrix a, with integer single indexes i and j and integer array multiple indexes is and js. The same typing rules apply for all multiple indexes.\n\n\n\nexample\nrow index\ncolumn index\nresult type\n\n\n\n\na[i]\nsingle\nn/a\nrow vector\n\n\na[is]\nmultiple\nn/a\nmatrix\n\n\na[i, j]\nsingle\nsingle\nreal\n\n\na[i, js]\nsingle\nmultiple\nrow vector\n\n\na[is, j]\nmultiple\nsingle\nvector\n\n\na[is, js]\nmultiple\nmultiple\nmatrix\n\n\n\nEvaluation of matrices with multiple indexes is defined to respect the following distributivity conditions.\nm[idxs1, idxs2][i, j] = m[idxs1[i], idxs2[j]]\nm[idxs, idx][j] = m[idxs[j], idx]\nm[idx, idxs][j] = m[idx, idxs[j]]\nEvaluation of arrays of matrices and arrays of vectors or row vectors is defined recursively, beginning with the array dimensions.\n\n\n\n\n\nStan provides a range of built in mathematical and statistical functions, which are documented in the built-in function documentation.\nExpressions in Stan may consist of the name of function followed by a sequence of zero or more argument expressions. For instance, log(2.0) is the expression of type real denoting the result of applying the natural logarithm to the value of the real literal 2.0.\nSyntactically, function application has higher precedence than any of the other operators, so that y + log(x) is interpreted as y + (log(x)).\n\n\nEach function has a type signature which determines the allowable type of its arguments and its return type. For instance, the function signature for the logarithm function can be expressed as\nreal log(real);\nand the signature for the lmultiply function is\nreal lmultiply(real, real);\nA function is uniquely determined by its name and its sequence of argument types. For instance, the following two functions are different functions.\nreal mean(array [] real);\nreal mean(vector);\nThe first applies to a one-dimensional array of real values and the second to a vector.\nThe identity conditions for functions explicitly forbids having two functions with the same name and argument types but different return types. This restriction also makes it possible to infer the type of a function expression compositionally by only examining the type of its subexpressions.\n\n\n\nConstants in Stan are nothing more than nullary (no-argument) functions. For instance, the mathematical constants \\(\\pi\\) and \\(e\\) are represented as nullary functions named pi() and e(). See the Stan Functions Reference built-in constants section for a list of built-in constants.\n\n\n\nBecause of integer to real type promotion, rules must be established for which function is called given a sequence of argument types. The scheme employed by Stan is the same as that used by C++, which resolves a function call to the function requiring the minimum number of type promotions.\nFor example, consider a situation in which the following two function signatures have been registered for foo.\nreal foo(real, real);\nint foo(int, int);\nThe use of foo in the expression foo(1.0, 1.0) resolves to foo(real, real), and thus the expression foo(1.0, 1.0) itself is assigned a type of real.\nBecause integers may be promoted to real values, the expression foo(1, 1) could potentially match either foo(real, real) or foo(int, int). The former requires two type promotions and the latter requires none, so foo(1, 1) is resolved to function foo(int, int) and is thus assigned the type int.\nThe expression foo(1, 1.0) has argument types (int, real) and thus does not explicitly match either function signature. By promoting the integer expression 1 to type real, it is able to match foo(real, real), and hence the type of the function expression foo(1, 1.0) is real.\nIn some cases (though not for any built-in Stan functions), a situation may arise in which the function referred to by an expression remains ambiguous. For example, consider a situation in which there are exactly two functions named bar with the following signatures.\nreal bar(real, int);\nreal bar(int, real);\nWith these signatures, the expression bar(1.0, 1) and bar(1, 1.0) resolve to the first and second of the above functions, respectively. The expression bar(1.0, 1.0) is illegal because real values may not be demoted to integers. The expression bar(1, 1) is illegal for a different reason. If the first argument is promoted to a real value, it matches the first signature, whereas if the second argument is promoted to a real value, it matches the second signature. The problem is that these both require one promotion, so the function name bar is ambiguous. If there is not a unique function requiring fewer promotions than all others, as with bar(1, 1) given the two declarations above, the Stan compiler will flag the expression as illegal.\n\n\n\nFor most of the distributions supported by Stan, there is a corresponding random-number generating function. These random number generators are named by the distribution with the suffix _rng. For example, a univariate normal random number can be generated by normal_rng(0, 1); only the parameters of the distribution, here a location (0) and scale (1) are specified because the variate is generated.\n\n\nThe use of random-number generating functions is restricted to the transformed data and generated quantities blocks; attempts to use them elsewhere will result in a parsing error with a diagnostic message. They may also be used in the bodies of user-defined functions whose names end in _rng.\nThis allows the random number generating functions to be used for simulation in general, and for Bayesian posterior predictive checking in particular.\n\n\n\nPosterior predictive checks typically use the parameters of the model to generate simulated data (at the individual and optionally at the group level for hierarchical models), which can then be compared informally using plots and formally by means of test statistics, to the actual data in order to assess the suitability of the model; see Chapter 6 of (Gelman et al. 2013) for more information on posterior predictive checks.\n\n\n\n\n\nStan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).\n\n\n\n\n\nThere are several expression constructions in Stan that act as higher-order functions.2\nThe higher-order functions and the signature of their argument functions are listed in the following pair of tables.\n\n\nHigher-order functions in Stan with their argument function types. The first group of arguments can be a function of parameters or data. The second group of arguments, consisting of a real and integer array in all cases, must be expressions involving only data and literals.\n\n\n\n\n\n\n\n\n\nfunction\nparameter or data args\ndata args\nreturn type\n\n\n\n\nalgebra_solver\nvector, vector\narray [] real, array [] real\nvector\n\n\nalgebra_solver_newton\nvector, vector\narray [] real, array [] real\nvector\n\n\nintegrate_1d,\nreal, real, array [] real\narray [] real, array [] real\nreal\n\n\nintegrate_ode_X,\nreal, array [] real, array [] real\narray [] real, array [] real\narray [] real\n\n\nmap_rect\nvector, vector\narray [] real, array [] real\nvector\n\n\n\nFor example, the integrate_ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  array [] real foo(real t,\n                    array [] real y,\n                    array [] real theta,\n                    array [] real x_r,\n                    array [] real x_i) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\narray[2] real y0;\nreal t0;\narray[T] real ts;\narray[1] real theta;\narray[0] real x_r;\narray[0] int x_i;\n// ...\narray[T, 2] real y_hat = integrate_ode_rk45(foo, y0, t0,\n                                              ts, theta, x_r, x_i);\nThe function argument is foo, the name of the user-defined function; as shown in the higher-order functions table, integrate_ode_rk45 takes a real array, a real, three more real arrays, and an integer array as arguments and returns 2D real array.\n\n\n\nVariadic Higher-order functions in Stan with their argument function types. The first group of arguments are restricted in type. The sequence of trailing arguments can be of any length with any types.\n\n\n\nfunction\nrestricted args\nreturn type\n\n\n\n\nsolve_X\nvector\nvector\n\n\node_X,\nvector, real, array [] real\nvector[]\n\n\nreduce_sum\narray[] T, T1, T2\nreal\n\n\n\nT, T1, and T2 can be any Stan type.\nFor example, the ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  vector foo(real t, vector y, real theta, vector beta,\n            array [] real x_i, int index) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\nvector[2] y0;\nreal t0;\narray[T] real ts;\nreal theta;\nvector[7] beta;\narray[10] int x_i;\nint index;\n// ...\nvector[2] y_hat[T] = ode_rk45(foo, y0, t0, ts, theta,\n                              beta, x_i, index);\nThe function argument is foo, the name of the user-defined function. As shown in the variadic higher-order functions table, ode_rk45 takes a real, a vector, a real, a real array, and a sequence of arguments whose types match those at the end of foo and returns an array of vectors.\n\n\n\nThe function argument to higher-order functions is always passed as the first argument. This function argument must be provided as the name of a user-defined or built-in function. No quotes are necessary.\n\n\n\nSome of the arguments to higher-order functions are restricted to data. This means they must be expressions containing only data variables, transformed data variables, or literals; the may contain arbitrary functions applied to data variables or literals, but must not contain parameters, transformed parameters, or local variables from any block other than transformed data.\nFor user-defined functions the qualifier data may be prepended to the type to restrict the argument to data-only variables.\n\n\n\n\nDerivatives of the log probability function defined by a model are used in several ways by Stan. The Hamiltonian Monte Carlo samplers, including NUTS, use gradients to guide updates. The BFGS optimizers also use gradients to guide search for posterior modes.\n\n\nUnlike evaluations in pure mathematics, evaluation of derivatives in Stan is done by applying the chain rule on an expression-by-expression basis, evaluating using floating-point arithmetic. As a result, models such as the following are problematic for inference involving derivatives.\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(sqrt(x - x), 1);\n}\nAlgebraically, the sampling statement in the model could be reduced to\n  x ~ normal(0, 1);\nand it would seem the model should produce unit normal draws for x. But rather than canceling, the expression sqrt(x -   x) causes a problem for derivatives. The cause is the mechanistic evaluation of the chain rule,\n\\[\n\\begin{array}{rcl}\n\\frac{d}{dx} \\sqrt{x - x}\n& = &\n\\frac{1}{2 \\sqrt{x - x}} \\times \\frac{d}{dx} (x - x)\n\\\\[4pt]\n& = &\n\\frac{1}{0} \\times (1 - 1)\n\\\\[4pt]\n& = &\n\\infty \\times 0\n\\\\[4pt]\n& = & \\mathrm{NaN}.\n\\end{array}\n\\]\nRather than the \\(x - x\\) canceling out, it introduces a 0 into the numerator and denominator of the chain-rule evaluation.\nThe only way to avoid this kind problem is to be careful to do the necessary algebraic reductions as part of the model and not introduce expressions like sqrt(x - x) for which the chain rule produces not-a-number values.\n\n\n\nThe best way to diagnose whether something is going wrong with the derivatives is to use the test-gradient option to the sampler or optimizer inputs; this option is available in both Stan and RStan (though it may be slow, because it relies on finite differences to make a comparison to the built-in automatic differentiation).\nFor example, compiling the above model to an executable sqrt-x-minus-x in CmdStan, the test can be run as\n&gt; ./sqrt-x-minus-x diagnose test=gradient\nwhich produces\n...\nTEST GRADIENT MODE\n\n Log probability=-0.393734\n\n param idx           value           model     finite diff           error\n         0       -0.887393             nan               0             nan\nEven though finite differences calculates the right gradient of 0, automatic differentiation follows the chain rule and produces a not-a-number output.",
+    "text": "An expression is the syntactic unit in a Stan program that denotes a value. Every expression in a well-formed Stan program has a type that is determined statically (at compile time), based only on the type of its variables and the types of the functions used in it. If an expressions type cannot be determined statically, the Stan compiler will report the location of the problem.\nThis chapter covers the syntax, typing, and usage of the various forms of expressions in Stan.\n\n\nThe simplest form of expression is a literal that denotes a primitive numerical value.\n\n\nInteger literals represent integers of type int. Integer literals are written in base 10 without any separators. Integer literals may contain a single negative sign. (The expression --1 is interpreted as the negation of the literal -1.)\nThe following list contains well-formed integer literals.\n0, 1, -1, 256, -127098, 24567898765\nInteger literals must have values that fall within the bounds for integer values (see the section on numerical data types).\nInteger literals may not contain decimal points (.). Thus the expressions 1. and 1.0 are of type real and may not be used where a value of type int is required.\n\n\n\nA number written with a period or with scientific notation is assigned to a the continuous numeric type real. Real literals are written in base 10 with a period (.) as a separator and optionally an exponent with optional sign. Examples of well-formed real literals include the following.\n0.0, 1.0, 3.14, -217.9387, 2.7e3, -2E-5, 1.23e+3.\nThe notation e or E followed by a positive or negative integer denotes a power of 10 to multiply. For instance, 2.7e3 and 2.7e+3 denote \\(2.7 \\times 10^3\\), whereas -2E-5 denotes \\(-2 \\times\n10^{-5}\\).\n\n\n\nA number followed by the character i denotes an imaginary number and is assigned to the numeric type complex. The number preceding i may be either a real or integer literal and determines the magnitude of the imaginary number. Examples of well-formed imaginary literals include the following.\n1i, 2i, -325.786i, 1e10i, 2.87e-10i.\nNote that the character i by itself is not a well-formed imaginary literal. The unit imaginary number must be written as 1i.\n\n\n\nStan does not include complex literals directly, but a real or integer literal can be added to an imaginary literal to derive an expression that behaves like a complex literal. Examples include the following.\n1 + 2i, -3.2e9 + 1e10i\nThese will be assigned the type complex, which is the result of adding a real or integer and a complex number. They will also function like literals in the sense that the C++ compiler is able to reduce them to a single complex constant at compile time.\n\n\n\n\nA variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, target\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.\n\n\n\n\n\nExpressions for the Stan container objects, namely arrays, vectors, row vectors, matrices, and tuples, can all be constructed using expressions.\n\n\nSquare brackets may be wrapped around a sequence of comma separated primitive expressions to produce a row vector expression. For example, the expression [ 1, 10, 100 ] denotes a row vector of three elements with real values 1.0, 10.0, and 100.0. Applying the transpose operator to a row vector expression produces a vector expression. This syntax provides a way declare and define small vectors a single line, as follows.\nrow_vector[2] rv2 =  [ 1, 2 ];\nvector[3] v3 = [ 3, 4, 5 ]';\nThe vector expression values may be compound expressions or variable names, so it is legal to write [ 2 * 3, 1 + 4] or [ x, y ], providing that x and y are primitive variables.\n\n\n\nA matrix expression consists of square brackets wrapped around a sequence of comma separated row vector expressions. This syntax provides a way declare and define a matrix in a single line, as follows.\nmatrix[3, 2] m1 = [ [ 1, 2 ], [ 3, 4 ], [5, 6 ] ];\nAny expression denoting a row vector can be used in a matrix expression. For example, the following code is valid:\nvector[2] vX = [ 1, 10 ]';\nrow_vector[2] vY = [ 100, 1000 ];\nmatrix[3, 2] m2 = [ vX', vY, [ 1, 2 ]  ];\n\n\n\nComplex vector expressions work the same way as real vector expressions. For example, the following are all legal Stan expressions and assignments.\ncomplex_vector[3] = [1 + 2i, 3 - 1.7i, 0]';\ncomplex_row_vector[2] = [12, -2i];\ncomplex_matrix[2, 3] = [[1 + 2i, 3 - 1.7i, 0],\n                        [3.9 - 1.234i, 176i, 1 + 1i]];\n\n\nThe empty expression [ ] is ambiguous and therefore is not allowed and similarly expressions such as [ [ ] ] or [ [ ], [ ] ] are not allowed.\n\n\n\n\nCurly braces may be wrapped around a sequence of expressions to produce an array expression. For example, the expression { 1, 10, 100 } denotes an integer array of three elements with values 1, 10, and 100. This syntax is particularly convenient to define small arrays in a single line, as follows.\narray[3] int a = { 1, 10, 100 };\nThe values may be compound expressions, so it is legal to write { 2 * 3, 1 + 4 }. It is also possible to write two dimensional arrays directly, as in the following example.\narray[2, 3] int b = { { 1, 2, 3 }, { 4, 5, 6 } };\nThis way, b[1] is { 1, 2, 3 } and b[2] is { 4, 5, 6 }.\nWhitespace is always interchangeable in Stan, so the above can be laid out as follows to more clearly indicate the row and column structure of the resulting two dimensional array.\narray[2, 3] int b = { { 1, 2, 3 },\n                { 4, 5, 6 } };\n\n\n\nAny type of expression may be used within braces to form an array expression. In the simplest case, all of the elements will be of the same type and the result will be an array of elements of that type. For example, the elements of the array can be vectors, in which case the result is an array of vectors.\nvector[3] b;\nvector[3] c;\n// ...\narray[2] vector[3] d = { b, c };\nThe elements may also be a mixture of int and real typed expressions, in which case the result is an array of real values.\narray[2] real b = { 1, 1.9 };\n\n\n\nStan uses parentheses around a comma-separated sequence of expressions to construct a tuple. For example, we can construct a 2-tuple as follows.\ntuple(int, vector[3]) xy = (42, [1, 2.9, -1.3]');\nThe expression 42 is of type int and the expression [1, 2.9, -1.3] is of type row_vector so that [1, 2.9, -1.3]' is of type vector and of size 3. The whole tuple expression (42, [1, 2.9, -1.3]') thus has a sized type of tuple(int, vector[3]) and an unsized type (e.g., for a function argument) of tuple(int, vector).\nStan does not support the Python notation with trailing commas, such as (1, 2, 3, ) for a 3-tuple.\n\n\n\nThere are some restrictions on how array expressions may be used that arise from their types being calculated bottom up and the basic data type and assignment rules of Stan.\n\n\nAlthough it is tempting to try to define a ragged array expression, all Stan data types are rectangular (or boxes or other higher-dimensional generalizations). Thus the following nested array expression will cause an error when it tries to create a non-rectangular array.\n{ { 1, 2, 3 }, { 4, 5 } }  // compile time error: size mismatch\nThis may appear to be OK, because it is creating a two-dimensional integer array (array[,] int) out of two one-dimensional array integer arrays (array[] int). But it is not allowed because the two one-dimensional arrays are not the same size. If the elements are array expressions, this can be diagnosed at compile time. If one or both expressions is a variable, then that won’t be caught until runtime.\n{ { 1, 2, 3 }, m }  // runtime error if m not size 3\n\n\n\nBecause there is no way to infer the type of the result, the empty array expression ({ }) is not allowed. This does not sacrifice expressive power, because a declaration is sufficient to initialize a zero-element array.\narray[0] int a;   // a is fully defined as zero element array\n\n\n\nThere is no way to declare or construct a zero-tuple or one-tuple in Stan. Tuples must be at least two elements long. The expression () does not pick out a zero-tuple—it is ill formed. Similarly, the expression (1) is of type int rather than a tuple.\n\n\n\n\n\nAny expression wrapped in parentheses is also an expression. Like in C++, but unlike in R, only the round parentheses, ( and ), are allowed. The square brackets [ and ] are reserved for array indexing and the curly braces { and } for grouping statements.\nWith parentheses it is possible to explicitly group subexpressions with operators. Without parentheses, the expression 1 + 2 * 3 has a subexpression 2 * 3 and evaluates to 7. With parentheses, this grouping may be made explicit with the expression 1 + (2 * 3). More importantly, the expression (1 + 2) *   3 has 1 + 2 as a subexpression and evaluates to 9.\n\n\n\nFor integer and real-valued expressions, Stan supports the basic binary arithmetic operations of addition (+), subtraction (-), multiplication (*) and division (/) in the usual ways.\nFor integer expressions, Stan supports the modulus (%) binary arithmetic operation. Stan also supports the unary operation of negation for integer and real-valued expressions. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3.0 + 0.14\n-15\n2 * 3 + 1\n(x - y) / 2.0\n(n * (n + 1)) / 2\nx / n\nm % n\nThe negation, addition, subtraction, and multiplication operations are extended to matrices, vectors, and row vectors. The transpose operation, written using an apostrophe (') is also supported for vectors, row vectors, and matrices. Return types for matrix operations are the smallest types that can be statically guaranteed to contain the result. The full set of allowable input types and corresponding return types is detailed in the list of functions.\nFor example, if y and mu are variables of type vector and Sigma is a variable of type matrix, then (y - mu)' * Sigma * (y - mu) is a well-formed expression of type real. The type of the complete expression is inferred working outward from the subexpressions. The subexpression(s) y - mu are of type vector because the variables y and mu are of type vector. The transpose of this expression, the subexpression (y - mu)' is of type row_vector. Multiplication is left associative and transpose has higher precedence than multiplication, so the above expression is equivalent to the following fully specified form (((y - mu)') * Sigma) * (y - mu).\nThe type of subexpression (y - mu)' * Sigma is inferred to be row_vector, being the result of multiplying a row vector by a matrix. The whole expression’s type is thus the type of a row vector multiplied by a (column) vector, which produces a real value.\nStan provides elementwise matrix multiplication (e.g., a .* b) and division (e.g., a ./ b) operations. These provide a shorthand to replace loops, but are not intrinsically more efficient than a version programmed with an elementwise calculations and assignments in a loop. For example, given declarations,\nvector[N] a;\nvector[N] b;\nvector[N] c;\nthe assignment,\nc = a .* b;\nproduces the same result with roughly the same efficiency as the loop\nfor (n in 1:N) {\n  c[n] = a[n] * b[n];\n}\nStan supports exponentiation (^) of integer and real-valued expressions. The return type of exponentiation is always a real-value. For example, assuming n and m are integer variables and x and y real variables, the following expressions are legal.\n3 ^ 2\n3.0 ^ -2\n3.0 ^ 0.14\nx ^ n\nn ^ x\nn ^ m\nx ^ y\nExponentiation is right associative, so the expression 2 ^ 3 ^ 4 is equivalent to the fully specified form 2 ^ (3 ^ 4).\n\n\nThe precedence and associativity of operators, as well as built-in syntax such as array indexing and function application is given in tabular form in the following table.\n\n\nStan’s unary, binary, and ternary operators, with their precedences, associativities, place in an expression, and a description. The last two lines list the precedence of function application and array, matrix, and vector indexing. The operators are listed in order of precedence, from least tightly binding to most tightly binding. The full set of legal arguments and corresponding result types are provided in the function documentation for the operators (i.e., operator*(int, int):int indicates the application of the multiplication operator to two integers, which returns an integer). Parentheses may be used to group expressions explicitly rather than relying on precedence and associativity.\n\n\n\n\n\n\n\n\n\n\nOp.\nPrec.\nAssoc.\nPlacement\nDescription\n\n\n\n\n? ~ :\n10\nright\nternary infix\nconditional\n\n\n||\n9\nleft\nbinary infix\nlogical or\n\n\n&&\n8\nleft\nbinary infix\nlogical and\n\n\n==\n7\nleft\nbinary infix\nequality\n\n\n!=\n7\nleft\nbinary infix\ninequality\n\n\n&lt;\n6\nleft\nbinary infix\nless than\n\n\n&lt;=\n6\nleft\nbinary infix\nless than or equal\n\n\n&gt;\n6\nleft\nbinary infix\ngreater than\n\n\n&gt;=\n6\nleft\nbinary infix\ngreater than or equal\n\n\n+\n5\nleft\nbinary infix\naddition\n\n\n-\n5\nleft\nbinary infix\nsubtraction\n\n\n*\n4\nleft\nbinary infix\nmultiplication\n\n\n.*\n4\nleft\nbinary infix\nelementwise multiplication\n\n\n/\n4\nleft\nbinary infix\n(right) division\n\n\n./\n4\nleft\nbinary infix\nelementwise division\n\n\n%\n4\nleft\nbinary infix\nmodulus\n\n\n\\\n3\nleft\nbinary infix\nleft division\n\n\n%/%\n3\nleft\nbinary infix\ninteger division\n\n\n!\n2\nn/a\nunary prefix\nlogical negation\n\n\n-\n2\nn/a\nunary prefix\nnegation\n\n\n+\n2\nn/a\nunary prefix\npromotion (no-op in Stan)\n\n\n^\n1\nright\nbinary infix\nexponentiation\n\n\n.^\n1\nright\nbinary infix\nelementwise exponentiation\n\n\n'\n0\nn/a\nunary postfix\ntransposition\n\n\n()\n0\nn/a\nprefix, wrap\nfunction application\n\n\n[]\n0\nleft\nprefix, wrap\narray, matrix indexing\n\n\n\nOther expression-forming operations, such as function application and subscripting bind more tightly than any of the arithmetic operations.\nThe precedence and associativity determine how expressions are interpreted. Because addition is left associative, the expression a + b + c is interpreted as (a + b) + c. Similarly, a / b * c is interpreted as (a / b) * c.\nBecause multiplication has higher precedence than addition, the expression a * b + c is interpreted as (a * b) + c and the expression a + b * c is interpreted as a + (b * c). Similarly, 2 * x + 3 * - y is interpreted as (2 * x) + (3 * (-y)).\nTransposition and exponentiation bind more tightly than any other arithmetic or logical operation. For vectors, row vectors, and matrices, -u' is interpreted as -(u'), u * v' as u* (v'), and u' * v as (u') * v. For integer and reals, -n ^ 3 is interpreted as -(n ^ 3).\n\n\n\n\n\n\n\nThe ternary conditional operator is unique in that it takes three arguments and uses a mixed syntax. If a is an expression of type int and b and c are expressions that can be converted to one another (e.g., compared with ==), then\na ? b : c\nis an expression of the promoted type of b and c. The only promotion allowed in Stan is integer -&gt; real -&gt; complex; e.g. if one argument is of type int and the other of type real, the conditional expression as a whole is of type real. In other cases, the arguments have to be of the same underlying Stan type (i.e., constraints don’t count, only the shape) and the conditional expression is of that type.\n\n\nThe conditional operator is the most loosely binding operator, so its arguments rarely require parentheses for disambiguation. For example,\na &gt; 0 || b &lt; 0 ? c + d : e - f\nis equivalent to the explicitly grouped version\n(a &gt; 0 || b &lt; 0) ? (c + d) : (e - f)\nThe latter is easier to read even if the parentheses are not strictly necessary.\n\n\n\nThe conditional operator is right associative, so that\na ? b : c ? d : e\nparses as if explicitly grouped as\na ? b : (c ? d : e)\nAgain, the explicitly grouped version is easier to read.\n\n\n\n\nStan’s conditional operator works very much like its C++ analogue. The first argument must be an expression denoting an integer. Typically this is a variable or a relation operator, as in the variable a in the example above. Then there are two resulting arguments, the first being the result returned if the condition evaluates to true (i.e., non-zero) and the second if the condition evaluates to false (i.e., zero). In the example above, the value b is returned if the condition evaluates to a non-zero value and c is returned if the condition evaluates to zero.\n\n\nThe key property of the conditional operator that makes it so useful in high-performance computing is that it only evaluates the returned subexpression, not the alternative expression. In other words, it is not like a typical function that evaluates its argument expressions eagerly in order to pass their values to the function. As usual, the saving is mostly in the derivatives that do not get computed rather than the unnecessary function evaluation itself.\n\n\n\nIf one return expression is a data value (an expression involving only constants and variables defined in the data or transformed data block), and the other is not, then the ternary operator will promote the data value to a parameter value. This can cause needless work calculating derivatives in some cases and be less efficient than a full if-then conditional statement. For example,\ndata {\n  array[10] real x;\n  // ...\n}\nparameters {\n  array[10] real z;\n  // ...\n}\nmodel {\n  y ~ normal(cond ? x : z, sigma);\n  // ...\n}\nwould be more efficiently (if not more transparently) coded as\nif (cond) {\n  y ~ normal(x, sigma);\n} else {\n  y ~ normal(z, sigma);\n}\nThe conditional statement, like the conditional operator, only evaluates one of the result statements. In this case, the variable x will not be promoted to a parameter and thus not cause any needless work to be carried out when propagating the chain rule during derivative calculations.\n\n\n\n\n\nStan arrays, matrices, vectors, and row vectors are all accessed using the same array-like notation. For instance, if x is a variable of type array [] real (a one-dimensional array of reals) then x[1] is the value of the first element of the array.\nSubscripting has higher precedence than any of the arithmetic operations. For example, alpha * x[1] is equivalent to alpha * (x[1]).\nMultiple subscripts may be provided within a single pair of square brackets. If x is of type array[,] real, a two-dimensional array, then x[2, 501] is of type real.\n\n\nThe subscripting operator also returns subarrays of arrays. For example, if x is of type array[,,] real, then x[2] is of type array[,] real, and x[2, 3] is of type array[] real. As a result, the expressions x[2, 3] and x[2][3] have the same meaning.\n\n\n\nIf Sigma is a variable of type matrix, then Sigma[1] denotes the first row of Sigma and has the type row_vector.\n\n\n\nStan supports mixed indexing of arrays and their vector, row vector or matrix values. For example, if m is of type matrix[ , ], a two-dimensional array of matrices, then m[1] refers to the first row of the array, which is a one-dimensional array of matrices. More than one index may be used, so that m[1, 2] is of type matrix and denotes the matrix in the first row and second column of the array. Continuing to add indices, m[1, 2, 3] is of type row_vector and denotes the third row of the matrix denoted by m[1, 2]. Finally, m[1, 2, 3, 4] is of type real and denotes the value in the third row and fourth column of the matrix that is found at the first row and second column of the array m.\n\n\n\n\nIn addition to single integer indexes, as described in the language indexing section, Stan supports multiple indexing. Multiple indexes can be integer arrays of indexes, lower bounds, upper bounds, lower and upper bounds, or simply shorthand for all of the indexes. A complete table of index types is given in the following table.\n\n\nTypes of indexes and examples with one-dimensional containers of size N and an integer array ii of type array [] real size K.\n\n\n\nindex type\nexample\nvalue\n\n\n\n\ninteger\na[11]\nvalue of a at index 11\n\n\ninteger array\na[ii]\na[ii[1]], …, a[ii[K]]\n\n\nlower bound\na[3:]\na[3], …, a[N]\n\n\nupper bound\na[:5]\na[1], …, a[5]\n\n\nrange\na[2:7]\na[2], …, a[7]\n\n\nall\na[:]\na[1], …, a[N]\n\n\nall\na[]\na[1], …, a[N]\n\n\n\n\n\n\nThe fundamental semantic rule for dealing with multiple indexes is the following. If idxs is a multiple index, then it produces an indexable position in the result. To evaluate that index position in the result, the index is first passed to the multiple index, and the resulting index used.\na[idxs, ...][i, ...] = a[idxs[i], ...][...]\nOn the other hand, if idx is a single index, it reduces the dimensionality of the output, so that\na[idx, ...] = a[idx][...]\nThe only issue is what happens with matrices and vectors. Vectors work just like arrays. Matrices with multiple row indexes and multiple column indexes produce matrices. Matrices with multiple row indexes and a single column index become (column) vectors. Matrices with a single row index and multiple column indexes become row vectors. The types are summarized in the following table.\n\n\nSpecial rules for reducing matrices based on whether the argument is a single or multiple index. Examples are for a matrix a, with integer single indexes i and j and integer array multiple indexes is and js. The same typing rules apply for all multiple indexes.\n\n\n\nexample\nrow index\ncolumn index\nresult type\n\n\n\n\na[i]\nsingle\nn/a\nrow vector\n\n\na[is]\nmultiple\nn/a\nmatrix\n\n\na[i, j]\nsingle\nsingle\nreal\n\n\na[i, js]\nsingle\nmultiple\nrow vector\n\n\na[is, j]\nmultiple\nsingle\nvector\n\n\na[is, js]\nmultiple\nmultiple\nmatrix\n\n\n\nEvaluation of matrices with multiple indexes is defined to respect the following distributivity conditions.\nm[idxs1, idxs2][i, j] = m[idxs1[i], idxs2[j]]\nm[idxs, idx][j] = m[idxs[j], idx]\nm[idx, idxs][j] = m[idx, idxs[j]]\nEvaluation of arrays of matrices and arrays of vectors or row vectors is defined recursively, beginning with the array dimensions.\n\n\n\n\n\nStan provides a range of built in mathematical and statistical functions, which are documented in the built-in function documentation.\nExpressions in Stan may consist of the name of function followed by a sequence of zero or more argument expressions. For instance, log(2.0) is the expression of type real denoting the result of applying the natural logarithm to the value of the real literal 2.0.\nSyntactically, function application has higher precedence than any of the other operators, so that y + log(x) is interpreted as y + (log(x)).\n\n\nEach function has a type signature which determines the allowable type of its arguments and its return type. For instance, the function signature for the logarithm function can be expressed as\nreal log(real);\nand the signature for the lmultiply function is\nreal lmultiply(real, real);\nA function is uniquely determined by its name and its sequence of argument types. For instance, the following two functions are different functions.\nreal mean(array [] real);\nreal mean(vector);\nThe first applies to a one-dimensional array of real values and the second to a vector.\nThe identity conditions for functions explicitly forbids having two functions with the same name and argument types but different return types. This restriction also makes it possible to infer the type of a function expression compositionally by only examining the type of its subexpressions.\n\n\n\nConstants in Stan are nothing more than nullary (no-argument) functions. For instance, the mathematical constants \\(\\pi\\) and \\(e\\) are represented as nullary functions named pi() and e(). See the Stan Functions Reference built-in constants section for a list of built-in constants.\n\n\n\nBecause of integer to real type promotion, rules must be established for which function is called given a sequence of argument types. The scheme employed by Stan is the same as that used by C++, which resolves a function call to the function requiring the minimum number of type promotions.\nFor example, consider a situation in which the following two function signatures have been registered for foo.\nreal foo(real, real);\nint foo(int, int);\nThe use of foo in the expression foo(1.0, 1.0) resolves to foo(real, real), and thus the expression foo(1.0, 1.0) itself is assigned a type of real.\nBecause integers may be promoted to real values, the expression foo(1, 1) could potentially match either foo(real, real) or foo(int, int). The former requires two type promotions and the latter requires none, so foo(1, 1) is resolved to function foo(int, int) and is thus assigned the type int.\nThe expression foo(1, 1.0) has argument types (int, real) and thus does not explicitly match either function signature. By promoting the integer expression 1 to type real, it is able to match foo(real, real), and hence the type of the function expression foo(1, 1.0) is real.\nIn some cases (though not for any built-in Stan functions), a situation may arise in which the function referred to by an expression remains ambiguous. For example, consider a situation in which there are exactly two functions named bar with the following signatures.\nreal bar(real, int);\nreal bar(int, real);\nWith these signatures, the expression bar(1.0, 1) and bar(1, 1.0) resolve to the first and second of the above functions, respectively. The expression bar(1.0, 1.0) is illegal because real values may not be demoted to integers. The expression bar(1, 1) is illegal for a different reason. If the first argument is promoted to a real value, it matches the first signature, whereas if the second argument is promoted to a real value, it matches the second signature. The problem is that these both require one promotion, so the function name bar is ambiguous. If there is not a unique function requiring fewer promotions than all others, as with bar(1, 1) given the two declarations above, the Stan compiler will flag the expression as illegal.\n\n\n\nFor most of the distributions supported by Stan, there is a corresponding random-number generating function. These random number generators are named by the distribution with the suffix _rng. For example, a univariate normal random number can be generated by normal_rng(0, 1); only the parameters of the distribution, here a location (0) and scale (1) are specified because the variate is generated.\n\n\nThe use of random-number generating functions is restricted to the transformed data and generated quantities blocks; attempts to use them elsewhere will result in a parsing error with a diagnostic message. They may also be used in the bodies of user-defined functions whose names end in _rng.\nThis allows the random number generating functions to be used for simulation in general, and for Bayesian posterior predictive checking in particular.\n\n\n\nPosterior predictive checks typically use the parameters of the model to generate simulated data (at the individual and optionally at the group level for hierarchical models), which can then be compared informally using plots and formally by means of test statistics, to the actual data in order to assess the suitability of the model; see Chapter 6 of (Gelman et al. 2013) for more information on posterior predictive checks.\n\n\n\n\n\nStan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\nTuples have the natural extension of the above rules, applied to all sub-types at once\n\nA tuple(U1, ..., UN) may be promoted to a tuple(T1, ..., TN) if every Un can be promoted to Tn for n in 1:N\n\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).\n\n\n\n\n\nThere are several expression constructions in Stan that act as higher-order functions.2\nThe higher-order functions and the signature of their argument functions are listed in the following pair of tables.\n\n\nHigher-order functions in Stan with their argument function types. The first group of arguments can be a function of parameters or data. The second group of arguments, consisting of a real and integer array in all cases, must be expressions involving only data and literals.\n\n\n\n\n\n\n\n\n\nfunction\nparameter or data args\ndata args\nreturn type\n\n\n\n\nalgebra_solver\nvector, vector\narray [] real, array [] real\nvector\n\n\nalgebra_solver_newton\nvector, vector\narray [] real, array [] real\nvector\n\n\nintegrate_1d,\nreal, real, array [] real\narray [] real, array [] real\nreal\n\n\nintegrate_ode_X,\nreal, array [] real, array [] real\narray [] real, array [] real\narray [] real\n\n\nmap_rect\nvector, vector\narray [] real, array [] real\nvector\n\n\n\nFor example, the integrate_ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  array [] real foo(real t,\n                    array [] real y,\n                    array [] real theta,\n                    array [] real x_r,\n                    array [] real x_i) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\narray[2] real y0;\nreal t0;\narray[T] real ts;\narray[1] real theta;\narray[0] real x_r;\narray[0] int x_i;\n// ...\narray[T, 2] real y_hat = integrate_ode_rk45(foo, y0, t0,\n                                              ts, theta, x_r, x_i);\nThe function argument is foo, the name of the user-defined function; as shown in the higher-order functions table, integrate_ode_rk45 takes a real array, a real, three more real arrays, and an integer array as arguments and returns 2D real array.\n\n\n\nVariadic Higher-order functions in Stan with their argument function types. The first group of arguments are restricted in type. The sequence of trailing arguments can be of any length with any types.\n\n\n\nfunction\nrestricted args\nreturn type\n\n\n\n\nsolve_X\nvector\nvector\n\n\node_X,\nvector, real, array [] real\nvector[]\n\n\nreduce_sum\narray[] T, T1, T2\nreal\n\n\n\nT, T1, and T2 can be any Stan type.\nFor example, the ode_rk45 function can be used to integrate differential equations in Stan:\nfunctions {\n  vector foo(real t, vector y, real theta, vector beta,\n            array [] real x_i, int index) {\n    // ...\n  }\n}\n// ...\nint&lt;lower=1&gt; T;\nvector[2] y0;\nreal t0;\narray[T] real ts;\nreal theta;\nvector[7] beta;\narray[10] int x_i;\nint index;\n// ...\nvector[2] y_hat[T] = ode_rk45(foo, y0, t0, ts, theta,\n                              beta, x_i, index);\nThe function argument is foo, the name of the user-defined function. As shown in the variadic higher-order functions table, ode_rk45 takes a real, a vector, a real, a real array, and a sequence of arguments whose types match those at the end of foo and returns an array of vectors.\n\n\n\nThe function argument to higher-order functions is always passed as the first argument. This function argument must be provided as the name of a user-defined or built-in function. No quotes are necessary.\n\n\n\nSome of the arguments to higher-order functions are restricted to data. This means they must be expressions containing only data variables, transformed data variables, or literals; the may contain arbitrary functions applied to data variables or literals, but must not contain parameters, transformed parameters, or local variables from any block other than transformed data.\nFor user-defined functions the qualifier data may be prepended to the type to restrict the argument to data-only variables.\n\n\n\n\nDerivatives of the log probability function defined by a model are used in several ways by Stan. The Hamiltonian Monte Carlo samplers, including NUTS, use gradients to guide updates. The BFGS optimizers also use gradients to guide search for posterior modes.\n\n\nUnlike evaluations in pure mathematics, evaluation of derivatives in Stan is done by applying the chain rule on an expression-by-expression basis, evaluating using floating-point arithmetic. As a result, models such as the following are problematic for inference involving derivatives.\nparameters {\n  real x;\n}\nmodel {\n  x ~ normal(sqrt(x - x), 1);\n}\nAlgebraically, the sampling statement in the model could be reduced to\n  x ~ normal(0, 1);\nand it would seem the model should produce unit normal draws for x. But rather than canceling, the expression sqrt(x -   x) causes a problem for derivatives. The cause is the mechanistic evaluation of the chain rule,\n\\[\n\\begin{array}{rcl}\n\\frac{d}{dx} \\sqrt{x - x}\n& = &\n\\frac{1}{2 \\sqrt{x - x}} \\times \\frac{d}{dx} (x - x)\n\\\\[4pt]\n& = &\n\\frac{1}{0} \\times (1 - 1)\n\\\\[4pt]\n& = &\n\\infty \\times 0\n\\\\[4pt]\n& = & \\mathrm{NaN}.\n\\end{array}\n\\]\nRather than the \\(x - x\\) canceling out, it introduces a 0 into the numerator and denominator of the chain-rule evaluation.\nThe only way to avoid this kind problem is to be careful to do the necessary algebraic reductions as part of the model and not introduce expressions like sqrt(x - x) for which the chain rule produces not-a-number values.\n\n\n\nThe best way to diagnose whether something is going wrong with the derivatives is to use the test-gradient option to the sampler or optimizer inputs; this option is available in both Stan and RStan (though it may be slow, because it relies on finite differences to make a comparison to the built-in automatic differentiation).\nFor example, compiling the above model to an executable sqrt-x-minus-x in CmdStan, the test can be run as\n&gt; ./sqrt-x-minus-x diagnose test=gradient\nwhich produces\n...\nTEST GRADIENT MODE\n\n Log probability=-0.393734\n\n param idx           value           model     finite diff           error\n         0       -0.887393             nan               0             nan\nEven though finite differences calculates the right gradient of 0, automatic differentiation follows the chain rule and produces a not-a-number output.",
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-    "text": "A variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, get_lp, increment_log_prob,\ntarget\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.",
+    "text": "A variable by itself is a well-formed expression of the same type as the variable. Variables in Stan consist of ASCII strings containing only the basic lower-case and upper-case Roman letters, digits, and the underscore (_) character. Variables must start with a letter (a--z and A--Z) and may not end with two underscores (__).\nExamples of legal variable identifiers are as follows.\na, a3, a_3, Sigma, my_cpp_style_variable, myCamelCaseVariable\nUnlike in R and BUGS, variable identifiers in Stan may not contain a period character.\n\n\nStan reserves many strings for internal use and these may not be used as the name of a variable. An attempt to name a variable after an internal string results in the stanc translator halting with an error message indicating which reserved name was used and its location in the model code.\n\n\nThe name of the model cannot be used as a variable within the model. This is usually not a problem because the default in bin/stanc is to append _model to the name of the file containing the model specification. For example, if the model is in file foo.stan, it would not be legal to have a variable named foo_model when using the default model name through bin/stanc. With user-specified model names, variables cannot match the model.\n\n\n\nThe following list contains reserved words for Stan’s programming language. Not all of these features are implemented in Stan yet, but the tokens are reserved for future use.\nfor, in, while, repeat, until, if, then, else,\ntrue, false, target, struct, typedef, export,\nauto, extern, var, static, lower, upper, offset,\nmultiplier\nVariables should not be named after types, either, and thus may not be any of the following.\nint, real, complex, vector, simplex, unit_vector,\nordered, positive_ordered, row_vector, matrix,\ncholesky_factor_corr, cholesky_factor_cov,\ncorr_matrix, cov_matrix, array\nThe following built in functions are also reserved and cannot be used as variable names:\nprint, reject, profile, target\nThe following block identifiers are reserved and cannot be used as variable names:\nfunctions, model, data, parameters, quantities,\ntransformed, generated\n\n\n\nVariable names will also conflict with the names of distributions suffixed with _lpdf, _lpmf, _lcdf, and _lccdf, _cdf, and _ccdf, such as normal_lcdf_log. No user-defined variable can take a name ending in _lupdf or _lupmf even if a corresponding _lpdf or _lpmf is not defined.\nUsing any of these variable names causes the stanc translator to halt and report the name and location of the variable causing the conflict.\n\n\n\nStan primarily generates code in C++, which features its own reserved words. It is legal to name a variable any of the following names, however doing so will lead to it being renamed _stan_NAME (e.g. _stan_public) behind the scenes (in the generated C++ code).\n\nalignas, alignof, and, and_eq, asm, bitand, bitor, bool,\ncase, catch, char, char16_t, char32_t, class, compl, const,\nconstexpr, const_cast, decltype, default, delete, do,\ndouble, dynamic_cast, enum, explicit, float, friend, goto,\ninline, long, mutable, namespace, new, noexcept, not, not_eq,\nnullptr, operator, or, or_eq, private, protected, public,\nregister, reinterpret_cast, short, signed, sizeof,\nstatic_assert, static_cast, switch, template, this, thread_local,\nthrow, try, typeid, typename, union, unsigned, using, virtual,\nvolatile, wchar_t, xor, xor_eq, fvar, STAN_MAJOR, STAN_MINOR,\nSTAN_PATCH, STAN_MATH_MAJOR, STAN_MATH_MINOR, STAN_MATH_PATCH\n\n\n\n\nThe legal characters for variable identifiers are given in the following table.\n\n\nThe alphanumeric characters and underscore in base ASCII are the only legal characters in Stan identifiers.\n\n\n\ncharacters\nASCII code points\n\n\n\n\na -- z\n97 – 122\n\n\nA -- Z\n65 – 90\n\n\n0 -- 9\n48 – 57\n\n\n_\n95\n\n\n\nAlthough not the most expressive character set, ASCII is the most portable and least prone to corruption through improper character encodings or decodings. Sticking to this range of ASCII makes Stan compatible with Latin-1 or UTF-8 encodings of these characters, which are byte-for-byte identical to ASCII.\n\n\n\nWithin comments, Stan can work with any ASCII-compatible character encoding, such as ASCII itself, UTF-8, or Latin1. It is up to user shells and editors to display them properly.",
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-    "text": "Stan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).",
+    "text": "Stan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.\n\n\nThe primitive implementation types for Stan are\nint, real, complex, vector, row_vector,  matrix, complex_vector,\ncomplex_row_vector, complex_matrix\nEvery basic declared type corresponds to a primitive type; the following table shows the mapping from types to their primitive types.\n\n\nThe table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.\n\n\n\ntype\nprimitive type\n\n\n\n\nint\nint\n\n\nreal\nreal\n\n\nvector\nvector\n\n\nsimplex\nvector\n\n\nunit_vector\nvector\n\n\nordered\nvector\n\n\npositive_ordered\nvector\n\n\nrow_vector\nrow_vector\n\n\nmatrix\nmatrix\n\n\ncov_matrix\nmatrix\n\n\ncorr_matrix\nmatrix\n\n\ncholesky_factor_cov\nmatrix\n\n\ncholesky_factor_corr\nmatrix\n\n\ncomplex_vector\ncomplex_vector\n\n\ncomplex_row_vector\ncomplex_row_vector\n\n\ncomplex_matrix\ncomplex_matrix\n\n\n\nA full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, array [] real has an array dimensionality of 1, int an array dimensionality of 0, and array [,,] int an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.\nRecall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.\narray[I, J, K] matrix[M, N] a;\nThe matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.\n\n\n\n\nStan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.\n\n\n\nThere are two basic promotion rules,\n\nint types may be promoted to real, and\nreal types may be promoted to complex.\n\nPlus, promotion is transitive, so that\n\nif type U can be promoted to type V and type V can be promoted to type T, then U can be promoted to T.\n\nThe first rule means that expressions of type int may be used anywhere an expression of type real is specified, namely in assignment or function argument passing. An integer is promoted to real by casting it in the underlying C++ code.\nThe remaining rules have to do with covariant typing rules, which say that a container of type U may be promoted to a container of the same shape of type T if U can be promoted to T. For vector and matrix types, this induces three rules,\n\nvector may be promoted to complex_vector,\nrow_vector may be promoted to complex_row_vector\nmatrix may be promoted to complex_matrix.\n\nFor array types, there’s a single rule\n\narray[...] U may be promoted to array[...] T if U can be promoted to T.\n\nFor example, this means array[,] int may be used where array [,] real or array [,] complex is required; as another example, array[] real may be used anywhere array[] complex is required.\nTuples have the natural extension of the above rules, applied to all sub-types at once\n\nA tuple(U1, ..., UN) may be promoted to a tuple(T1, ..., TN) if every Un can be promoted to Tn for n in 1:N\n\n\n\nAn integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real. Imaginary literals such as -17i are of type complex. the expression 7 - 2i acts like a complex literal, but technically it combines a real literal 7 and an imaginary literal 2i through subtraction.\n\n\n\nThe type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.\nThere is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.1\n\n\n\nIf x is an expression of total dimensionality greater than or equal to \\(N\\), then the type of expression e[i1, i2, ..., iN] is the same as that of e[i1][i2]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then\n\nif e is an expression of type array[i1, i2, ..., iN] T and k, i1, …, iN are expressions of type int, then e[k] is an expression of type array[i2, ..., iN] T,\nif e is an expression of type array[i] T with i and k expressions of type int, then e[k] is of type T,\nif e has implementation type vector or row_vector, dimensionality 0, then e[i] has implementation type real,\nif e has implementation type matrix, then e[i] has type row_vector,\nif e has implementation type complex_vector or complex_row_vector and i is an expression of type int, then e[i] is an expression of type complex, and\nif e has implementation type complex_matrix, and i is an expression of type int, then e[i] is an expression of type complex_row_vector.\n\n\n\n\nIf f is the name of a function and e1,...,eN are expressions for \\(N \\geq 0\\), then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.\nIn looking up functions, binary operators like real * real are defined as operator*(real, real) in the documentation and index.\nIn matching a function definition, all of the promotion rules are in play (integers may be promoted to reals, reals to complex, and containers may be promoted if their types are promoted). For example, arguments of type int may be promoted to type real or complex if necessary (see the subsection on type promotion in the function application section, a real argument will be promoted to complex if necessary, a vector will be promoted to complex_vector if necessary, and so on.\nIn general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector, vector).",
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-    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Features]. The Stan compiler can automatically update many of these on the behalf of the user for at least one version after they are removed.\nAny feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).\n\n\nDeprecated:The distribution lkj_cov is deprecated.\nReplacement: Replace lkj_cov_lpdf(...) with an lkj_corr distribution on the correlation matrix and independent lognormal distributions on the scales. That is, replace\ncov_matrix[K] Sigma;\n// ...\nSigma ~ lkj_cov(mu, tau, eta);\nwith\ncorr_matrix[K] Omega;\nvector&lt;lower=0&gt;[K] sigma;\n// ...\nOmega ~ lkj_corr(eta);\nsigma ~ lognormal(mu, tau);\n// ...\ncov_matrix[K] Sigma;\nSigma &lt;- quad_form_diag(Omega, sigma);\nThe variable Sigma may be defined as a local variable in the model block or as a transformed parameter. An even more efficient transform would use Cholesky factors rather than full correlation matrix types.\nScheduled Removal: Stan 3.0 or later.\n\n\n\nSeveral built-in Stan functions have been deprecated. Consult the functions reference for more information.",
+    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting with Stan 2.29, minor (syntax-level) deprecations can be removed 3 versions after release; e.g., syntax deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in Removed Features. The Stan compiler can automatically update many of these on the behalf of the user for at least one version after they are removed.\nAny feature which changes semantic meaning (such as the upgraded ODE solver interface) will not be removed until a major version change (e.g., Stan 3.0).\n\n\nDeprecated:The distribution lkj_cov is deprecated.\nReplacement: Replace lkj_cov_lpdf(...) with an lkj_corr distribution on the correlation matrix and independent lognormal distributions on the scales. That is, replace\ncov_matrix[K] Sigma;\n// ...\nSigma ~ lkj_cov(mu, tau, eta);\nwith\ncorr_matrix[K] Omega;\nvector&lt;lower=0&gt;[K] sigma;\n// ...\nOmega ~ lkj_corr(eta);\nsigma ~ lognormal(mu, tau);\n// ...\ncov_matrix[K] Sigma;\nSigma &lt;- quad_form_diag(Omega, sigma);\nThe variable Sigma may be defined as a local variable in the model block or as a transformed parameter. An even more efficient transform would use Cholesky factors rather than full correlation matrix types.\nScheduled Removal: Stan 3.0 or later.\n\n\n\nSeveral built-in Stan functions have been deprecated. Consult the functions reference for more information.",
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     "href": "reference-manual/blocks.html",
     "title": "Program Blocks",
     "section": "",
-    "text": "A Stan program is organized into a sequence of named blocks, the bodies of which consist of variable declarations, followed in the case of some blocks with statements.\n\n\nThe full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n\\slshape (initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.\n\n\n\n\nStatistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y[N] is a modeled data vector. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.\n\n\n\nThe rest of this chapter will lay out the details of each block in order, starting with the data block in this section.\n\n\nThe data block is for the declaration of variables that are read in as data. With the current model executable, each Markov chain of draws will be executed in a different process, and each such process will read the data exactly once.3\nData variables are not transformed in any way. The format for data files or data in memory depends on the interface; see the user’s guides and interface documentation for PyStan, RStan, and CmdStan for details.\n\n\n\nThe data block does not allow statements.\n\n\n\nEach variable’s value is validated against its declaration as it is read. For example, if a variable sigma is declared as real&lt;lower=0&gt;, then trying to assign it a negative value will raise an error. As a result, data type errors will be caught as early as possible. Similarly, attempts to provide data of the wrong size for a compound data structure will also raise an error.\n\n\n\n\nThe transformed data block is for declaring and defining variables that do not need to be changed when running the program.\n\n\nFor the transformed data block, variables are all declared in the variable declarations and defined in the statements. There is no reading from external sources and no transformations performed.\nVariables declared in the data block may be used to declare transformed variables.\n\n\n\nThe statements in a transformed data block are used to define (provide values for) variables declared in the transformed data block. Assignments are only allowed to variables declared in the transformed data block.\nThese statements are executed once, in order, right after the data is read into the data variables. This means they are executed once per chain.\nVariables declared in the data block may be used in statements in the transformed data block.\n\n\nThe statements in the transformed data block are designed to be executed once and have a deterministic result. Therefore, log probability is not accumulated and sampling statements may not be used.\n\n\n\n\nAny constraints on variables declared in the transformed data block are checked after the statements are executed. If any defined variable violates its constraints, Stan will halt with a diagnostic error message.\n\n\n\n\nThe variables declared in the parameters program block correspond directly to the variables being sampled by Stan’s samplers (HMC and NUTS). From a user’s perspective, the parameters in the program block are the parameters being sampled by Stan.\nVariables declared as parameters cannot be directly assigned values. So there is no block of statements in the parameters program block. Variable quantities derived from parameters may be declared in the transformed parameters or generated quantities blocks, or may be defined as local variables in any statement blocks following their declaration.\nThere is a substantial amount of computation involved for parameter variables in a Stan program at each leapfrog step within the HMC or NUTS samplers, and a bit more computation along with writes involved for saving the parameter values corresponding to a sample.\n\n\nStan’s two samplers, standard Hamiltonian Monte Carlo (HMC) and the adaptive No-U-Turn sampler (NUTS), are most easily (and often most effectively) implemented over a multivariate probability density that has support on all of \\(\\mathbb{R}^n\\). To do this, the parameters defined in the parameters block must be transformed so they are unconstrained.\nIn practice, the samplers keep an unconstrained parameter vector in memory representing the current state of the sampler. The model defined by the compiled Stan program defines an (unnormalized) log probability function over the unconstrained parameters. In order to do this, the log probability function must apply the inverse transform to the unconstrained parameters to calculate the constrained parameters defined in Stan’s parameters program block. The log Jacobian of the inverse transform is then added to the accumulated log probability function. This then allows the Stan model to be defined in terms of the constrained parameters.\nIn some cases, the number of parameters is reduced in the unconstrained space. For instance, a \\(K\\)-simplex only requires \\(K-1\\) unconstrained parameters, and a \\(K\\)-correlation matrix only requires \\(\\binom{K}{2}\\) unconstrained parameters. This means that the probability function defined by the compiled Stan program may have fewer parameters than it would appear from looking at the declarations in the parameters program block.\nThe probability function on the unconstrained parameters is defined in such a way that the order of the parameters in the vector corresponds to the order of the variables defined in the parameters program block. The details of the specific transformations are provided in the variable transforms chapter.\n\n\n\nHamiltonian Monte Carlo requires the gradient of the (unnormalized) log probability function with respect to the unconstrained parameters to be evaluated during every leapfrog step. There may be one leapfrog step per sample or hundreds, with more being required for models with complex posterior distribution geometries.\nGradients are calculated behind the scenes using Stan’s algorithmic differentiation library. The time to compute the gradient does not depend directly on the number of parameters, only on the number of subexpressions in the calculation of the log probability. This includes the expressions added from the transforms’ Jacobians.\nThe amount of work done by the sampler does depend on the number of unconstrained parameters, but this is usually dwarfed by the gradient calculations.\n\n\n\nIn the basic Stan compiled program, there is a file to which the values of variables are written for each draw. The constrained versions of the variables are written in the order they are defined in the parameters block. In order to do this, the transformed parameter, model, and generated quantities statements must also be executed.\n\n\n\n\nThe transformed parameters program block consists of optional variable declarations followed by statements. After the statements are executed, the constraints on the transformed parameters are validated. Any variable declared as a transformed parameter is part of the output produced for draws.\nAny variable that is defined wholly in terms of data or transformed data should be declared and defined in the transformed data block. Defining such quantities in the transformed parameters block is legal, but much less efficient than defining them as transformed data.\n\n\nLike the constraints on data, the constraints on transformed parameters is meant to catch programming errors as well as convey programmer intent. They are not automatically transformed in such a way as to be satisfied. What will happen if a transformed parameter does not match its constraint is that the current parameter values will be rejected. This can cause Stan’s algorithms to hang or to devolve to random walks. It is not intended to be a way to enforce ad hoc constraints in Stan programs. See the section on reject statements for further discussion of the behavior of reject statements.\n\n\n\n\nThe model program block consists of optional variable declarations followed by statements. The variables in the model block are local variables and are not written as part of the output.\nLocal variables may not be defined with constraints because there is no well-defined way to have them be both flexible and easy to validate.\nThe statements in the model block typically define the model. This is the block in which probability (sampling notation) statements are allowed. These are typically used when programming in the BUGS idiom to define the probability model.\n\n\n\nThe generated quantities program block is rather different than the other blocks. Nothing in the generated quantities block affects the sampled parameter values. The block is executed only after a sample has been generated.\nAmong the applications of posterior inference that can be coded in the generated quantities block are\n\nforward sampling to generate simulated data for model testing,\ngenerating predictions for new data,\ncalculating posterior event probabilities, including multiple comparisons, sign tests, etc.,\ncalculating posterior expectations,\ntransforming parameters for reporting,\napplying full Bayesian decision theory,\ncalculating log likelihoods, deviances, etc. for model comparison.\n\nParameter estimates, predictions, statistics, and event probabilities calculated directly using plug-in estimates. Stan automatically provides full Bayesian inference by producing draws from the posterior distribution of any calculated event probabilities, predictions, or statistics.\nWithin the generated quantities block, the values of all other variables declared in earlier program blocks (other than local variables) are available for use in the generated quantities block.\nIt is more efficient to define a variable in the generated quantities block instead of the transformed parameters block. Therefore, if a quantity does not play a role in the model, it should be defined in the generated quantities block.\nAfter the generated quantities statements are executed, the constraints on the declared generated quantity variables are validated.\nAll variables declared as generated quantities are printed as part of the output. Variables declared in nested blocks are local variables, not generated quantities, and thus won’t be printed. For example:\ngenerated quantities {\n  int a; // added to the output\n\n  {\n    int b; // not added to the output\n  }\n}",
+    "text": "A Stan program is organized into a sequence of named blocks, the bodies of which consist of variable declarations, followed in the case of some blocks with statements.\n\n\nThe full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n(initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.\n\n\n\n\nStatistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y is an array of modeled data. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.\n\n\n\nThe rest of this chapter will lay out the details of each block in order, starting with the data block in this section.\n\n\nThe data block is for the declaration of variables that are read in as data. With the current model executable, each Markov chain of draws will be executed in a different process, and each such process will read the data exactly once.3\nData variables are not transformed in any way. The format for data files or data in memory depends on the interface; see the user’s guides and interface documentation for PyStan, RStan, and CmdStan for details.\n\n\n\nThe data block does not allow statements.\n\n\n\nEach variable’s value is validated against its declaration as it is read. For example, if a variable sigma is declared as real&lt;lower=0&gt;, then trying to assign it a negative value will raise an error. As a result, data type errors will be caught as early as possible. Similarly, attempts to provide data of the wrong size for a compound data structure will also raise an error.\n\n\n\n\nThe transformed data block is for declaring and defining variables that do not need to be changed when running the program.\n\n\nFor the transformed data block, variables are all declared in the variable declarations and defined in the statements. There is no reading from external sources and no transformations performed.\nVariables declared in the data block may be used to declare transformed variables.\n\n\n\nThe statements in a transformed data block are used to define (provide values for) variables declared in the transformed data block. Assignments are only allowed to variables declared in the transformed data block.\nThese statements are executed once, in order, right after the data is read into the data variables. This means they are executed once per chain.\nVariables declared in the data block may be used in statements in the transformed data block.\n\n\nThe statements in the transformed data block are designed to be executed once and have a deterministic result. Therefore, log probability is not accumulated and sampling statements may not be used.\n\n\n\n\nAny constraints on variables declared in the transformed data block are checked after the statements are executed. If any defined variable violates its constraints, Stan will halt with a diagnostic error message.\n\n\n\n\nThe variables declared in the parameters program block correspond directly to the variables being sampled by Stan’s samplers (HMC and NUTS). From a user’s perspective, the parameters in the program block are the parameters being sampled by Stan.\nVariables declared as parameters cannot be directly assigned values. So there is no block of statements in the parameters program block. Variable quantities derived from parameters may be declared in the transformed parameters or generated quantities blocks, or may be defined as local variables in any statement blocks following their declaration.\nThere is a substantial amount of computation involved for parameter variables in a Stan program at each leapfrog step within the HMC or NUTS samplers, and a bit more computation along with writes involved for saving the parameter values corresponding to a sample.\n\n\nStan’s two samplers, standard Hamiltonian Monte Carlo (HMC) and the adaptive No-U-Turn sampler (NUTS), are most easily (and often most effectively) implemented over a multivariate probability density that has support on all of \\(\\mathbb{R}^n\\). To do this, the parameters defined in the parameters block must be transformed so they are unconstrained.\nIn practice, the samplers keep an unconstrained parameter vector in memory representing the current state of the sampler. The model defined by the compiled Stan program defines an (unnormalized) log probability function over the unconstrained parameters. In order to do this, the log probability function must apply the inverse transform to the unconstrained parameters to calculate the constrained parameters defined in Stan’s parameters program block. The log Jacobian of the inverse transform is then added to the accumulated log probability function. This then allows the Stan model to be defined in terms of the constrained parameters.\nIn some cases, the number of parameters is reduced in the unconstrained space. For instance, a \\(K\\)-simplex only requires \\(K-1\\) unconstrained parameters, and a \\(K\\)-correlation matrix only requires \\(\\binom{K}{2}\\) unconstrained parameters. This means that the probability function defined by the compiled Stan program may have fewer parameters than it would appear from looking at the declarations in the parameters program block.\nThe probability function on the unconstrained parameters is defined in such a way that the order of the parameters in the vector corresponds to the order of the variables defined in the parameters program block. The details of the specific transformations are provided in the variable transforms chapter.\n\n\n\nHamiltonian Monte Carlo requires the gradient of the (unnormalized) log probability function with respect to the unconstrained parameters to be evaluated during every leapfrog step. There may be one leapfrog step per sample or hundreds, with more being required for models with complex posterior distribution geometries.\nGradients are calculated behind the scenes using Stan’s algorithmic differentiation library. The time to compute the gradient does not depend directly on the number of parameters, only on the number of subexpressions in the calculation of the log probability. This includes the expressions added from the transforms’ Jacobians.\nThe amount of work done by the sampler does depend on the number of unconstrained parameters, but this is usually dwarfed by the gradient calculations.\n\n\n\nIn the basic Stan compiled program, there is a file to which the values of variables are written for each draw. The constrained versions of the variables are written in the order they are defined in the parameters block. In order to do this, the transformed parameter, model, and generated quantities statements must also be executed.\n\n\n\n\nThe transformed parameters program block consists of optional variable declarations followed by statements. After the statements are executed, the constraints on the transformed parameters are validated. Any variable declared as a transformed parameter is part of the output produced for draws.\nAny variable that is defined wholly in terms of data or transformed data should be declared and defined in the transformed data block. Defining such quantities in the transformed parameters block is legal, but much less efficient than defining them as transformed data.\n\n\nLike the constraints on data, the constraints on transformed parameters is meant to catch programming errors as well as convey programmer intent. They are not automatically transformed in such a way as to be satisfied. What will happen if a transformed parameter does not match its constraint is that the current parameter values will be rejected. This can cause Stan’s algorithms to hang or to devolve to random walks. It is not intended to be a way to enforce ad hoc constraints in Stan programs. See the section on reject statements for further discussion of the behavior of reject statements.\n\n\n\n\nThe model program block consists of optional variable declarations followed by statements. The variables in the model block are local variables and are not written as part of the output.\nLocal variables may not be defined with constraints because there is no well-defined way to have them be both flexible and easy to validate.\nThe statements in the model block typically define the model. This is the block in which probability (sampling notation) statements are allowed. These are typically used when programming in the BUGS idiom to define the probability model.\n\n\n\nThe generated quantities program block is rather different than the other blocks. Nothing in the generated quantities block affects the sampled parameter values. The block is executed only after a sample has been generated.\nAmong the applications of posterior inference that can be coded in the generated quantities block are\n\nforward sampling to generate simulated data for model testing,\ngenerating predictions for new data,\ncalculating posterior event probabilities, including multiple comparisons, sign tests, etc.,\ncalculating posterior expectations,\ntransforming parameters for reporting,\napplying full Bayesian decision theory,\ncalculating log likelihoods, deviances, etc. for model comparison.\n\nParameter estimates, predictions, statistics, and event probabilities calculated directly using plug-in estimates. Stan automatically provides full Bayesian inference by producing draws from the posterior distribution of any calculated event probabilities, predictions, or statistics.\nWithin the generated quantities block, the values of all other variables declared in earlier program blocks (other than local variables) are available for use in the generated quantities block.\nIt is more efficient to define a variable in the generated quantities block instead of the transformed parameters block. Therefore, if a quantity does not play a role in the model, it should be defined in the generated quantities block.\nAfter the generated quantities statements are executed, the constraints on the declared generated quantity variables are validated.\nAll variables declared as generated quantities are printed as part of the output. Variables declared in nested blocks are local variables, not generated quantities, and thus won’t be printed. For example:\ngenerated quantities {\n  int a; // added to the output\n\n  {\n    int b; // not added to the output\n  }\n}",
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-    "text": "The full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n\\slshape (initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.",
+    "text": "The full set of named program blocks is exemplified in the following skeletal Stan program.\nfunctions {\n  // ... function declarations and definitions ...\n}\ndata {\n  // ... declarations ...\n}\ntransformed data {\n   // ... declarations ... statements ...\n}\nparameters {\n   // ... declarations ...\n}\ntransformed parameters {\n   // ... declarations ... statements ...\n}\nmodel {\n   // ... declarations ... statements ...\n}\ngenerated quantities {\n   // ... declarations ... statements ...\n}\nThe function-definition block contains user-defined functions. The data block declares the required data for the model. The transformed data block allows the definition of constants and transforms of the data. The parameters block declares the model’s parameters — the unconstrained version of the parameters is what’s sampled or optimized. The transformed parameters block allows variables to be defined in terms of data and parameters that may be used later and will be saved. The model block is where the log probability function is defined. The generated quantities block allows derived quantities based on parameters, data, and optionally (pseudo) random number generation.\n\n\nAll of the blocks are optional. A consequence of this is that the empty string is a valid Stan program, although it will trigger a warning message from the Stan compiler. The Stan program blocks that occur must occur in the order presented in the skeletal program above. Within each block, both declarations and statements are optional, subject to the restriction that the declarations come before the statements.\n\n\n\nThe variables declared in each block have scope over all subsequent statements. Thus a variable declared in the transformed data block may be used in the model block. But a variable declared in the generated quantities block may not be used in any earlier block, including the model block. The exception to this rule is that variables declared in the model block are always local to the model block and may not be accessed in the generated quantities block; to make a variable accessible in the model and generated quantities block, it must be declared as a transformed parameter.\nVariables declared as function parameters have scope only within that function definition’s body, and may not be assigned to (they are constant).\n\n\n\nFunctions defined in the function block may be used in any appropriate block. Most functions can be used in any block and applied to a mixture of parameters and data (including constants or program literals).\nRandom-number-generating functions are restricted to transformed data and generated quantities blocks, and within user-defined functions ending in _rng; such functions are suffixed with _rng. Log-probability modifying functions to blocks where the log probability accumulator is in scope (transformed parameters and model); such functions are suffixed with _lp.\nDensity functions defined in the program may be used in sampling statements.\n\n\n\nThe variables declared in the data and parameters block are treated differently than other variables in that they are automatically defined by the context in which they are used. This is why there are no statements allowed in the data or parameters block.\nThe variables in the data block are read from an external input source such as a file or a designated R data structure. The variables in the parameters block are read from the sampler’s current parameter values (either standard HMC or NUTS). The initial values may be provided through an external input source, which is also typically a file or a designated R data structure. In each case, the parameters are instantiated to the values for which the model defines a log probability function.\n\n\n\nThe transformed data and transformed parameters block behave similarly to each other. Both allow new variables to be declared and then defined through a sequence of statements. Because variables scope over every statement that follows them, transformed data variables may be defined in terms of the data variables.\nBefore generating any draws, data variables are read in, then the transformed data variables are declared and the associated statements executed to define them. This means the statements in the transformed data block are only ever evaluated once.1\nTransformed parameters work the same way, being defined in terms of the parameters, transformed data, and data variables. The difference is the frequency of evaluation. Parameters are read in and (inverse) transformed to constrained representations on their natural scales once per log probability and gradient evaluation. This means the inverse transforms and their log absolute Jacobian determinants are evaluated once per leapfrog step. Transformed parameters are then declared and their defining statements executed once per leapfrog step.\n\n\n\nThe generated quantity variables are defined once per sample after all the leapfrog steps have been completed. These may be random quantities, so the block must be rerun even if the Metropolis adjustment of HMC or NUTS rejects the update proposal.\n\n\n\nA table summarizing the point at which variables are read, written, and defined is given in the block actions table.\nBlock Actions Table.  The read, write, transform, and evaluate actions and periodicities listed in the last column correspond to the Stan program blocks in the first column. The middle column indicates whether the block allows statements. The last row indicates that parameter initialization requires a read and transform operation applied once per chain.\n\n\n\nblock\nstatement\naction / period\n\n\n\n\ndata\nno\nread / chain\n\n\ntransformed data\nyes\nevaluate / chain\n\n\nparameters\nno\ninv. transform, Jacobian / leapfrog\n\n\n \n \ninv. transform, write / sample\n\n\ntransformed parameters\nyes\nevaluate / leapfrog\n\n\n \n \nwrite / sample\n\n\nmodel\nyes\nevaluate / leapfrog step\n\n\ngenerated quantities\nyes\neval / sample\n\n\n \n \nwrite / sample\n\n\n(initialization)\nn/a\nread, transform / chain\n\n\n\nVariable Declaration Table.  This table indicates where variables that are not basic data or parameters should be declared, based on whether it is defined in terms of parameters, whether it is used in the log probability function defined in the model block, and whether it is printed. The two lines marked with asterisks (\\(*\\)) should not be used as there is no need to print a variable every iteration that does not depend on the value of any parameters.\n\n\n\nparam depend\nin target\nsave\ndeclare in\n\n\n\n\n+\n+\n+\ntransformed parameters\n\n\n+\n+\n-\nmodel (local)\n\n\n+\n-\n+\ngenerated quantities\n\n\n+\n-\n-\ngenerated quantities (local)\n\n\n-\n+\n+\ntransformed data   and generated quantities\n\n\n-\n+\n-\ntransformed data\n\n\n-\n-\n+\ngenerated quantities\n\n\n-\n-\n-\ntransformed data (local)\n\n\n\nAnother way to look at the variables is in terms of their function. To decide which variable to use, consult the charts in the variable declaration table. The last line has no corresponding location, as there is no need to print a variable every iteration that does not depend on parameters.2\nThe rest of this chapter provides full details on when and how the variables and statements in each block are executed.",
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-    "text": "Statistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y[N] is a modeled data vector. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.",
+    "text": "Statistical Variable Taxonomy Table.  Variables of the kind indicated in the left column must be declared in one of the blocks declared in the right column.\n\n\n\n\n\n\n\nvariable kind\ndeclaration block\n\n\n\n\nconstants\ndata, transformed data\n\n\nunmodeled data\ndata, transformed data\n\n\nmodeled data\ndata, transformed data\n\n\nmissing data\nparameters, transformed parameters\n\n\nmodeled parameters\nparameters, transformed parameters\n\n\nunmodeled parameters\ndata, transformed data\n\n\nderived quantities\ntransformed data, transformed parameters, generated quantities\n\n\nloop indices\nloop statement\n\n\n\nPage 366 of (Gelman and Hill 2007) provides a taxonomy of the kinds of variables used in Bayesian models. The table of kinds of variables contains Gelman and Hill’s taxonomy along with a missing-data kind along with the corresponding locations of declarations and definitions in Stan.\nConstants can be built into a model as literals, data variables, or as transformed data variables. If specified as variables, their definition must be included in data files. If they are specified as transformed data variables, they cannot be used to specify the sizes of elements in the data block.\nThe following program illustrates various variables kinds, listing the kind of each variable next to its declaration.\ndata {\n  int&lt;lower=0&gt; N;           // unmodeled data\n  array[N] real y;          // modeled data\n  real mu_mu;               // config. unmodeled param\n  real&lt;lower=0&gt; sigma_mu;   // config. unmodeled param\n}\ntransformed data {\n  real&lt;lower=0&gt; alpha;      // const. unmodeled param\n  real&lt;lower=0&gt; beta;       // const. unmodeled param\n  alpha = 0.1;\n  beta = 0.1;\n}\nparameters {\n  real mu_y;                // modeled param\n  real&lt;lower=0&gt; tau_y;      // modeled param\n}\ntransformed parameters {\n  real&lt;lower=0&gt; sigma_y;    // derived quantity (param)\n  sigma_y = pow(tau_y, -0.5);\n}\nmodel {\n  tau_y ~ gamma(alpha, beta);\n  mu_y ~ normal(mu_mu, sigma_mu);\n  for (n in 1:N) {\n    y[n] ~ normal(mu_y, sigma_y);\n  }\n}\ngenerated quantities {\n  real variance_y;       // derived quantity (transform)\n  variance_y = sigma_y * sigma_y;\n}\nIn this example, y is an array of modeled data. Although it is specified in the data block, and thus must have a known value before the program may be run, it is modeled as if it were generated randomly as described by the model.\nThe variable N is a typical example of unmodeled data. It is used to indicate a size that is not part of the model itself.\nThe other variables declared in the data and transformed data block are examples of unmodeled parameters, also known as hyperparameters. Unmodeled parameters are parameters to probability densities that are not themselves modeled probabilistically. In Stan, unmodeled parameters that appear in the data block may be specified on a per-model execution basis as part of the data read. In the above model, mu_mu and sigma_mu are configurable unmodeled parameters.\nUnmodeled parameters that are hard coded in the model must be declared in the transformed data block. For example, the unmodeled parameters alpha and beta are both hard coded to the value 0.1. To allow such variables to be configurable based on data supplied to the program at run time, they must be declared in the data block, like the variables mu_mu and sigma_mu.\nThis program declares two modeled parameters, mu and tau_y. These are the location and precision used in the normal model of the values in y. The heart of the model will be sampling the values of these parameters from their posterior distribution.\nThe modeled parameter tau_y is transformed from a precision to a scale parameter and assigned to the variable sigma_y in the transformed parameters block. Thus the variable sigma_y is considered a derived quantity — its value is entirely determined by the values of other variables.\nThe generated quantities block defines a value variance_y, which is defined as a transform of the scale or deviation parameter sigma_y. It is defined in the generated quantities block because it is not used in the model. Making it a generated quantity allows it to be monitored for convergence (being a non-linear transform, it will have different autocorrelation and hence convergence properties than the deviation itself).\nIn later versions of Stan which have random number generators for the distributions, the generated quantities block will be usable to generate replicated data for model checking.\nFinally, the variable n is used as a loop index in the model block.",
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-    "text": "This chapter describes Stan’s built-in function that take various types of arguments and return integer values.\n\n\nStan’s arithmetic is based on standard double-precision C++ integer and floating-point arithmetic. If the arguments to an arithmetic operator are both integers, as in 2 + 2, integer arithmetic is used. If one argument is an integer and the other a floating-point value, as in 2.0 + 2 and 2 + 2.0, then the integer is promoted to a floating point value and floating-point arithmetic is used.\nInteger arithmetic behaves slightly differently than floating point arithmetic. The first difference is how overflow is treated. If the sum or product of two integers overflows the maximum integer representable, the result is an undesirable wraparound behavior at the bit level. If the integers were first promoted to real numbers, they would not overflow a floating-point representation. There are no extra checks in Stan to flag overflows, so it is up to the user to make sure it does not occur.\nSecondly, because the set of integers is not closed under division and there is no special infinite value for integers, integer division implicitly rounds the result. If both arguments are positive, the result is rounded down. For example, 1 / 2 evaluates to 0 and 5 / 3 evaluates to 1.\nIf one of the integer arguments to division is negative, the latest C++ specification ( C++11), requires rounding toward zero. This would have 1 / 2 and -1 / 2 evaluate to 0, -7 / 2 evaluate to -3, and 7 / 2 evaluate to 3. Before the C++11 specification, the behavior was platform dependent, allowing rounding up or down. All compilers recent enough to be able to deal with Stan’s templating should follow the C++11 specification, but it may be worth testing if you are not sure and plan to use integer division with negative values.\nUnlike floating point division, where 1.0 / 0.0 produces the special positive infinite value, integer division by zero, as in 1 / 0, has undefined behavior in the C++ standard. For example, the clang++ compiler on Mac OS X returns 3764, whereas the g++ compiler throws an exception and aborts the program with a warning. As with overflow, it is up to the user to make sure integer divide-by-zero does not occur.\n\n\nOperators are described using the C++ syntax. For instance, the binary operator of addition, written X + Y, would have the Stan signature int operator+(int, int) indicating it takes two real arguments and returns a real value. As noted previously, the value of integer division is platform-dependent when rounding is platform dependent before C++11; the descriptions below provide the C++11 definition.\n \n\nint operator+(int x, int y) The sum of the addends x and y \\[\\begin{equation*} \\text{operator+}(x,y) = (x + y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator-(int x, int y) The difference between the minuend x and subtrahend y \\[\\begin{equation*}\n\\text{operator-}(x,y) = (x - y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator*(int x, int y) The product of the factors x and y \\[\\begin{equation*} \\text{operator*}(x,y) = (x\n\\times y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator/(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator/}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\] deprecated; - use operator%/% instead.\nAvailable since 2.0, deprecated in 2.24\n \n\nint operator%/%(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator\\%/\\%}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\]\nAvailable since 2.24\n \n\nint operator%(int x, int y) x modulo y, which is the positive remainder after dividing x by y. If both x and y are non-negative, so is the result; otherwise, the sign of the result is platform dependent. \\[\\begin{equation*} \\mathrm{operator\\%}(x, y) \\ =\n\\ x \\ \\text{mod} \\ y \\ = \\ x - y * \\lfloor x / y \\rfloor \\end{equation*}\\]\nAvailable since 2.13\n\n\n\n \n\nint operator-(int x) The negation of the subtrahend x \\[\\begin{equation*} \\text{operator-}(x) = -x \\end{equation*}\\]\nAvailable since 2.0\n \n\nT operator-(T x) Vectorized version of operator-. If T x is a (possibly nested) array of integers, -x is the same shape array where each individual integer is negated.\nAvailable since 2.31\n \n\nint operator+(int x) This is a no-op. \\[\\begin{equation*} \\text{operator+}(x) = x \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n\n \n\nT abs(T x) The absolute value of x.\nThis function works elementwise over containers such as vectors. Given a type T which is int, or an array of ints, abs returns the same type where each element has had its absolute value taken.\nAvailable since 2.0, vectorized in 2.30\n \n\nint int_step(int x)\n \n\nint int_step(real x) Return the step function of x as an integer, \\[\\begin{equation*} \\mathrm{int\\_step}(x)\n= \\begin{cases} 1 & \\text{if } x &gt; 0 \\\\ 0 & \\text{if } x \\leq 0 \\text{\nor } x \\text{ is } NaN \\end{cases} \\end{equation*}\\] Warning: int_step(0) and int_step(NaN) return 0 whereas step(0) and step(NaN) return 1.\nSee the warning in section step functions about the dangers of step functions applied to anything other than data.\nAvailable since 2.0\n\n\n\n \n\nint min(int x, int y) Return the minimum of x and y. \\[\\begin{equation*} \\text{min}(x, y) = \\begin{cases} x &\n\\text{if } x &lt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n \n\nint max(int x, int y) Return the maximum of x and y. \\[\\begin{equation*} \\text{max}(x, y) = \\begin{cases} x &\n\\text{if } x &gt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n \n\nint size(int x)\n \n\nint size(real x)\nReturn the size of x which for scalar-valued x is 1\nAvailable since 2.26\n\n\n\nIt is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see [Rounding functions]) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
+    "text": "This chapter describes Stan’s built-in function that take various types of arguments and return integer values.\n\n\nStan’s arithmetic is based on standard double-precision C++ integer and floating-point arithmetic. If the arguments to an arithmetic operator are both integers, as in 2 + 2, integer arithmetic is used. If one argument is an integer and the other a floating-point value, as in 2.0 + 2 and 2 + 2.0, then the integer is promoted to a floating point value and floating-point arithmetic is used.\nInteger arithmetic behaves slightly differently than floating point arithmetic. The first difference is how overflow is treated. If the sum or product of two integers overflows the maximum integer representable, the result is an undesirable wraparound behavior at the bit level. If the integers were first promoted to real numbers, they would not overflow a floating-point representation. There are no extra checks in Stan to flag overflows, so it is up to the user to make sure it does not occur.\nSecondly, because the set of integers is not closed under division and there is no special infinite value for integers, integer division implicitly rounds the result. If both arguments are positive, the result is rounded down. For example, 1 / 2 evaluates to 0 and 5 / 3 evaluates to 1.\nIf one of the integer arguments to division is negative, the latest C++ specification ( C++11), requires rounding toward zero. This would have 1 / 2 and -1 / 2 evaluate to 0, -7 / 2 evaluate to -3, and 7 / 2 evaluate to 3. Before the C++11 specification, the behavior was platform dependent, allowing rounding up or down. All compilers recent enough to be able to deal with Stan’s templating should follow the C++11 specification, but it may be worth testing if you are not sure and plan to use integer division with negative values.\nUnlike floating point division, where 1.0 / 0.0 produces the special positive infinite value, integer division by zero, as in 1 / 0, has undefined behavior in the C++ standard. For example, the clang++ compiler on Mac OS X returns 3764, whereas the g++ compiler throws an exception and aborts the program with a warning. As with overflow, it is up to the user to make sure integer divide-by-zero does not occur.\n\n\nOperators are described using the C++ syntax. For instance, the binary operator of addition, written X + Y, would have the Stan signature int operator+(int, int) indicating it takes two real arguments and returns a real value. As noted previously, the value of integer division is platform-dependent when rounding is platform dependent before C++11; the descriptions below provide the C++11 definition.\n \n\nint operator+(int x, int y) The sum of the addends x and y \\[\\begin{equation*} \\text{operator+}(x,y) = (x + y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator-(int x, int y) The difference between the minuend x and subtrahend y \\[\\begin{equation*}\n\\text{operator-}(x,y) = (x - y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator*(int x, int y) The product of the factors x and y \\[\\begin{equation*} \\text{operator*}(x,y) = (x\n\\times y) \\end{equation*}\\]\nAvailable since 2.0\n \n\nint operator/(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator/}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\] deprecated; - use operator%/% instead.\nAvailable since 2.0, deprecated in 2.24\n \n\nint operator%/%(int x, int y) The integer quotient of the dividend x and divisor y \\[\\begin{equation*}\n\\text{operator\\%/\\%}(x,y) = \\begin{cases} \\lfloor x / y \\rfloor & \\text{if\n} x / y \\geq 0 \\\\ - \\lfloor \\text{floor}(-x / y) \\rfloor & \\text{if }\nx / y &lt; 0. \\end{cases} \\end{equation*}\\]\nAvailable since 2.24\n \n\nint operator%(int x, int y) x modulo y, which is the positive remainder after dividing x by y. If both x and y are non-negative, so is the result; otherwise, the sign of the result is platform dependent. \\[\\begin{equation*} \\mathrm{operator\\%}(x, y) \\ =\n\\ x \\ \\text{mod} \\ y \\ = \\ x - y * \\lfloor x / y \\rfloor \\end{equation*}\\]\nAvailable since 2.13\n\n\n\n \n\nint operator-(int x) The negation of the subtrahend x \\[\\begin{equation*} \\text{operator-}(x) = -x \\end{equation*}\\]\nAvailable since 2.0\n \n\nT operator-(T x) Vectorized version of operator-. If T x is a (possibly nested) array of integers, -x is the same shape array where each individual integer is negated.\nAvailable since 2.31\n \n\nint operator+(int x) This is a no-op. \\[\\begin{equation*} \\text{operator+}(x) = x \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n\n \n\nT abs(T x) The absolute value of x.\nThis function works elementwise over containers such as vectors. Given a type T which is int, or an array of ints, abs returns the same type where each element has had its absolute value taken.\nAvailable since 2.0, vectorized in 2.30\n \n\nint int_step(int x)\n \n\nint int_step(real x) Return the step function of x as an integer, \\[\\begin{equation*} \\mathrm{int\\_step}(x)\n= \\begin{cases} 1 & \\text{if } x &gt; 0 \\\\ 0 & \\text{if } x \\leq 0 \\text{\nor } x \\text{ is } NaN \\end{cases} \\end{equation*}\\] Warning: int_step(0) and int_step(NaN) return 0 whereas step(0) and step(NaN) return 1.\nSee the warning in section step functions about the dangers of step functions applied to anything other than data.\nAvailable since 2.0\n\n\n\n \n\nint min(int x, int y) Return the minimum of x and y. \\[\\begin{equation*} \\text{min}(x, y) = \\begin{cases} x &\n\\text{if } x &lt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n \n\nint max(int x, int y) Return the maximum of x and y. \\[\\begin{equation*} \\text{max}(x, y) = \\begin{cases} x &\n\\text{if } x &gt; y\\\\ y & \\text{otherwise} \\end{cases} \\end{equation*}\\]\nAvailable since 2.0\n\n\n\n \n\nint size(int x)\n \n\nint size(real x)\nReturn the size of x which for scalar-valued x is 1\nAvailable since 2.26\n\n\n\nIt is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see Rounding functions) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
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-    "text": "It is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see [Rounding functions]) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
+    "text": "It is possible to cast real numbers to integers as long as the real value is data. See data only qualifiers in the Stan Reference Manual.\n \n\nint to_int(data real x)\nReturn the value x truncated to an integer. This will throw an error if the value of x is too big to represent as a 32-bit signed integer.\nThis is similar to trunc (see Rounding functions) but the return type is of type int. For example, to_int(3.9) is 3, and to_int(-3.9) is -3.\nAvailable since 2.31\n \n\nI to_int(data T x)\nThe vectorized version of to_int. This function accepts a (possibly nested) array of reals and returns an array of the same shape where each element has been truncated to an integer.\nAvailable since 2.31",
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     "title": "Deprecated Functions",
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-    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of get_lp or multiply_log) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in [Removed Functions]. The Stan compiler can automatically update these on the behalf of the user for the entire deprecation window and at least one version following the removal.\n\n\nDeprecated: Using / with two integer arguments is interpreted as integer floor division, such that\n\\[ 1 / 2 = 0 \\]\nThis is deprecated due to its confusion with real-valued division, where\n\\[ 1.0 / 2.0 = 0.5 \\]\nReplacement: Use the integer division operator operator%/% instead.\n\n\n\nThese ODE integrator functions have been replaced by those described in Ordinary Differential Equation (ODE) Solvers.\n\n\nA system of ODEs is specified as an ordinary function in Stan within the functions block. The ODE system function must have this function signature:\narray[] real ode(real time, array[] real state, array[] real theta,\n                 array[] real x_r, array[] int x_i);\nThe ODE system function should return the derivative of the state with respect to time at the time provided. The length of the returned real array must match the length of the state input into the function.\nThe arguments to this function are:\n\ntime, the time to evaluate the ODE system\nstate, the state of the ODE system at the time specified\ntheta, parameter values used to evaluate the ODE system\nx_r, data values used to evaluate the ODE system\nx_i, integer data values used to evaluate the ODE system.\n\nThe ODE system function separates parameter values, theta, from data values, x_r, for efficiency in computing the gradients of the ODE.\n\n\n\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i, real rel_tol, real abs_tol, int max_num_steps) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the Adams-Moulton method.\nAvailable since 2.23, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the Adams-Moulton method with additional control parameters for the solver.\nAvailable since 2.23, deprecated in 2.24\n\n\n\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n\n\n\nThe arguments to the ODE solvers in both the stiff and non-stiff cases are as follows.\n\node: function literal referring to a function specifying the system of differential equations with signature:\n\n(real, array[] real, array[] real, data array[] real, data array[] int):array[] real\nThe arguments represent (1) time, (2) system state, (3) parameters, (4) real data, and (5) integer data, and the return value contains the derivatives with respect to time of the state,\n\ninitial_state: initial state, type array[] real,\ninitial_time: initial time, type int or real,\ntimes: solution times, type array[] real,\ntheta: parameters, type array[] real,\ndata x_r: real data, type array[] real, data only, and\ndata x_i: integer data, type array[] int, data only.\n\nFor more fine-grained control of the ODE solvers, these parameters can also be provided:\n\ndata rel_tol: relative tolerance for the ODE solver, type real, data only,\ndata abs_tol: absolute tolerance for the ODE solver, type real, data only, and\ndata max_num_steps: maximum number of steps to take in the ODE solver, type int, data only.\n\n\n\nThe return value for the ODE solvers is an array of type array[,] real, with values consisting of solutions at the specified times.\n\n\n\nThe sizes must match, and in particular, the following groups are of the same size:\n\nstate variables passed into the system function, derivatives returned by the system function, initial state passed into the solver, and rows of the return value of the solver,\nsolution times and number of rows of the return value of the solver,\nparameters, real data and integer data passed to the solver will be passed to the system function\n\n\n\n\n\n\nThese algebraic solver functions have been replaced by those described in Algebraic Equation Solvers..\n\n\nAn algebraic system is specified as an ordinary function in Stan within the function block. The algebraic system function must have this signature:\n vector algebra_system(vector y, vector theta,\n                              data array[] real x_r, array[] int x_i)\nThe algebraic system function should return the value of the algebraic function which goes to 0, when we plug in the solution to the algebraic system.\nThe argument of this function are:\n\ny, the unknowns we wish to solve for\ntheta, parameter values used to evaluate the algebraic system\nx_r, data values used to evaluate the algebraic system\nx_i, integer data used to evaluate the algebraic system\n\nThe algebraic system function separates parameter values, theta, from data values, x_r, for efficiency in propagating the derivatives through the algebraic system.\n\n\n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm.\nAvailable since 2.17, deprecated in 2.31\n \n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm with additional control parameters for the solver.\nAvailable since 2.17, deprecated in 2.31\nNote: In future releases, the function algebra_solver will be deprecated and replaced with algebra_solver_powell.\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using Newton’s method.\nAvailable since 2.24, deprecated in 2.31\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using Newton’s method with additional control parameters for the solver.\nAvailable since 2.24, deprecated in 2.31\n\n\nThe arguments to the algebraic solvers are as follows:\n\nalgebra_system: function literal referring to a function specifying the system of algebraic equations with signature (vector, vector, array[] real, array[] int):vector. The arguments represent (1) unknowns, (2) parameters, (3) real data, and (4) integer data, and the return value contains the value of the algebraic function, which goes to 0 when we plug in the solution to the algebraic system,\ny_guess: initial guess for the solution, type vector,\ntheta: parameters only, type vector,\nx_r: real data only, type array[] real, and\nx_i: integer data only, type array[] int.\n\nFor more fine-grained control of the algebraic solver, these parameters can also be provided:\n\nrel_tol: relative tolerance for the algebraic solver, type real, data only,\nfunction_tol: function tolerance for the algebraic solver, type real, data only,\nmax_num_steps: maximum number of steps to take in the algebraic solver, type int, data only.\n\n\n\n\nThe return value for the algebraic solver is an object of type vector, with values which, when plugged in as y make the algebraic function go to 0.\n\n\n\nCertain sizes have to be consistent. The initial guess, return value of the solver, and return value of the algebraic function must all be the same size.\nThe parameters, real data, and integer data will be passed from the solver directly to the system function.",
+    "text": "This appendix lists currently deprecated functionality along with how to replace it.\nStarting in Stan 2.29, deprecated functions with drop in replacements (such as the renaming of get_lp or multiply_log) will be removed 3 versions later e.g., functions deprecated in Stan 2.20 will be removed in Stan 2.23 and placed in Removed Functions. The Stan compiler can automatically update these on the behalf of the user for the entire deprecation window and at least one version following the removal.\n\n\nDeprecated: Using / with two integer arguments is interpreted as integer floor division, such that\n\\[ 1 / 2 = 0 \\]\nThis is deprecated due to its confusion with real-valued division, where\n\\[ 1.0 / 2.0 = 0.5 \\]\nReplacement: Use the integer division operator operator%/% instead.\n\n\n\nThese ODE integrator functions have been replaced by those described in Ordinary Differential Equation (ODE) Solvers.\n\n\nA system of ODEs is specified as an ordinary function in Stan within the functions block. The ODE system function must have this function signature:\narray[] real ode(real time, array[] real state, array[] real theta,\n                 array[] real x_r, array[] int x_i);\nThe ODE system function should return the derivative of the state with respect to time at the time provided. The length of the returned real array must match the length of the state input into the function.\nThe arguments to this function are:\n\ntime, the time to evaluate the ODE system\nstate, the state of the ODE system at the time specified\ntheta, parameter values used to evaluate the ODE system\nx_r, data values used to evaluate the ODE system\nx_i, integer data values used to evaluate the ODE system.\n\nThe ODE system function separates parameter values, theta, from data values, x_r, for efficiency in computing the gradients of the ODE.\n\n\n\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_rk45(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i, real rel_tol, real abs_tol, int max_num_steps) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, array[] real x_r, array[] int x_i) Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the Adams-Moulton method.\nAvailable since 2.23, deprecated in 2.24\n \n\narray[,] real integrate_ode_adams(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the Adams-Moulton method with additional control parameters for the solver.\nAvailable since 2.23, deprecated in 2.24\n\n\n\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method.\nAvailable since 2.10, deprecated in 2.24\n \n\narray[,] real integrate_ode_bdf(function ode, array[] real initial_state, real initial_time, array[] real times, array[] real theta, data array[] real x_r, data array[] int x_i, data real rel_tol, data real abs_tol, data int max_num_steps) Solves the ODE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.\nAvailable since 2.10, deprecated in 2.24\n\n\n\nThe arguments to the ODE solvers in both the stiff and non-stiff cases are as follows.\n\node: function literal referring to a function specifying the system of differential equations with signature:\n\n(real, array[] real, array[] real, data array[] real, data array[] int):array[] real\nThe arguments represent (1) time, (2) system state, (3) parameters, (4) real data, and (5) integer data, and the return value contains the derivatives with respect to time of the state,\n\ninitial_state: initial state, type array[] real,\ninitial_time: initial time, type int or real,\ntimes: solution times, type array[] real,\ntheta: parameters, type array[] real,\ndata x_r: real data, type array[] real, data only, and\ndata x_i: integer data, type array[] int, data only.\n\nFor more fine-grained control of the ODE solvers, these parameters can also be provided:\n\ndata rel_tol: relative tolerance for the ODE solver, type real, data only,\ndata abs_tol: absolute tolerance for the ODE solver, type real, data only, and\ndata max_num_steps: maximum number of steps to take in the ODE solver, type int, data only.\n\n\n\nThe return value for the ODE solvers is an array of type array[,] real, with values consisting of solutions at the specified times.\n\n\n\nThe sizes must match, and in particular, the following groups are of the same size:\n\nstate variables passed into the system function, derivatives returned by the system function, initial state passed into the solver, and rows of the return value of the solver,\nsolution times and number of rows of the return value of the solver,\nparameters, real data and integer data passed to the solver will be passed to the system function\n\n\n\n\n\n\nThese algebraic solver functions have been replaced by those described in Algebraic Equation Solvers..\n\n\nAn algebraic system is specified as an ordinary function in Stan within the function block. The algebraic system function must have this signature:\n vector algebra_system(vector y, vector theta,\n                              data array[] real x_r, array[] int x_i)\nThe algebraic system function should return the value of the algebraic function which goes to 0, when we plug in the solution to the algebraic system.\nThe argument of this function are:\n\ny, the unknowns we wish to solve for\ntheta, parameter values used to evaluate the algebraic system\nx_r, data values used to evaluate the algebraic system\nx_i, integer data used to evaluate the algebraic system\n\nThe algebraic system function separates parameter values, theta, from data values, x_r, for efficiency in propagating the derivatives through the algebraic system.\n\n\n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm.\nAvailable since 2.17, deprecated in 2.31\n \n\nvector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm with additional control parameters for the solver.\nAvailable since 2.17, deprecated in 2.31\nNote: In future releases, the function algebra_solver will be deprecated and replaced with algebra_solver_powell.\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i) Solves the algebraic system, given an initial guess, using Newton’s method.\nAvailable since 2.24, deprecated in 2.31\n \n\nvector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps) Solves the algebraic system, given an initial guess, using Newton’s method with additional control parameters for the solver.\nAvailable since 2.24, deprecated in 2.31\n\n\nThe arguments to the algebraic solvers are as follows:\n\nalgebra_system: function literal referring to a function specifying the system of algebraic equations with signature (vector, vector, array[] real, array[] int):vector. The arguments represent (1) unknowns, (2) parameters, (3) real data, and (4) integer data, and the return value contains the value of the algebraic function, which goes to 0 when we plug in the solution to the algebraic system,\ny_guess: initial guess for the solution, type vector,\ntheta: parameters only, type vector,\nx_r: real data only, type array[] real, and\nx_i: integer data only, type array[] int.\n\nFor more fine-grained control of the algebraic solver, these parameters can also be provided:\n\nrel_tol: relative tolerance for the algebraic solver, type real, data only,\nfunction_tol: function tolerance for the algebraic solver, type real, data only,\nmax_num_steps: maximum number of steps to take in the algebraic solver, type int, data only.\n\n\n\n\nThe return value for the algebraic solver is an object of type vector, with values which, when plugged in as y make the algebraic function go to 0.\n\n\n\nCertain sizes have to be consistent. The initial guess, return value of the solver, and return value of the algebraic function must all be the same size.\nThe parameters, real data, and integer data will be passed from the solver directly to the system function.",
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-    "text": "CmdStan can approximate the posterior distribution using variational inference. The approximation is a Gaussian in the unconstrained variable space.\nStan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) Kucukelbir et al. (2017). ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using stochastic gradient ascent. The measures of convergence are similar to the relative tolerance scheme of Stan’s optimization algorithms.\nThe algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run with method=variational via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\nThe console output includes a notice that this algorithm is considered to be experimental:\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n\n\nStan implements two variational algorithms. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.\n\n\n\n\n\niter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv. Lines 1 - 28 contain configuration information:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nThe column header row is:\nlp__,log_p__,log_g__,theta\nThe stepsize adaptation information is:\n# Stepsize adaptation complete.\n# eta = 1\nThe reported mean variational approximations information is:\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...",
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-    "text": "Stan implements two variational algorithms. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.",
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-    "text": "iter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.",
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-    "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv. Lines 1 - 28 contain configuration information:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nThe column header row is:\nlp__,log_p__,log_g__,theta\nThe stepsize adaptation information is:\n# Stepsize adaptation complete.\n# eta = 1\nThe reported mean variational approximations information is:\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...",
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-    "text": "CmdStan translates Stan programs to C++ using the Stan compiler program which is included in the CmdStan release bin directory as program stanc. One can view the complete stanc documentation in the Stan User’s Guide.\nAs of release 2.22, the CmdStan Stan to C++ compiler is written in OCaml. This compiler is called “stanc3” and has has its own repository https://github.com/stan-dev/stanc3, from which pre-built binaries for Linux, Mac, and Windows can be downloaded.\n\n\nBefore the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.\n\n\n\nThe Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
+    "text": "The CmdStan stansummary program reports statistics for one or more sampler chains over all sampler and model parameters and quantities of interest. The statistics reported include both summary statistics of the estimates and diagnostic statistics on the sampler chains, reported in the following order:\n\nMean - sample mean\nMCSE - Monte Carlo Standard Error, a measure of the amount of noise in the sample\nStdDev - sample standard deviation\nQuantiles - default 5%, 50%, 95%\nN_eff - effective sample size - the number of independent draws in the sample\nN_eff/S - the number of independent draws per second\nR_hat - \\(\\hat{R}\\) statistic, a measure of chain equilibrium, must be within \\(0.05\\) of \\(1.0\\).\n\nWhen reviewing the stansummary output, it is important to check the final three output columns first - these are the diagnostic statistics on chain convergence and number of independent draws in the sample. A \\(\\hat{R}\\) statistic of greater than \\(1.05\\) indicates that the chain has not converged and therefore the sample is not drawn from the posterior, thus the estimates of the mean and all other summary statistics are invalid.\nEstimation by sampling produces an approximate value for the model parameters; the MCSE statistic indicates the amount of noise in the estimate. Therefore MCSE column is placed next to the sample mean column, in order to make it easy to compare this sample with others.\nFor more information, see the Posterior Analysis chapter of the Stan Reference Manual which describes both the theory and practice of MCMC estimation techniques. The summary statistics and the algorithms used to compute them are described in sections Notation for samples and Effective Sample Size.\n\n\nThe CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary\n\n\n\nThe stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].\n\n\n\n\nThe stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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-    "text": "Before the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.",
+    "text": "The CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary",
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-    "text": "The Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
+    "text": "The stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].",
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-    "text": "NOTE: Although the RDump format is still supported, I/O with JSON is faster and recommended. See the chapter on JSON for more details.\nRDump format can be used to represent values for Stan variables. This format was introduced in SPLUS and is used in R, JAGS, and in BUGS (but with a different ordering).\nA dump file is structured as a sequence of variable definitions. Each variable is defined in terms of its dimensionality and its values. There are three kinds of variable declarations: - scalars - sequences - general arrays\n\n\nDump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.\n\n\n\nA simple scalar value can be thought of as having an empty list of dimensions. Its declaration in the dump format follows the SPLUS assignment syntax. For example, the following would constitute a valid dump file defining a single scalar variable y with value \\(17.2\\):\ny &lt;- 17.2\n\n\n\nOne-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)\n\n\n\nFor more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))\n\n\n\nThe dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;\n\n\n\n\nAt this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.\n\n\n\nThere is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.\n\n\n\n\nIn order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}\n\n\n\nThe line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. For example, the following declaration of a \\(2 \\times 2 \\times 3\\) array is valid.\ny &lt;-\nstructure(c(1,2,3,\n4,5,6,7,8,9,10,11,\n12), .Dim = c(2,2,\n3))\nBecause there are no decimal points in the values, the resulting dump file may be used for three-dimensional array variables declared as int or real.\n\n\n\nA more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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-    "text": "One-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)",
+    "text": "The stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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-    "text": "For more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))",
+    "text": "The output from all CmdStan methods is in CSV format. A Stan CSV file is a data table where the columns are the method and model parameters and quantities of interest. Each row contains one record’s worth of data in plain-text format using the comma character (‘,’) as the field delimiter (hence the name).\nFor the Stan CSV files, data is strictly numerical, however, possible values include both positive and negative infinity and “Not-a-Number” which are represented as the strings NaN, inf, +inf, -inf. All other values are written in decimal notation with at most 6 digits of precision.\nStan CSV files have a header row containing the column names. They also make extensive use of CSV comments, i.e., lines which begin with the # character. In addition to initial and final comment rows, some methods also put comment rows in the middle of the data table, which makes it difficult to use many of the commonly used CSV parser packages.\n\n\nThe data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3\n\n\n\nThe sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation\n\n\n\n\n\n\nConfig as comments\nHeader row\nPenalized maximum likelihood estimate\n\n\n\n\n\nConfig as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior\n\n\n\n\n\nHeader row\nQuantities of interest\n\n\n\n\n\nHeader row\nGradients",
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-    "text": "The dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;",
+    "text": "The data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3",
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-    "text": "At this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.",
+    "text": "The sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation",
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-    "text": "There is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.",
+    "text": "Config as comments\nHeader row\nPenalized maximum likelihood estimate",
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-    "text": "In order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}",
+    "text": "Config as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior",
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-    "text": "The line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. For example, the following declaration of a \\(2 \\times 2 \\times 3\\) array is valid.\ny &lt;-\nstructure(c(1,2,3,\n4,5,6,7,8,9,10,11,\n12), .Dim = c(2,2,\n3))\nBecause there are no decimal points in the values, the resulting dump file may be used for three-dimensional array variables declared as int or real.",
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-    "text": "A more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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-    "text": "Variational Inference using Pathfinder\nThe CmdStan method pathfinder uses the Pathfinder algorithm of Zhang et al. (2022). Pathfinder is a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the negative inverse Hessian estimates produced by the L-BFGS optimizer. Pathfinder returns draws from the Gaussian approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior.\nPathfinder differs from the ADVI method in that it uses quasi-Newton optimization on the log posterior instead of stochastic gradient descent (SGD) on the Monte Carlo computation of the evidence lower bound (ELBO). Pathfinder’s approach is both faster and more stable than that of ADVI. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors.\nA single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes. By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws.\nThe following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli pathfinder data file=bernoulli.data.R\nExecuting this command prints both output to the console and csv files.\nThe first part of the console output reports on the configuration used.\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\nid = 1 (Default)\ndata\n  file = examples/bernoulli/bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 1995513073 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n  sig_figs = -1 (Default)\n  profile_file = profile.csv (Default)\nnum_threads = 1 (Default)\nThe rest of the output describes the progression of the algorithm.\nBy default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, the uses importance resampling on the set of returned draws to produce the specified number of draws.\nPath [1] :Initial log joint density = -11.543343\nPath [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00\nPath [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)\nPath [2] :Initial log joint density = -7.443345\nPath [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00\nPath [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)\nPath [3] :Initial log joint density = -18.986308\nPath [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00\nPath [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)\nPath [4] :Initial log joint density = -8.304453\nPath [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00\nPath [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)\nTotal log probability function evaluations:8404\nPathfinder outputs a StanCSV file file which contains the importance resampled draws from multi-path Pathfinder. The initial CSV comment rows contain the complete set of CmdStan configuration options. Next is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs. See section Pathfinder Method for details.\n\n\n\n\n\n Back to topReferences\n\nZhang, Lu, Bob Carpenter, Andrew Gelman, and Aki Vehtari. 2022. “Pathfinder: Parallel Quasi-Newton Variational Inference.” Journal of Machine Learning Research 23 (306): 1–49. http://jmlr.org/papers/v23/21-0889.html.",
+    "text": "print (deprecated): MCMC Output Analysis\nThe print utility is deprecated, but is still available until CmdStan v3.0. It has been replaced by the stansummary utility.\n\n\n\n\n Back to top",
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     "text": "Stan provides three ways of parallelizing execution of a Stan model:\n\nmulti-threading with Intel Threading Building Blocks (TBB),\nmulti-processing with Message Passing Interface (MPI) and\nmanycore processing with OpenCL.\n\n\n\nIn order to exploit multi-threading in a Stan model, the models must be rewritten to use the reduce_sum and map_rect functions. For instructions on how to rewrite Stan models to use these functions see Stan’s User guide chapter on parallelization, the reduce_sum case study or the Multithreading and Map-Reduce tutorial.\n\n\nOnce a model is rewritten to use the above-mentioned functions, the model must be compiled with the STAN_THREADS makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable threading with TBB:\nSTAN_THREADS=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nBefore running a multi-threaded model, we need to specify the maximum number of threads the program can run (total threads for all chains). This is done by setting the num_threads argument. Valid values for num_threads are positive integers and -1. If num_threads is set to -1, all available cores will be used.\nGenerally, this number should not exceed the number of available cores for best performance.\nExample:\n./model sample data file=data.json num_threads=4 ...\nWhen the model is compiled with STAN_THREADS we can sample with multiple chains with a single executable (see section running multiple chains for cases when this is available). When running multiple chains num_threads is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:\n./model sample num_chains=2 data file=data.json num_threads=8 ...\n\n\n\n\nIn order to use multi-processing with MPI in a Stan model, the models must be rewritten to use the map_rect function. By using MPI, the model can be parallelized across multiple cores or a cluster. MPI with Stan is supported on MacOS and Linux.\n\n\nCompiling and running Stan models with MPI requires that the system has an MPI implementation installed. For Unix systems the most commonly used implementations are MPICH and OpenMPI.\n\n\n\nOnce a model is rewritten to use map_rect, additional makefile flags must be written to the make/local. These are:\n\nSTAN_MPI: Enables the use of MPI with Stan if true.\nCXX: The name of the MPI C++ compiler wrapper. Typically mpicxx.\nTBB_CXX_TYPE: The C++ compiler the MPI wrapper wraps. Typically gcc on Linux and clang on macOS.\n\nAn example of make/local on Linux:\nSTAN_MPI=true\nCXX=mpicxx\nTBB_CXX_TYPE=gcc\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_MPI is run using an MPI launcher. The MPI standard suggests using mpiexec, but a vendor wrapper for the launcher like mpirun can also be used. The launcher is supplied the path to the built executable and the number of processes to start: -n X for mpiexec or -np X for mpirun where X is replaced by the integer representing the number of processes.\nExample for running a model with six processes:\nmpiexec -n 6 path/to/model sample data file=data.json ...\n\n\n\n\n\n\nOpenCL is supported on most modern CPUs and GPUs. In order to run OpenCL-enabled Stan models, an OpenCL runtime for the target device must be installed. This subsection lists installation instructions for OpenCL runtimes of the commonly-found devices.\nIn order to check if any OpenCL-enabled device and its runtime is already present use the clinfo tool. On Linux, clinfo can typically be installed with the default package manager (for example sudo apt-get install clinfo on Ubuntu). For Windows, pre-built clinfo binary can be found here.\nAlso use clinfo to verify successful installation of OpenCL runtimes.\n\n\n\nLinux:\nInstall the NVIDIA GPU driver and the NVIDIA CUDA Toolkit. On Ubuntu the commands to install both is:\nsudo apt update\nsudo apt install nvidia-driver-460 nvidia-cuda-toolkit\nReplace the driver version (460 in the above case) with the lastest number at the time of installation.\nWindows:\nInstall the NVIDIA GPU Driver and CUDA Toolkit.\n\n\n\n\n\nLinux:\nInstall Radeon Software for Linux available here.\nWindows:\nWe recommend installing the open source OCL-SDK.\n\n\n\n\nInstall the open source PoCL.\n\n\n\nFollow Intel’s install instructions given here (requires registration).\n\n\n\n\nIn order to enable the OpenCL backend the model must be compiled with the STAN_OPENCL makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable parallelization with OpenCL:\nSTAN_OPENCL=true\nIf you are using OpenCL with an integrated GPU you also need to add the INTEGRATED_OPENCL flag, as the sharing of memory between CPU and GPU is slightly different with integrated graphics:\nINTEGRATED_OPENCL=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_OPENCL can also be supplied the OpenCL platform and device IDs of the target device. These IDs determine the device on which to run the OpenCL-supported functions on. You can list the devices on your system using the clinfo program. If the system has one GPU and no OpenCL CPU runtime, the platform and device IDs of the GPU are typically 0. In that case you can also omit the OpenCL IDs as the default 0 IDs are used in that case.\nWe supply these IDs when starting the executable as shown below:\npath/to/model sample data file=data.json opencl platform=0 device=1",
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     "section": "",
     "text": "In order to exploit multi-threading in a Stan model, the models must be rewritten to use the reduce_sum and map_rect functions. For instructions on how to rewrite Stan models to use these functions see Stan’s User guide chapter on parallelization, the reduce_sum case study or the Multithreading and Map-Reduce tutorial.\n\n\nOnce a model is rewritten to use the above-mentioned functions, the model must be compiled with the STAN_THREADS makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable threading with TBB:\nSTAN_THREADS=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nBefore running a multi-threaded model, we need to specify the maximum number of threads the program can run (total threads for all chains). This is done by setting the num_threads argument. Valid values for num_threads are positive integers and -1. If num_threads is set to -1, all available cores will be used.\nGenerally, this number should not exceed the number of available cores for best performance.\nExample:\n./model sample data file=data.json num_threads=4 ...\nWhen the model is compiled with STAN_THREADS we can sample with multiple chains with a single executable (see section running multiple chains for cases when this is available). When running multiple chains num_threads is the maximum number of threads that can be used by all the chains combined. The exact number of threads that will be used for each chain at a given point in time is determined by the TBB scheduler. The following example start 2 chains with 8 total threads available:\n./model sample num_chains=2 data file=data.json num_threads=8 ...",
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     "section": "",
     "text": "In order to use multi-processing with MPI in a Stan model, the models must be rewritten to use the map_rect function. By using MPI, the model can be parallelized across multiple cores or a cluster. MPI with Stan is supported on MacOS and Linux.\n\n\nCompiling and running Stan models with MPI requires that the system has an MPI implementation installed. For Unix systems the most commonly used implementations are MPICH and OpenMPI.\n\n\n\nOnce a model is rewritten to use map_rect, additional makefile flags must be written to the make/local. These are:\n\nSTAN_MPI: Enables the use of MPI with Stan if true.\nCXX: The name of the MPI C++ compiler wrapper. Typically mpicxx.\nTBB_CXX_TYPE: The C++ compiler the MPI wrapper wraps. Typically gcc on Linux and clang on macOS.\n\nAn example of make/local on Linux:\nSTAN_MPI=true\nCXX=mpicxx\nTBB_CXX_TYPE=gcc\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_MPI is run using an MPI launcher. The MPI standard suggests using mpiexec, but a vendor wrapper for the launcher like mpirun can also be used. The launcher is supplied the path to the built executable and the number of processes to start: -n X for mpiexec or -np X for mpirun where X is replaced by the integer representing the number of processes.\nExample for running a model with six processes:\nmpiexec -n 6 path/to/model sample data file=data.json ...",
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     "section": "",
     "text": "OpenCL is supported on most modern CPUs and GPUs. In order to run OpenCL-enabled Stan models, an OpenCL runtime for the target device must be installed. This subsection lists installation instructions for OpenCL runtimes of the commonly-found devices.\nIn order to check if any OpenCL-enabled device and its runtime is already present use the clinfo tool. On Linux, clinfo can typically be installed with the default package manager (for example sudo apt-get install clinfo on Ubuntu). For Windows, pre-built clinfo binary can be found here.\nAlso use clinfo to verify successful installation of OpenCL runtimes.\n\n\n\nLinux:\nInstall the NVIDIA GPU driver and the NVIDIA CUDA Toolkit. On Ubuntu the commands to install both is:\nsudo apt update\nsudo apt install nvidia-driver-460 nvidia-cuda-toolkit\nReplace the driver version (460 in the above case) with the lastest number at the time of installation.\nWindows:\nInstall the NVIDIA GPU Driver and CUDA Toolkit.\n\n\n\n\n\nLinux:\nInstall Radeon Software for Linux available here.\nWindows:\nWe recommend installing the open source OCL-SDK.\n\n\n\n\nInstall the open source PoCL.\n\n\n\nFollow Intel’s install instructions given here (requires registration).\n\n\n\n\nIn order to enable the OpenCL backend the model must be compiled with the STAN_OPENCL makefile flag. The flag can be supplied in the make call but we recommend writing the flag to the make/local file.\nAn example of the contents of make/local to enable parallelization with OpenCL:\nSTAN_OPENCL=true\nIf you are using OpenCL with an integrated GPU you also need to add the INTEGRATED_OPENCL flag, as the sharing of memory between CPU and GPU is slightly different with integrated graphics:\nINTEGRATED_OPENCL=true\nThe model is then compiled as normal:\nmake path/to/model\n\n\n\nThe Stan model compiled with STAN_OPENCL can also be supplied the OpenCL platform and device IDs of the target device. These IDs determine the device on which to run the OpenCL-supported functions on. You can list the devices on your system using the clinfo program. If the system has one GPU and no OpenCL CPU runtime, the platform and device IDs of the GPU are typically 0. In that case you can also omit the OpenCL IDs as the default 0 IDs are used in that case.\nWe supply these IDs when starting the executable as shown below:\npath/to/model sample data file=data.json opencl platform=0 device=1",
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-    "text": "Optimization\nThe CmdStan executable can run Stan’s optimization algorithms which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability.\nThe executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.\n&gt; ./bernoulli optimize data file=bernoulli.data.json\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:\n ./bernoulli optimize data file=bernoulli.data.json\nmethod = optimize\n  optimize\n    algorithm = lbfgs (Default)\n      lbfgs\n        init_alpha = 0.001 (Default)\n        tol_obj = 1e-12 (Default)\n        tol_rel_obj = 10000 (Default)\n        tol_grad = 1e-08 (Default)\n        tol_rel_grad = 10000000 (Default)\n        tol_param = 1e-08 (Default)\n        history_size = 5 (Default)\n    iter = 2000 (Default)\n    save_iterations = 0 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 87122538 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nThe second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The alpha value is for step size used. In the final state the change in parameters was roughly \\(0.002\\) and the length of the gradient roughly 3e-05 (\\(0.00003\\)).\nInitial log joint probability = -6.85653\n    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes \n       5      -5.00402    0.00184936   3.35074e-05           1           1        8   \nOptimization terminated normally: \n  Convergence detected: relative gradient magnitude is below tolerance\nThe output from optimization is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n...\nFollowing the config information, are two lines of output: the CSV headers and the recorded values:\nlp__,theta\n-5.00402,0.200003\nNote that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with lp__ and the model parameter theta. The maximum log probability is -5.0 and the posterior mode for theta is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file. Running the optimizer with save_iterations=1 writes both the initial log joint probability and values for all iterations to the output CSV file.\nRunning the example model with option save_iterations=1, i.e., the command\n&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json\nproduces CSV file output rows:\nlp__,theta\n-6.85653,0.493689\n-6.10128,0.420936\n-5.02953,0.22956\n-5.00517,0.206107\n-5.00403,0.200299\n-5.00402,0.200003\n\n\n\n\n Back to top",
+    "text": "The sample method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on MCMC Sampling.\n\n\nTo generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory.\nThe full set of configuration options available for the sample method is available by using the sample help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.\nHere, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\n&gt; ls\n  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n...\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)\n\n\n\nEach execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\nWhen the example model bernoulli.stan is run via the command line with all default arguments, the following configuration is displayed:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)\n\n\n\nAt every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.\n\n\n\nThe adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.\n\n\n\n\n\nThe algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).\n\n\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.\n\n\n\nA Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThe preferred way of using multiple chains is to run them all from the same executable using the num_chains argument. There is also the option to use the Unix or DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with STAN_THREADS=true, these will either run in parallel or one after the other.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\nFor example, this will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nThis will produce samples in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially.\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\n\nThe stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)\n\n\n\nNote: Many of these examples can be simplified by using the num_chains argument.\nWhen the num_chains argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following extended examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
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-    "text": "The sample method provides Bayesian inference over the model conditioned on data using Hamiltonian Monte Carlo (HMC) sampling. By default, the inference engine used is the No-U-Turn sampler (NUTS), an adaptive form of Hamiltonian Monte Carlo sampling. For details on HMC and NUTS, see the Stan Reference Manual chapter on MCMC Sampling.\nThe full set of configuration options available for the sample method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default sample method configuration options:\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n\n\nAt every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.\n\n\n\nThe adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.\n\n\n\n\n\nThe algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).\n\n\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.\n\n\n\nAs described in the quickstart section on parallelism, the preferred way to run multiple chains is to use the num_chains argument.\nThis will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with STAN_THREADS=true.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\n\n\n\nNote: Many of these examples can be simplified by using the num_chains argument.\nThe Quickstart Guide MCMC Sampling chapter section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.\nThis creates multiple copies of the model process which will all load the data.\nTo run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=my_model.data.json \\\n                  output file=output_${i}.csv &\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. The ampersand (&) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows the corresponding loop is:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=my_model.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
+    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory.\nThe full set of configuration options available for the sample method is available by using the sample help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the sampler console output and in the output CSV file’s comments.\nHere, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\n&gt; ls\n  bernoulli  bernoulli.data.json  bernoulli.data.R  bernoulli.stan\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n...\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)",
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+    "text": "Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\nWhen the example model bernoulli.stan is run via the command line with all default arguments, the following configuration is displayed:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n#     save_warmup = 0 (Default)\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000 (Default)\n#       delta = 0.800000 (Default)\n#       kappa = 0.750000 (Default)\n#       t0 = 10.000000 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#       save_metric = 0 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1.000000 (Default)\n#         stepsize_jitter = 0.000000 (Default)\n#     num_chains = 1 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)",
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     "text": "At every sampler iteration, the sampler returns a set of estimates for all parameters and quantities of interest in the model. During warmup, the NUTS algorithm adjusts the HMC algorithm parameters metric and stepsize in order to efficiently sample from typical set, the neighborhood substantial posterior probability mass through which the Markov chain will travel in equilibrium. After warmup, the fixed metric and stepsize are used to produce a set of draws.\nThe following keyword-value arguments control the total number of iterations:\n\nnum_samples\nnum_warmup\nsave_warmup\nthin\n\nThe values for arguments num_samples and num_warmup must be a non-negative integer. The default value for both is \\(1000\\).\nFor well-specified models and data, the sampler may converge faster and this many warmup iterations may be overkill. Conversely, complex models which have difficult posterior geometries may require more warmup iterations in order to arrive at good values for the step size and metric.\nThe number of sampling iterations to runs depends on the effective sample size (EFF) reported for each parameter and the desired precision of your estimates. An EFF of at least 100 is required to make a viable estimate. The precision of your estimate is \\(\\sqrt{N}\\); therefore every additional decimal place of accuracy increases this by a factor of 10.\nArgument save_warmup takes values \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., warmup draws are not saved to the output file. When the value is \\(1\\), the warmup draws are written to the CSV output file directly after the CSV header line.\nArgument thin controls the number of draws from the posterior written to the output file. Some users familiar with older approaches to MCMC sampling might be used to thinning to eliminate an expected autocorrelation in the samples. HMC is not nearly as susceptible to this autocorrelation problem and thus thinning is generally not required nor advised, as HMC can produce anticorrelated draws, which increase the effective sample size beyond the number of draws from the posterior. Thinning should only be used in circumstances where storage of the samples is limited and/or RAM for later processing the samples is limited.\nThe value of argument thin must be a positive integer. When thin is set to value \\(N\\), every \\(N^{th}\\) iteration is written to the output file. Should the value of thin exceed the specified number of iterations, the first iteration is saved to the output. This is because the iteration counter starts from zero and whenever the counter modulo the value of thin equals zero, the iteration is saved to the output file. Since zero modulo any positive integer is zero, the first iteration is always saved. When num_sampling=M and thin=N, the number of iterations written to the output CSV file will be ceiling(M/N). If save_warmup=1, thinning is applied to the warmup iterations as well.",
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     "text": "The adapt keyword is used to specify non-default options for the sampler adaptation schedule and settings.\nAdaptation can be turned off by setting sub-argument engaged to value \\(0\\). If engaged=0, no adaptation will be done, and all other adaptation sub-arguments will be ignored. Since the default argument is engaged=1, this keyword-value pair can be omitted from the command.\nThere are two sets of adaptation sub-arguments: step size optimization parameters and the warmup schedule. These are described in detail in the Reference Manual section Automatic Parameter Tuning.\nThe boolean sub-argument save_metric was added in Stan version 2.34. When save_metric=1 (true), the adapted stepsize and metric are output as JSON at the end of adaptation. The saved metric file name is the output file basename with the suffix _metric.json, e.g., if using the default output filename output.csv, the saved metric file will be output_metric.json. This metric file can be reused in subsequent sampler runs as the initial metric, via sampler argument metric_file.\n\n\nThe Stan User’s Guide section on model conditioning and curvature provides a discussion of adaptation and stepsize issues. The Stan Reference Manual section on HMC algorithm parameters explains the NUTS-HMC adaptation schedule and the tuning parameters for setting the step size.\nThe following keyword-value arguments control the settings used to optimize the step size:\n\ndelta - The target Metropolis acceptance rate. The default value is \\(0.8\\). Its value must be strictly between \\(0\\) and \\(1\\). Increasing the default value forces the algorithm to use smaller step sizes. This can improve sampling efficiency (effective sample size per iteration) at the cost of increased iteration times. Raising the value of delta will also allow some models that would otherwise get stuck to overcome their blockages.  Models with difficult posterior geometries may required increasing the delta argument closer to \\(1\\); we recommend first trying to raise it to \\(0.9\\) or at most \\(0.95\\). Values about \\(0.95\\) are strong indication of bad geometry; the better solution is to change the model geometry through reparameterization which could yield both more efficient and faster sampling.\ngamma - Adaptation regularization scale. Must be a positive real number, default value is \\(0.05\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nkappa - Adaptation relaxation exponent. Must be a positive real number, default value is \\(0.75\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\nt_0 - Adaptation iteration offset. Must be a positive real number, default value is \\(10\\). This is a parameter of the Nesterov dual-averaging algorithm. We recommend always using the default value.\n\n\n\n\nWhen adaptation is engaged, the warmup schedule is specified by sub-arguments, all of which take positive integers as values:\n\ninit_buffer - The number of iterations spent tuning the step size at the outset of adaptation.\nwindow - The initial number of iterations devoted to tune the metric, will be doubled successively.\nterm_buffer - The number of iterations used to re-tune the step size once the metric has been tuned.\n\nThe specified values may be modified slightly in order to ensure alignment between the warmup schedule and total number of warmup iterations.\nThe following figure is taken from the Stan Reference Manual, where label “I” correspond to init_buffer, the initial “II” corresponds to window, and the final “III” corresponds to term_buffer:\nWarmup Epochs Figure.  Adaptation during warmup occurs in three stages: an initial fast adaptation interval (I), a series of expanding slow adaptation intervals (II), and a final fast adaptation interval (III). For HMC, both the fast and slow intervals are used for adapting the step size, while the slow intervals are used for learning the (co)variance necessitated by the metric. Iteration numbering starts at 1 on the left side of the figure and increases to the right.",
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     "text": "The algorithm keyword-value pair specifies the algorithm used to generate the sample. There are two possible values: hmc, which generates from an HMC-driven Markov chain; and fixed_param which generates a new sample without changing the state of the Markov chain. The default argument is algorithm=hmc.\n\n\nIf a model doesn’t specify any parameters, then argument algorithm=fixed_param is mandatory.\nThe fixed parameter sampler generates a new sample without changing the current state of the Markov chain. This can be used to write models which generate pseudo-data via calls to RNG functions in the transformed data and generated quantities blocks.\n\n\n\nAll HMC algorithms have three parameters:\n\nstep size\nmetric\nintegration time - the number of steps taken along the Hamiltonian trajectory\n\nSee the Stan Reference Manual section on HMC algorithm parameters for further details.\n\n\nThe HMC algorithm simulates the evolution of a Hamiltonian system. The step size parameter controls the resolution of the sampler. Low step sizes can get HMC samplers unstuck that would otherwise get stuck with higher step sizes.\nThe following keyword-value arguments control the step size:\n\nstepsize - How far to move each time the Hamiltonian system evolves forward. Must be a positive real number, default value is \\(1\\).\nstepsize_jitter - Allows step size to be “jittered” randomly during sampling to avoid any poor interactions with a fixed step size and regions of high curvature. Must be a real value between \\(0\\) and \\(1\\). The default value is \\(0\\). Setting stepsize_jitter to \\(1\\) causes step sizes to be selected in the range of \\(0\\) to twice the adapted step size. Jittering below the adapted value will increase the number of steps required and will slow down sampling, while jittering above the adapted value can cause premature rejection due to simulation error in the Hamiltonian dynamics calculation. We strongly recommend always using the default value.\n\n\n\n\nAll HMC implementations in Stan utilize quadratic kinetic energy functions which are specified up to the choice of a symmetric, positive-definite matrix known as a mass matrix or, more formally, a metric Betancourt (2017).\nThe metric argument specifies the choice of Euclidean HMC implementations:\n\nmetric=unit specifies unit metric (diagonal matrix of ones).\nmetric=diag_e specifies a diagonal metric (diagonal matrix with positive diagonal entries). This is the default value.\nmetric=dense_e specifies a dense metric (a dense, symmetric positive definite matrix).\n\nBy default, the metric is estimated during warmup. However, when metric=diag_e or metric=dense_e, an initial guess for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a single variable inv_metric. For a diag_e metric the inv_metric value must be a vector of positive values, one for each parameter in the system. For a dense_e metric, inv_metric value must be a positive-definite square matrix with number of rows and columns equal to the number of parameters in the model.\nThe metric_file option can be used with and without adaptation enabled. If adaptation is enabled, the provided metric will be used as the initial guess in the adaptation process. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess.\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n\n\n\nThe total integration time is determined by the argument engine which take possible values:\n\nnuts - the No-U-Turn Sampler which dynamically determines the optimal integration time.\nstatic - an HMC sampler which uses a user-specified integration time.\n\nThe default argument is engine=nuts.\nThe NUTS sampler generates a proposal by starting at an initial position determined by the parameters drawn in the last iteration. It then evolves the initial system both forwards and backwards in time to form a balanced binary tree. The algorithm is iterative; at each iteration the tree depth is increased by one, doubling the number of leapfrog steps thus effectively doubling the computation time. The algorithm terminates in one of two ways: either the NUTS criterion (i.e., a U-turn in Euclidean space on a subtree) is satisfied for a new subtree or the completed tree; or the depth of the completed tree hits the maximum depth allowed.\nWhen engine=nuts, the subargument max_depth can be used to control the depth of the tree. The default argument is max_depth=10. In the case where a model has a difficult posterior from which to sample, max_depth should be increased to ensure that that the NUTS tree can grow as large as necessary.\nWhen the argument engine=static is specified, the user must specify the integration time via keyword int_time which takes as a value a positive number. The default value is \\(2\\pi\\).",
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     "text": "The output keyword sub-argument diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file which contains sampler information for each draw, and the gradients on the unconstrained scale and log probabilities for all parameters in the model. By default, no auxiliary output file is produced.",
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-    "text": "As described in the quickstart section on parallelism, the preferred way to run multiple chains is to use the num_chains argument.\nThis will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. As noted in the quickstart guide, this will be done in parallel if the model was compiled with STAN_THREADS=true.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.",
+    "text": "A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has approximately converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThe preferred way of using multiple chains is to run them all from the same executable using the num_chains argument. There is also the option to use the Unix or DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine. This will run multiple chains of MCMC from the same executable, which can save on memory usage due to only needing one copy of the model and data. Depending on whether the model was compiled with STAN_THREADS=true, these will either run in parallel or one after the other.\nThe num_chains argument changes the meanings of several other arguments when it is greater than 1 (the default). Many arguments are now interpreted as a “template” which is used for each chain.\nFor example, when num_chains=2, the argument output file=foo.csv no longer produces a file foo.csv, but instead produces two files, foo_1.csv and foo_2.csv. If you also supply id=5, the files produced will be foo_5.csv and foo_6.csv – id=5 gives the id of the first chain, and the remaining chains are sequential from there.\nThis also applies to input files, like those used for initialization. For example, if num_chains=3 and init=bar.json will first look for bar_1.json. If it exists, it will use bar_1.json for the first chain, bar_2.json for the second, and so on. If bar_1.json does not exist, it falls back to looking for bar.json, and if it exists, uses the same initial values for each chain. The numbers in these filenames are also based on the id argument, which defaults to 1.\nFor example, this will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nThis will produce samples in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially.\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16",
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-    "text": "Note: Many of these examples can be simplified by using the num_chains argument.\nThe Quickstart Guide MCMC Sampling chapter section on multiple chains also showed how to run multiple chains given a model and data, using the minimal required command line options: the method, the name of the data file, and a chain-specific name for the output file.\nThis creates multiple copies of the model process which will all load the data.\nTo run 4 chains in parallel on Mac OS and Linux, the syntax in both bash and zsh is the same:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=my_model.data.json \\\n                  output file=output_${i}.csv &\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. The ampersand (&) pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows the corresponding loop is:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=my_model.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
+    "text": "The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on data consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)",
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+    "text": "Note: Many of these examples can be simplified by using the num_chains argument.\nWhen the num_chains argument is not available or is undesirable for whatever reason, built-in tools in the system shell can be used.\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS. The expression %i is the loop index.\nIn the following extended examples, we focus on just the nested sampler command for Unix.\n\n\nFor reproducibility, we specify the same RNG seed across all chains and use the chain id argument to specify the RNG offset.\nThe RNG seed is specified by random seed=&lt;int&gt; and the offset is specified by id=&lt;loop index&gt;, so the call to the sampler is:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            random seed=12345 id=${i}\n\n\n\nThe warmup and sampling iteration keyword-value arguments must follow the sample keyword. The call to the sampler which overrides the default warmup and sampling iterations is:\n./my_model sample num_warmup=500 num_sampling=500 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nTo save warmup draws as part of the Stan CSV output file, use the keyword-value argument save_warmup=1. This must be grouped with the other sample keyword sub-arguments.\n./my_model sample num_warmup=500 num_sampling=500 save_warmup=1 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nBy default, all parameters are initialized on an unconstrained scale to random draws from a uniform distribution over the range \\([{-2}, 2]\\). To initialize some or all parameters to good starting points on the constrained scale from a data file in JSON or Rdump format, use the keyword-value argument init=&lt;filepath&gt;:\n./my_model sample init=my_param_inits.json data file=my_model.data.json \\\n           output file=output_${i}.csv\nTo verify that the specified values will be used by the sampler, you can run the sampler with option algorithm=fixed_param, so that the initial values are used to generate the sample. Since this generates a set of identical draws, setting num_warmp=0 and num_samples=1 saves unnecessary iterations. As the output values are also on the constrained scale, the set of reported values will match the set of specified initial values.\nFor example, if we run the example Bernoulli model with specified initial value for parameter “theta”:\n{ \"theta\" : 0.5 }\nvia command:\n./bernoulli sample algorithm=fixed_param num_warmup=0 num_samples=1 \\\n            init=bernoulli.init.json data file=bernoulli.data.json\nThe resulting output CSV file contains a single draw:\nlp__,accept_stat__,theta\n0,0,0.5\n#\n#  Elapsed Time: 0 seconds (Warm-up)\n#                0 seconds (Sampling)\n#                0 seconds (Total)\n#\n\n\n\nAn initial estimate for the metric can be specified with the metric_file argument whose value is the filepath to a JSON or Rdump file which contains a variable inv_metric. The metric_file option can be used with and without adaptation enabled.\nBy default, the metric is estimated during warmup adaptation. If the initial guess is good, then adaptation should not change it much. If the metric is no good, then the adaptation will override the initial guess. For example, the JSON file bernoulli.diag_e.json, contents\n{ \"inv_metric\" : [0.296291] }\ncan be used as the initial metric as follows:\n../my_model sample algorithm=hmc metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nIf adaptation is disabled, both the metric_file and stepsize arguments should be specified.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nThe resulting output CSV file will contain the following set of comment lines:\n# Adaptation terminated\n# Step size = 0.9\n# Diagonal elements of inverse mass matrix:\n# 0.296291\nAs of Stan versione 2.34, the adapted metric can be saved in JSON format, via sub-argument save_metric, described above. This allows for no or minimal adaptation starting from this file. It is still necessary to specify the stepsize argument as well as the metric_file arguments; the former is the value of the stepsize element in the saved metric file, and the later is the metric file path.\n\n\n\nThe keyword-value arguments for these settings are grouped together under the adapt keyword which itself is a sub-argument of the sample keyword.\nModels with difficult posterior geometries may required increasing the delta argument closer to \\(1\\).\n./my_model sample adapt delta=0.95 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nTo skip adaptation altogether, use the keyword-value argument engaged=0. Disabling adaptation disables both metric and stepsize adaptation, so a stepsize should be provided along with a metric to enable efficient sampling.\n../my_model sample adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\nEven with adaptation disabled, it is still advisable to run warmup iterations in order to allow the initial parameter values to be adjusted to estimates which fall within the typical set.\nTo skip warmup altogether requires specifying both num_warmup=0 and adapt engaged=0.\n../my_model sample num_warmup=0 adapt engaged=0 \\\n            algorithm=hmc stepsize=0.9 \\\n            metric_file=bernoulli.diag_e.json \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nModels with difficult posterior geometries may required increasing the max_depth argument from its default value \\(10\\). This requires specifying a series of keyword-argument pairs:\n./my_model sample adapt delta=0.95 \\\n            algorithm=hmc engine=nuts max_depth=15 \\\n            data file=my_model.data.json \\\n            output file=output_${i}.csv\n\n\n\nThe output keyword sub-argument diagnostic_file=&lt;filepath&gt; write the sampler parameters and gradients of all model parameters for each draw to a CSV file:\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            diagnostic_file=diagnostics_${i}.csv\n\n\n\nThe output keyword sub-argument refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is \\(100\\) iterations. The progress updates look like:\nIteration:    1 / 2000 [  0%]  (Warmup)\nIteration:  100 / 2000 [  5%]  (Warmup)\nIteration:  200 / 2000 [ 10%]  (Warmup)\nIteration:  300 / 2000 [ 15%]  (Warmup)\nFor simple models which fit quickly, such updates can be annoying; to suppress them altogether, set refresh=0. This only turns off the Iteration: messages; the configuration and timing information are still written to the terminal.\n./my_model sample data file=my_model.data.json \\\n            output file=output_${i}.csv \\\n            refresh=0\nFor complicated models which take a long time to fit, setting the refresh rate to a low number, e.g. \\(10\\) or even \\(1\\), provides a way to more closely monitor the sampler.\n\n\n\nThe CmdStan argument parser requires keeping sampler config sub-arguments together; interleaving sampler config with the inputs, outputs, inits, RNG seed and chain id config results in an error message such as the following:\n./bernoulli sample data file=bernoulli.data.json adapt delta=0.95\nadapt is either mistyped or misplaced.\nPerhaps you meant one of the following valid configurations?\n  method=sample sample adapt\n  method=variational variational adapt\nFailed to parse arguments, terminating Stan\nThe following example provides a template for a call to the sampler which specifies input data, initial parameters, initial step-size and metric, adaptation, output, and RNG initialization.\n./my_model sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100 \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           random seed=12345 id=${i}\nThe keywords sample, data, output, and random are the top-level argument groups. Within the sample config arguments, the keyword adapt groups the adaptation algorithm parameters and the keyword-value algorithm=hmc groups the NUTS-HMC parameters.\nThe top-level groups can be freely ordered with respect to one another. The following is also a valid command:\n./my_model random seed=12345 id=${i} \\\n           data file=my_model.data.json \\\n           output file=output_${i}.csv refresh=10 \\\n           sample num_warmup=2000 \\\n           init=my_param_inits.json \\\n           algorithm=hmc engine=nuts max_depth=15 \\\n           metric=dense_e metric_file=my_metric.json \\\n           stepsize=0.6555 \\\n           adapt delta=0.95 init_buffer=100 \\\n           window=50 term_buffer=100",
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@@ -4014,7 +3971,7 @@
     "section": "",
     "text": "The laplace method produces a sample from a normal approximation centered at the mode of a distribution in the unconstrained space. If the mode is a maximum a posteriori (MAP) estimate, the samples provide an estimate of the mean and standard deviation of the posterior distribution. If the mode is a maximum likelihood estimate (MLE), the sample provides an estimate of the standard error of the likelihood. In general, the posterior mode in the unconstrained space doesn’t correspond to the mean (nor mode) in the constrained space, and thus the sample is needed to infer the mean as well as the standard deviation. (See this case study for a visual illustration.)\nThis is computationally inexpensive compared to exact Bayesian inference with MCMC. The goodness of this estimate depends on both the estimate of the mode and how much the true posterior in the unconstrained space resembles a Gaussian.\n\n\nThis method takes 2 arguments:\n\njacobian - Whether or not the Jacobian adjustment should be included in the gradient. The default value is 1 (include adjustment). (Note: in optimization, the default value is 0, for historical reasons.)\nmode - Input file of parameters values on the constrained scale. When Stan’s optimize method is used to estimate the modal values, the value of boolean argument jacobian should be 0 if optimize was run with default settings, i.e., the input is the MLE estimate; if optimize was run with argument jacobian=1, then the laplace method default setting, jacobian=1, should be used.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the laplace method is reported at the beginning of the output file as CSV comments.\nOutput columns log_p__ and log_q__, the unnormalized log density and the unnormalized density of the Laplace approximation, respectively. These can be used for diagnostics and importance sampling.\nOutput columns for all model parameters on the constrained scale.\n\n\n\n\nTo get an approximate estimate of the mode and standard deviation of the example Bernoulli model given the example dataset:\n\nfind the MAP estimate by running optimization with argument jacobian=1\nrun the Laplace estimator using the MAP estimate as the mode argument.\n\nBecause the default output file name from all methods is output.csv, a more informative name is used for the output of optimization. We run the commands from the CmdStan home directory. This results in a sample with mean 2.7 and standard deviation 0.12. In comparison, running the NUTS-HMC sampler results in mean 2.6 and standard deviation 0.12.\n./examples/bernoulli/bernoulli optimize jacobian=1 \\\n  data file=examples/bernoulli/bernoulli.data.json \\\n  output file=bernoulli_optimize_lbfgs.csv random seed=1234\n\n\n./examples/bernoulli/bernoulli laplace mode=bernoulli_optimize_lbfgs.csv \\\n data file=examples/bernoulli/bernoulli.data.json random seed=1234\nThe header and first few data rows of the output sample are shown below.\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-12-20 01:01:14 UTC\n# method = laplace\n#   laplace\n#     mode = bernoulli_lbfgs.csv\n#     jacobian = 1 (Default)\n#     draws = 1000 (Default)\n# id = 1 (Default)\n# data\n#   file = examples/bernoulli/bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 875960551 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 v2.31.0-7-g20444266\n# stancflags =\nlog_p__,log_q__,theta\n-9.4562,-2.33997,0.0498545\n-6.9144,-0.0117349,0.182898\n-7.18171,-0.746034,0.376428\n...",
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     "text": "This method takes 2 arguments:\n\njacobian - Whether or not the Jacobian adjustment should be included in the gradient. The default value is 1 (include adjustment). (Note: in optimization, the default value is 0, for historical reasons.)\nmode - Input file of parameters values on the constrained scale. When Stan’s optimize method is used to estimate the modal values, the value of boolean argument jacobian should be 0 if optimize was run with default settings, i.e., the input is the MLE estimate; if optimize was run with argument jacobian=1, then the laplace method default setting, jacobian=1, should be used.",
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     "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the laplace method is reported at the beginning of the output file as CSV comments.\nOutput columns log_p__ and log_q__, the unnormalized log density and the unnormalized density of the Laplace approximation, respectively. These can be used for diagnostics and importance sampling.\nOutput columns for all model parameters on the constrained scale.",
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     "text": "To get an approximate estimate of the mode and standard deviation of the example Bernoulli model given the example dataset:\n\nfind the MAP estimate by running optimization with argument jacobian=1\nrun the Laplace estimator using the MAP estimate as the mode argument.\n\nBecause the default output file name from all methods is output.csv, a more informative name is used for the output of optimization. We run the commands from the CmdStan home directory. This results in a sample with mean 2.7 and standard deviation 0.12. In comparison, running the NUTS-HMC sampler results in mean 2.6 and standard deviation 0.12.\n./examples/bernoulli/bernoulli optimize jacobian=1 \\\n  data file=examples/bernoulli/bernoulli.data.json \\\n  output file=bernoulli_optimize_lbfgs.csv random seed=1234\n\n\n./examples/bernoulli/bernoulli laplace mode=bernoulli_optimize_lbfgs.csv \\\n data file=examples/bernoulli/bernoulli.data.json random seed=1234\nThe header and first few data rows of the output sample are shown below.\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-12-20 01:01:14 UTC\n# method = laplace\n#   laplace\n#     mode = bernoulli_lbfgs.csv\n#     jacobian = 1 (Default)\n#     draws = 1000 (Default)\n# id = 1 (Default)\n# data\n#   file = examples/bernoulli/bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 875960551 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 v2.31.0-7-g20444266\n# stancflags =\nlog_p__,log_q__,theta\n-9.4562,-2.33997,0.0498545\n-6.9144,-0.0117349,0.182898\n-7.18171,-0.746034,0.376428\n...",
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-    "text": "There are a few ways that you can install CmdStan. Depending on your operating system and your level of expertise, you can either:\n\nUse the conda package management system to install a pre-built version of CmdStan along with the required dependencies. Recommended for Windows users.\nInstall the source code from GitHub CmdStan repository. This requires a modern C++ compiler and toolchain. See the C++ Toolchain section for further details.\n\n\n\nWith conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.\n\n\n\n\nInstallation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)\n\n\n\n\nTo check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\ \n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.\n\n\n\nUpdates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.\n\n\n\n\nCompiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.\n\n\n\n\n\n\nCmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g., \n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
+    "text": "There are a few ways that you can install CmdStan. Depending on your operating system and your level of expertise, you can either:\n\nUse the conda package management system to install a pre-built version of CmdStan along with the required dependencies. Recommended for Windows users.\nInstall the source code from GitHub CmdStan repository. This requires a modern C++ compiler and toolchain. See the C++ Toolchain section for further details.\n\n\n\nWith conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.\n\n\n\n\nInstallation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)\n\n\n\n\nTo check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\\n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.\n\n\n\nUpdates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.\n\n\n\n\nCompiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.\n\n\n\n\n\n\nCmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g.,\n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
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-    "text": "With conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries) detailed below under [Source installation]. The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.",
+    "text": "With conda, you can install CmdStan from the conda-forge channel. This will install a pre-built version of CmdStan along with the required dependencies (i.e. a C++ compiler, a version of Make, and required libraries). The conda installation is designed so one can use the R or Python bindings to CmdStan seamlessly. Additionally, it provides the command cmdstan_model to activate the CmdStan makefile from anywhere.\nNote: This requires that conda has been installed already on your machine. You can either install miniconda, a free, minimal installer for conda or you can get the full Anaconda system which provides graphical installer wizards for MacOS and Windows users.\nWe recommend installing CmdStan in a new conda environment:\n conda create -n stan -c conda-forge cmdstan\nThis command creates a new conda environment named stan and downloads and installs the cmdstan package as well as CmdStan and the required C++ toolchain.\nTo install into an existing conda environment, use the conda install command instead of create:\n conda install -c conda-forge cmdstan\nWhichever installation method you use, afterwards you must activate the new environment or deactivate/activate the existing one. For example, if you installed cmdstan into a new environment stan, run the command\n conda activate stan\nBy default, the latest release of CmdStan is installed. If you require a specific release of CmdStan, CmdStan versions 2.26.1 and newer can be installed by specifying cmdstan==VERSION in the install command. For example to install an earlier version of CmdStan into your current conda environment, run the following command, then re-activate the environment\nconda install -c conda-forge cmdstan=2.27.0\n\n\nA Conda environment is a directory that contains a specific collection of Conda packages. To see the locations of your conda environments, use the command\n conda info -e\nThe shell environment variable CONDA_PREFIX points to the active conda environment (if any). Both CmdStan and the C++ toolchain are installed into the bin subdirectory of the conda environment directory, i.e., $CONDA_PREFIX/bin/cmdstan (Linux, MacOS), %CONDA_PREFIX%\\bin\\cmdstan (Windows).\nPlease report conda-specific install problems directly to the conda-forge issue tracker, here.",
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     "text": "Installation from GitHub consists of the following steps:\n\nVerify that you have a modern C++ toolchain. See the C++ Toolchain section for details.\nDownload the CmdStan source code from GitHub\nBuild the CmdStan libraries and executables\nCheck the installation by compiling and running the CmdStan example model bernoulli.stan.\n\n\n\nThe GitHub source code is divided into sub-modules, each in its own repository. The CmdStan repo contains just the cmdstan module; the Stan inference engine algorithms and Stan math library functions are specified as submodules and stored in the GitHub repositories stan and math, respectively.\nA CmdStan release is compressed tarfile which contains CmdStan and the Stan and math library submodules. The most recent CmdStan release is always available as https://github.com/stan-dev/cmdstan/releases/latest. A CmdStan release is versioned by major, minor, patch numbers, e.g., “2.29.2”. Please ensure you download a tarfile which is named “cmdstan-&lt;version-number” rather than using the “Source Code” links at the bottom of the release. These are automatically generated by GitHub and do not contain the required submodules. The release tarfile unpacks into a directory named “cmdstan-”, e.g. “cmdstan-2.29.2”.\nBy cloning the CmdStan repository with argument --recursive, Git automatically initializes and updates each submodule in the repository, including nested submodules if any of the submodules in the repository have submodules themselves. The following command will download the source code from the current development branch of CmdStan into a directory named cmdstan:\n&gt; git clone https://github.com/stan-dev/cmdstan.git --recursive\nThroughout this manual, we refer to this top-level CmdStan source directory as &lt;cmdstan-home&gt;. This directory contains the following subdirectories:\n\ndirectory cmdstan/stan contains the sub-module stan (https://github.com/stan-dev/stan)\ndirectory cmdstan/stan/lib/stan_math contains the sub-module math (https://github.com/stan-dev/math)\n\n\n\n\nBuilding CmdStan involves preparing a set of executable programs and compiling the command line interface and supporting libraries. The CmdStan tools are:\n\nstanc: the Stan compiler (translates Stan language to C++).\nstansummary: a basic posterior analysis tool. The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler. For all parameters and quantities of interest in the Stan program, stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\ndiagnose: a basic sampler diagnostic tool which checks for indications that the HMC sampler was unable to sample from the full posterior.\n\nCmdStan releases include pre-built binaries of the Stan language compiler (https://github.com/stan-dev/stanc3): bin/linux-stanc, bin/mac-stanc and bin/windows-stanc. The CmdStan makefile build task copies the appropriate binary to bin/stanc. For CmdStan installations which have been cloned of downloaded from the CmdStan GitHub repository, the makefile task will download the appropriate OS-specific binary from the stanc3 repository’s nightly release.\nSteps to build CmdStan:\n\nOpen a command-line terminal window and change directories to the CmdStan home directory.\nRun the makefile target build which instantiates the CmdStan utilities and compiles all necessary C++ libraries.\n\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make build    # on Windows use mingw32-make\nIf your computer has multiple cores and sufficient ram, the build process can be parallelized by providing the -j option. For example, to build on 4 cores, type:\n&gt; make -j4 build    # on Windows use mingw32-make\nWhen make build is successful, the directory &lt;cmdstan-home&gt;/bin/ will contain the executables stanc, stansummary, and diagnose (on Windows, corresponding .exe files) and the final lines of console output will show the version of CmdStan that has just been built, e.g.:\n--- CmdStan v2.29.2 built ---\nWarning: The Make program may take 10+ minutes and consume 2+ GB of memory to build CmdStan.\nWindows only: CmdStan requires that the Intel TBB library, which is built by the above command, can be found by the Windows system. This requires that the directory &lt;cmdstan-home&gt;/stan/lib/stan_math/lib/tbb is part of the PATH environment variable. See these instructions for details on changing the PATH. To permanently make this setting for the current user, you may execute:\n&gt; mingw32-make install-tbb\nAfter changing the PATH environment variable, you must open an new shell in order for the new environment variable settings to take effect. (This is not necessary on Mac and Linux systems because they can use the absolute path to the Intel TBB library when linking into Stan programs.)",
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-    "text": "To check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\ \n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.",
+    "text": "To check that the CmdStan installation is complete and in working order, run the following series of commands from the folder which CmdStan was installed.\nOn Linux and macOS:\n# compile the example\n&gt; make examples/bernoulli/bernoulli\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli sample\\\n  data file=examples/bernoulli/bernoulli.data.json\n\n# default output written to file `output.csv`,\n# default num_samples is 1000, output file should have approx. 1050 lines\n&gt; ls -l output.csv\n\n# run the `bin/stansummary utility to summarize parameter estimates\n&gt; bin/stansummary output.csv\nOn Windows:\n# compile the example\n&gt; mingw32-make examples/bernoulli/bernoulli.exe\n\n# fit to provided data (results of 10 trials, 2 out of 10 successes)\n&gt; ./examples/bernoulli/bernoulli.exe sample data file=examples/bernoulli/bernoulli.data.json\n\n# run the `bin/stansummary.exe utility to summarize parameter estimates\n&gt; bin/stansummary.exe output.csv\nThe sample data in file bernoulli.json.data specifies 2 out of 10 successes, therefore the range mean(theta)\\(\\pm\\)sd(theta) should include 0.2.",
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     "section": "",
     "text": "Updates to CmdStan, changes in compiler options, or updates to the C++ toolchain may result in errors when trying to compile a Stan program. Often, these problems can be resolved by removing the existing CmdStan binaries and recompiling. To do this, you must run the makefile commands from the &lt;cmdstan-home&gt; directory:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make clean-all    # on Windows use mingw32-make\n&gt; make build\n\n\nThis section contains solutions to problems reported on https://discourse.mc-stan.org\nCompiler error message about PCH file\nTo speed up compilation, the Stan makefile pre-compiles parts of the core Stan library. If these pre-compiled files are out of sync with the compiled model, the compiler will complain, e.g.:\nerror: PCH file uses an older PCH format that is no longer supported\nIn this case, clean and rebuild CmdStan, as shown in the previous section.\nWindows: ‘mingw32-make’ is not recognised\nIf the C++ toolchain has been installed but not properly registered, then the call to mingw32-make will result in error message:\n'mingw32-make' is not recognised as an internal or external command\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nWindows: ‘g++’ or ‘cut’ is not recognized\nThe CmdStan makefile uses a few shell utilities which might not be present in Windows, resulting in the error message:\n'cut' is not recognized as an internal or external command,\noperable program or batch file.\nTo fix this, ensure you have followed the steps for adding the toolchain to your PATH and installing the additional utilities covered in the configuration instructions\nSpaces in paths to CmdStan or model\nBoth make and mingw32-make can fail when dealing with files in folders with a space somewhere in their file path. Particularly on Windows, this can be an issue when CmdStan, or the models you are trying to build, are placed in the One Drive folder.\nUnfortunately, the errors created by this situation are not alwas informative. Some errors you may see are:\nmingw32-make: *** INTERNAL: readdir: Invalid argument\nmake: *** [make/program:50: x.hpp] Error 2\nIf the (fully-expanded) folder path to CmdStan or the model you are trying to build contains a space, we recommend trying a different location if you encounter any issues during building.",
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     "section": "",
     "text": "Compiling a Stan program requires a modern C++ compiler and the GNU Make build utility (a.k.a. “gmake”). These vary by operating system.\n\n\nThe required C++ compiler is g++ 4.9 3. On most systems the GNU Make utility is pre-installed and is the default make utility. There is usually a pre-installed C++ compiler as well, however, it may not be new enough. To check, run commands:\ng++ --version\nmake --version\nIf these are at least at g++ version 4.9.3 or later and make version 3.81 or later, no additional installations are necessary. It may still be desirable to update the C++ compiler g++, because later versions are faster.\nTo install the latest version of these tools (or upgrade an older version), use the following commands or their equivalent for your distribution, install via the commands:\nsudo apt install g++\nsudo apt install make\nIf you can’t run sudo, you will need to ask your sysadmin or cluster administrator to install these tools for you.\n\n\n\nTo install a C++ development environment on a Mac, use Apple’s Xcode development environment https://developer.apple.com/xcode/.\nFrom the Xcode home page View in Mac App Store.\n\nFrom the App Store, click Install, enter an Apple ID, and wait for Xcode to finish installing.\nOpen the Xcode application, click top-level menu Preferences, click top-row button Downloads, click button for Components, click on the Install button to the right of the Command Line Tools entry, then wait for it to finish installing.\nClick the top-level menu item Xcode, then click item Quit Xcode to quit.\n\nTo test, open the Terminal application and enter:\nclang++ --version\nmake --version\nIf you have installed XCode, but don’t have make, you can install the XCode command-line tools via command:\nxcode-select --install\n\n\n\nThe Windows toolchain consists of programs g++, the C++ compiler, and mingw32-make, the GNU Make utility. To check if these are present, open a command shell 1 and type:\ng++ --version\nmingw32-make --version\nCmdStan is known compatible with the RTools40, RTools42, and RTools43 toolchains. These require slightly different steps to configure, so please follow the appropriate steps below. All toolchains will require updating your PATH variable, See these instructions for details on changing the PATH if you are unfamiliar. The following instructions will assume that the default installation directory was used, so be sure to update the paths accordingly if you have chosen a different directory.\n\n\nRTools40 provides both a standard g++-8 toolchain and a g++-10 Universal C Runtime (UCRT) toolchain. Note the that newer g++-10 UCRT toolchain is only available for 64-bit systems, whereas the g++-8 toolchain is available for both. Additionally the UCRT is only natively supported on Windows 10 and newer, older systems will require a Microsoft update\n\n\nDownload the installer and complete the prompts to install.\nNext, you need to add the location of the toolchain to your PATH environment variable so that it can be called from the command line. Add the following lines to your PATH:\nC:\\rtools40\\usr\\bin\n\n# Add only one of the below\nC:\\rtools40\\mingw32\\bin   # 32-bit g++-8\nC:\\rtools40\\mingw64\\bin   # 64-bit g++-8\nC:\\rtools40\\ucrt64\\bin    # 64-bit g++-10 (UCRT)\nCmdStan additionally needs the mingw32-make utility, which you can install using RTools. Navigate to the installation directory (e.g., C:\\rtools40) and launch the msys2.exe file. Execute the appropriate command below to install mingw32-make for your selected toolchain:\npacman -Sy mingw-w64-i686-make            # 32-bit g++-8\npacman -Sy mingw-w64-x86_64-make          # 64-bit g++-8\npacman -Sy mingw-w64-ucrt-x86_64-make     # 64-bit g++-10 (UCRT)\n\n\n\n\nBoth RTools42 & RTools43 provide 64-bit UCRT toolchains, where RTools42 provides g++-10 and RTools43 provides g++-12. The installation/configuration is identical for both toolchains.\n\n\nDownload the installer for your preferred toolchain and complete the prompts for installation:\n\nRTools42\nRTools43\n\nNext, you need to add the toolchain directory to your PATH variable. Add the appropriate lines from below:\n# RTools42\nC:\\rtools42\\usr\\bin\nC:\\rtools42\\ucrt64\\bin\n\n# RTools43\nC:\\rtools43\\usr\\bin\nC:\\rtools43\\ucrt64\\bin\nNext, you need to install the mingw32-make utility and some additional compiler dependencies. Navigate to the installation directory of the toolchain and launch the msys2.exe file. Execute the below commands to install the needed dependencies:\npacman -Sy mingw-w64-ucrt-x86_64-make mingw-w64-ucrt-x86_64-gcc\n32-bit Builds\nCmdStan defaults to a 64-bit build. On a 32-bit operating system, you must specify the make variable BIT=32 as part of the make command, described in the next section.",
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-    "text": "CmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g., \n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
+    "text": "CmdStan relies on the GNU Make utility to build both the Stan model executables and the CmdStan tools.\nGNU Make builds executable programs and libraries from source code by reading files called Makefiles which specify how to derive the target program. A Makefile consists of a set of recursive rules where each rule specifies a target, its dependencies, and the specific operations required to build the target. Specifying dependencies for a target provides a way to control the build process so that targets which depend on other files will be updated as needed only when there are changes to those other files. Thus Make provides an efficient way to manage complex software.\nThe CmdStan Makefile is in the &lt;cmdstan-home&gt; directory and is named makefile. This is one of the default GNU Makefile names, which allows you to omit the -f makefile argument to the Make command. Because the CmdStan Makefile includes several other Makefiles, Make only works properly when invoked from the &lt;cmdstan-home&gt; directory; attempts to use this Makefile from another directory by specifying the full path to the file makefile won’t work. For example, trying to call Make from another directory by specifying the full path the the makefile results in the following set of error messages:\nmake -f ~/github/stan-dev/cmdstan/makefile\n/Users/mitzi/github/stan-dev/cmdstan/makefile:58: make/stanc: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:59: make/program: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:60: make/tests: No such file or directory\n/Users/mitzi/github/stan-dev/cmdstan/makefile:61: make/command: No such file or directory\nmake: *** No rule to make target `make/command'.  Stop.\nThe conda-forge cmdstan package provides a solution to this problem via cmdstan_model command which lets you run the CmdStan makefile from anywhere to compile a Stan model.\nMakefile syntax allows general pattern rules based on file suffixes. Stan programs must be stored in files with suffix .stan; the CmdStan makefile rules specify how to transform the Stan source code into a binary executable. For example, to compile the Stan program my_program.stan in directory ../my_dir/, the make target is ../my_dir/my_program or ../my_dir/my_program.exe (on Windows).\nTo call Make, you invoke the utility name, either make or mingw32-make, followed by, in order:\n\nzero or more Make program options, then specify any Make variables as a series of\nzero of more Make variables, described below\nzero or more target names; the set of names is determined by the Makefile rules.\n\nmake &lt;flags&gt; &lt;variables&gt; &lt;targets&gt;\nMakefile Variables\nMake targets can be preceded by any number of Makefile variable name=value pairs. For example, to compile ../my_dir/my_program.stan for an OpenCL (GPU) machine, set the makefile variable STAN_OPENCL to TRUE:\n&gt; make STAN_OPENCL=TRUE ../my_dir/my_program    # on Windows use mingw32-make\nMakefile variables can also be set by creating a file named local in the CmdStan make subdirectory which contains a list of &lt;VARIABLE&gt;=&lt;VALUE&gt; pairs, one per line. For example, if you are working on a 32-bit machine, you would put the line BIT=32 into the file &lt;cmdstan-home&gt;/make/local so that all CmdStan programs and Stan models compile properly.\nThe complete set of Makefile variables can be found in file &lt;cmdstan-home&gt;/cmdstan/stan/lib/stan_math/make/compiler_flags.\nMake Targets\nWhen invoked without any arguments at all, Make prints a help message:\n&gt; make    # on Windows use mingw32-make\n--------------------------------------------------------------------------------\nCmdStan v2.33.1 help\n\n  Build CmdStan utilities:\n    &gt; make build\n\n    This target will:\n    1. Install the Stan compiler bin/stanc from stanc3 binaries.\n    2. Build the print utility bin/print (deprecated; will be removed in v3.0)\n    3. Build the stansummary utility bin/stansummary\n    4. Build the diagnose utility bin/diagnose\n    5. Build all libraries and object files compile and link an executable Stan program\n\n    Note: to build using multiple cores, use the -j option to make, e.g.,\n    for 4 cores:\n    &gt; make build -j4\n\n\n  Build a Stan program:\n\n    Given a Stan program at foo/bar.stan, build an executable by typing:\n    &gt; make foo/bar\n\n    This target will:\n    1. Install the Stan compiler (bin/stanc), as needed.\n    2. Use the Stan compiler to generate C++ code, foo/bar.hpp.\n    3. Compile the C++ code using cc . to generate foo/bar\n\n  Additional make options:\n    STANCFLAGS: defaults to \"\". These are extra options passed to bin/stanc\n      when generating C++ code. If you want to allow undefined functions in the\n      Stan program, either add this to make/local or the command line:\n          STANCFLAGS = --allow_undefined\n    USER_HEADER: when STANCFLAGS has --allow_undefined, this is the name of the\n      header file that is included. This defaults to \"user_header.hpp\" in the\n      directory of the Stan program.\n    STANC3_VERSION: When set, uses that tagged version specified; otherwise, downloads\n      the nightly version.\n    STAN_CPP_OPTIMS: Turns on additonal compiler flags for performance.\n    STAN_NO_RANGE_CHECKS: Removes the range checks from the model for performance.\n\n\n  Example - bernoulli model: examples/bernoulli/bernoulli.stan\n\n    1. Build the model:\n       &gt; make examples/bernoulli/bernoulli\n    2. Run the model:\n       &gt; examples/bernoulli/bernoulli sample data file=examples/bernoulli/bernoulli.data.R\n    3. Look at the samples:\n       &gt; bin/stansummary output.csv\n\n\n  Clean CmdStan:\n\n    Remove the built CmdStan tools:\n    &gt; make clean-all\n\n--------------------------------------------------------------------------------",
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     "text": "Footnotes\n\n\nTo open a Windows command shell, first open the Start Menu, (usually in the lower left of the screen), select option All Programs, then option Accessories, then program Command Prompt. Alternatively, enter [Windows+r] (both keys together on the keyboard), and enter cmd into the text field that pops up in the Run window, then press [Return] on the keyboard to run.↩︎",
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-    "text": "Generating Quantities of Interest from a Fitted Model\nThe generated quantities block computes quantities of interest (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:\n\ngenerate simulated data for model testing by forward sampling\ngenerate predictions for new data\ncalculate posterior event probabilities, including multiple comparisons, sign tests, etc.\ncalculating posterior expectations\ntransform parameters for reporting\napply full Bayesian decision theory\ncalculate log likelihoods, deviances, etc. for model comparison\n\nThe generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the generate_quantities method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block. See the Stan User’s Guide section Stand-alone generated quantities and ongoing prediction.\nTo illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; theta_rep;\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter estimates from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv consists of just the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2135140492 (Default)\n# output\n#   file = bernoulli_ppc.csv\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\ntheta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10\n0.2,0,0,1,0,0,0,0,0,1,0\n0.3,1,0,0,1,0,1,0,0,0,0\n0.8,1,0,1,1,1,1,1,1,1,0\n0.1,0,0,0,0,0,1,0,0,0,0\n0.3,0,0,0,0,0,0,1,1,1,0\nNote: the only relevant analysis of the resulting CSV output is computing per-column statistics; this can easily be done in Python, R, Excel or similar, or you can use the CmdStanPy and CmdStanR interfaces which provide a better user experience for this workflow.\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  real&lt;lower=0, upper=1&gt; theta_rep;\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ntheta_cp,theta_rep,y_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10\n0.102391,0,0,0,0,0,0,0,0,0,0,0\n0.519567,0.2,0,1,0,0,1,0,0,0,0,0\n0.544634,0.6,1,0,0,0,0,1,1,1,1,1\n0.167651,0,0,0,0,0,0,0,0,0,0,0\n0.167651,0.1,1,0,0,0,0,0,0,0,0,0\n\n\n\n\n Back to top",
+    "text": "The generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. Instead, you write a modified version of the original Stan program and add a generated quantities block or modify the existing one which specifies how to compute the new quantities of interest. Running the generate_quantities method on the new program together with sampler outputs (i.e., a set of draws) from the fitted model runs the generated quantities block of the new program using the the existing sample by plugging in the per-draw parameter estimates for the computations in the generated quantities block.\nThis method requires sub-argument fitted_params which takes as its value an existing Stan CSV file that contains a parameter values from an equivalent model, i.e., a model with the same parameters block, conditioned on the same data.\nThe generated quantities block computes quantities of interest (QOIs) based on the data, transformed data, parameters, and transformed parameters. It can be used to:\n\ngenerate simulated data for model testing by forward sampling\ngenerate predictions for new data\ncalculate posterior event probabilities, including multiple comparisons, sign tests, etc.\ncalculate posterior expectations\ntransform parameters for reporting\napply full Bayesian decision theory\ncalculate log likelihoods, deviances, etc. for model comparison\n\nFor an overview of the uses of this feature, see the Stan User’s Guide section on Stand-alone generated quantities and ongoing prediction.\n\n\nTo illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter values from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv contains only the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2983956445 (Default)\n# output\n#   file = output.csv (Default)\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep\n1,1,1,0,0,0,1,1,0,1,0.6\n1,1,0,1,0,0,1,0,1,0,0.5\n1,0,1,1,1,1,1,1,0,1,0.8\n0,1,0,1,0,1,0,1,0,0,0.4\n1,0,0,0,0,0,0,0,0,0,0.1\n0,0,0,0,0,1,1,1,0,0,0.3\n0,0,1,0,1,0,0,0,0,0,0.2\n1,0,1,0,1,1,0,1,1,0,0.6\n...\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep\n0,1,1,0,1,0,0,1,1,0,0.545679,0.5\n1,1,1,1,1,1,0,1,1,0,0.527164,0.8\n1,1,1,1,0,1,1,1,1,0,0.529116,0.8\n1,0,1,1,1,1,0,0,1,0,0.478844,0.6\n0,1,0,0,0,0,1,0,1,0,0.238793,0.3\n0,0,0,0,0,1,1,0,0,0,0.258294,0.2\n1,1,1,0,0,0,0,0,0,0,0.258465,0.3\n\n\n\nThe fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \\\n(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)",
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-    "text": "The --allow-undefined flag can be passed to the call to stanc, which will allow undefined functions in the Stan language to be parsed without an error. We can then include a definition of the function in a C++ header file.\nThis requires specifying two makefile variables:\n\nSTANCFLAGS=--allow-undefined\nUSER_HEADER=&lt;header_file.hpp&gt;, where &lt;header_file.hpp&gt; is the name of a header file that defines a function with the same name and a compatible signature. This function can appear in the global namespace or in the model namespace, which is defined as the name of the model (either the file name, or the --name argument to stanc) followed by _namespace.\n\nThis is an advanced feature which is only recommended to users familiar with the internals of Stan’s Math library. Most existing C++ code will need to be modified to work with Stan, to varying degrees.\nAs an example, consider the following variant of the Bernoulli example\nfunctions {\n  real make_odds(data real theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n}\nmodel {\n  theta ~ beta(1, 1); // uniform prior on interval 0, 1\n  y ~ bernoulli(theta);\n}\ngenerated quantities {\n  real odds;\n  odds = make_odds(theta);\n}\nHere the make_odds function is declared but not defined, which would ordinarily result in a parser error. However, if you put STANCFLAGS = --allow-undefined into the make/local file or into the stanc call, then the stanc compiler will translate this program to C++, but the generated C++ code will not compile unless you write a file such as examples/bernoulli/make_odds.hpp with the following lines\n#include &lt;ostream&gt;\n\ndouble make_odds(const double& theta, std::ostream *pstream__) {\n  return theta / (1 - theta);\n}\nThe signature for this function needs to fulfill all the usages in the C++ class emitted by stanc. The pstream__ argument is mandatory in the signature but need not be used if your function does not print any output. Because make_odds was declared with a data argument and only used in generated quantites, a signature which accepts and returns double is acceptable. Functions which will have parameters passed as input in the transformed parameters or model blocks will require the ability to accept Stan’s autodiff types. If you wish to autodiff through this function, the simplest option is to make it a template, like\ntemplate &lt;typename T&gt;\nT make_odds(const T &theta, std::ostream *pstream__)\n{\n    return theta / (1 - theta);\n}\nGiven the above, the following make invocation should work\n&gt; make STANCFLAGS=--allow-undefined USER_HEADER=examples/bernoulli/make_odds.hpp examples/bernoulli/bernoulli # on Windows add .exe\nAlternatively, you could put STANCFLAGS and USER_HEADER into the make/local file instead of specifying them on the command-line.\nIf the function were more complicated and involved functions in the Stan Math Library, then you would need to add #include &lt;stan/model/model_header.hpp&gt; and prefix the function calls with stan::math::.\n\n\nExternal C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.\n\n\n\nSome functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
+    "text": "To illustrate how this works we use the generate_quantities method to do posterior predictive checks using the estimate of theta given the example bernoulli model and data, following the posterior predictive simulation procedure in the Stan User’s Guide.\nWe write a program bernoulli_ppc.stan which contains the following generated quantities block, with comments to explain the procedure:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nThe rest of the program is the same as in bernoulli.stan.\nThe generate_method requires the sub-argument fitted_params which takes as its value the name of a Stan CSV file. The per-draw parameter values from the fitted_params file will be used to run the generated quantities block.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe output file bernoulli_ppc.csv contains only the values for the variables declared in the generated quantities block, i.e., theta_rep and the elements of y_sim:\n# model = bernoulli_ppc_model\n# method = generate_quantities\n#   generate_quantities\n#     fitted_params = bernoulli_fit.csv\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2983956445 (Default)\n# output\n#   file = output.csv (Default)\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_rep\n1,1,1,0,0,0,1,1,0,1,0.6\n1,1,0,1,0,0,1,0,1,0,0.5\n1,0,1,1,1,1,1,1,0,1,0.8\n0,1,0,1,0,1,0,1,0,0,0.4\n1,0,0,0,0,0,0,0,0,0,0.1\n0,0,0,0,0,1,1,1,0,0,0.3\n0,0,1,0,1,0,0,0,0,0,0.2\n1,0,1,0,1,1,0,1,1,0,0.6\n...\nGiven the current implementation, to see the fitted parameter values for each draw, create a copy variable in the generated quantities block, e.g.:\ngenerated quantities {\n  array[N] int y_sim;\n  // use current estimate of theta to generate new sample\n  for (n in 1:N) {\n    y_sim[n] = bernoulli_rng(theta);\n  }\n  real&lt;lower=0, upper=1&gt; theta_cp = theta;\n  // estimate theta_rep from new sample\n  real&lt;lower=0, upper=1&gt; theta_rep = sum(y_sim) * 1.0 / N;\n}\nNow the output is slightly more interpretable: theta_cp is the same as the theta used to generate the values y_sim[1] through y_sim[1]. Comparing columns theta_cp and theta_rep allows us to see how the uncertainty in our estimate of theta is carried forward into our predictions:\ny_sim.1,y_sim.2,y_sim.3,y_sim.4,y_sim.5,y_sim.6,y_sim.7,y_sim.8,y_sim.9,y_sim.10,theta_cp,theta_rep\n0,1,1,0,1,0,0,1,1,0,0.545679,0.5\n1,1,1,1,1,1,0,1,1,0,0.527164,0.8\n1,1,1,1,0,1,1,1,1,0,0.529116,0.8\n1,0,1,1,1,1,0,0,1,0,0.478844,0.6\n0,1,0,0,0,0,1,0,1,0,0.238793,0.3\n0,0,0,0,0,1,1,0,0,0,0.258294,0.2\n1,1,1,0,0,0,0,0,0,0,0.258465,0.3",
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-    "text": "External C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.",
+    "text": "The fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contents of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] \\\n(in 'bernoulli_ppc.stan', line 5, column 2 to column 30)",
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-    "text": "Some functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
+    "text": "The following is a simple, complete Stan program for a Bernoulli model of binary data.1 The model assumes the binary observed data y[1],...,y[N] are i.i.d. with Bernoulli chance-of-success theta.\ndata { \n  int&lt;lower=0&gt; N; \n  array[N] int&lt;lower=0, upper=1&gt; y;\n} \nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n} \nmodel {\n  theta ~ beta(1, 1);  // uniform prior on interval 0,1\n  y ~ bernoulli(theta);\n}\nThe input data file contains definitions for the two variables N and y which are specified in the data block of program bernoulli.stan (above).\nA data set of N=10 observations is included in the example Bernoulli model directory in both JSON notation and Rdump data format where 8 out of 10 trials had outcome 0 (failure) and 2 trials had outcome 1 (success). In JSON, this data is:\n{\n    \"N\" : 10,\n    \"y\" : [0,1,0,0,0,0,0,0,0,1]\n}",
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-    "text": "Footnotes\n\n\nDetails of programming in the Stan Math style are omitted from this section, it is presented only as an example↩︎",
+    "text": "Footnotes\n\n\nThe model is available with the CmdStan distribution at the path &lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan.↩︎",
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-    "text": "Error messages and return codes\nCmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. The signal number is retcode - 128.\n\nCommon sources of program error are:\n\nMissing or ill-formed input data. All variables declared in the data block must be supplied in the input data file.\nModel parameters cannot be initialized due to model misspecification.\nIndexing errors. The Stan language provides indexing and slicing operations. Indexing errors are easy to make and difficult to debug. The compiler and runtime detect index out of bounds errors; they cannot detect the semantic errors such as indexing into the wrong variables or incorrect index updates.\n\n\n\n\n\n Back to top"
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     "text": "Diagnosing HMC by Comparison of Gradients\nCmdStan has a basic diagnostic feature that will calculate the gradients of the initial state and compare them with gradients calculated by finite differences. Discrepancies between the two indicate that there is a problem with the model or initial states or else there is a bug in Stan.\nTo allow for the possibility of adding other kinds of diagnostic tests, the diagnose method argument configuration has subargument test which currently only takes value gradient. There are two available gradient test configuration arguments:\n\nepsilon - The finite difference step size. Must be a positive real number. Default value is \\(1^{-6}\\)\nerror - The error threshold. Must be a positive real number. Default value is \\(1^{-6}\\)\n\nTo run on the different platforms with the default configuration, use one of the following.\nMac OS and Linux\n&gt; ./my_model diagnose data file=my_data\nWindows\n&gt; my_model diagnose data file=my_data\nTo relax the test threshold, specify the error argument as follows:\n&gt; ./my_model diagnose test=gradient error=0.0001 data file=my_data\nTo see how this works, we run diagnostics on the example bernoulli model:\n&gt; ./bernoulli diagnose data file=bernoulli.data.R\nExecuting this command prints output to the console and as a series of comment lines to the output csv file. The console output is:\nmethod = diagnose\n  diagnose\n    test = gradient (Default)\n      gradient\n        epsilon = 9.9999999999999995e-07 (Default)\n        error = 9.9999999999999995e-07 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 2152196153 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n\nTEST GRADIENT MODE\n\n Log probability=-8.42814\n\n param idx           value           model     finite diff           error\n         0       0.0361376         -3.1084         -3.1084    -2.37554e-10\nThe same information is printed to the output file as csv comments, i.e., each line is prefixed with a pound sign #.\n\n\n\n\n Back to top",
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     "text": "A CmdStan executable is built from the Stan model concept and the CmdStan command line parser. The command line argument syntax consists of sets of keywords and keyword-value pairs. Arguments are grouped by the following keywords:\n\nmethod - specifies the kind of inference done on the model. Each kind of inference requires further configuration via sub-arguments. The method argument is required. It can be specified overtly as the a keyword-value pair method=&lt;inference&gt; or implicitly as one of the following:\n\nsample - obtain a sample from the posterior using HMC\noptimize - penalized maximum likelihood estimation\nvariational - automatic variational inference\ngenerate_quantities - run model’s generated quantities block on existing sample to obtain new quantities of interest.\nlog_prob - compute the log probability and gradient of the model for one set of parameters.\ndiagnose - compute and compare sampler gradient calculations to finite differences.\n\ndata - specifies the input data file, if any.\noutput - specifies program outputs, both disk files and terminal window outputs.\ninit - specifies initial values for the model parameters, if any.\nrandom - specifies the seed for the pseudo-random number.\n\nThe remainder of this chapter covers the general configuration options used for all processing. The following chapters cover the per-inference configuration options.\n\n\nThe values for all variables declared in the data block of the model are read in from an input data file in either JSON or Rdump format. The syntax for the input data argument is:\ndata file=&lt;filepath&gt;\nThe keyword data must be followed directly by the keyword-value pair file=&lt;filepath&gt;. If the model doesn’t declare any data variables, this argument is ignored.\nThe input data file must contain definitions for all data variables declared in the data block. If one or more data block variables are missing from the input data file, the program prints an error message to stderr and returns a non-zero return code. For example, the model bernoulli.stan defines two data variables N and y. If the input data file doesn’t include both variables, or if the data variable doesn’t match the declared type and dimensions, the program will exit with an error message at the point where it first encounters missing data.\nFor example if the input data file doesn’t include the definition for variable y, the executable exits with the following message:\nException: variable does not exist; processing stage=data initialization; variable name=y; base type=int (in 'examples/bernoulli/bernoulli.stan', line 3, column 2 to column 28)\n\n\n\nThe output keyword is used to specify non-default options for output files and messages written to the terminal window. The output keyword takes several keyword-value pair sub-arguments.\nThe keyword value pair file=&lt;filepath&gt; specifies the location of the Stan CSV output file. If unspecified, the output file is written to a file named output.csv in the current working directory.\nThe keyword value pair diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file. By default, no auxiliary output file is produced. This option is only valid for the iterative algorithms sample and variational.\nThe keyword value pair refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is 100 iterations.\nThe keyword value pair sig_figs=&lt;int&gt; specifies the number of significant digits for all numerical values in the output files. Allowable values are between 1 and 18, which is the maximum amount of precision available for 64-bit floating point arithmetic. The default value is 6.   Note: increasing sig_figs above the default will increase the size of the output CSV files accordingly.\nThe keyword value pair profile_file=&lt;filepath&gt; specifies the location of the output file for profiling data. If the model uses no profiling, the output profile file is not produced. If the model uses profiling and profile_file is unspecified, the profiling data is written to a file named profile.csv in the current working directory.\nThe keyword value pair save_cmdstan_config=&lt;int&gt; specifies whether to save the configuration options used to run the program to a file named &lt;output file&gt;_config.json alongside the other output files. The default value is 0, which means the configuration file is not saved. The contents of this file are similar to the comments in the Stan CSV file, but should be more portable across versions and easier to parse.\n\n\n\nInitialization is only applied to parameters defined in the parameters block. By default, all parameters are initialized to random draws from a uniform distribution over the range \\([-2, 2]\\). These values are on the unconstrained scale, so must be inverse transformed back to satisfy the constraints declared for parameters. Because zero is chosen to be a reasonable default initial value for most parameters, the interval around zero provides a fairly diffuse starting point. For instance, unconstrained variables are initialized randomly in \\((-2, 2)\\), variables constrained to be positive are initialized roughly in \\((0.14, 7.4)\\), variables constrained to fall between 0 and 1 are initialized with values roughly in \\((0.12, 0.88)\\).\nThe initialization argument is specified as keyword-value pair with keyword init. The value can be one of the following:\n\npositive real number \\(x\\). All parameters will be initialized to random draws from a uniform distribution over the range \\([-x, x]\\).\n\\(0\\) - All parameters will be initialized to zero values on the unconstrained scale. The transforms are arranged in such a way that zero initialization provides reasonable variable initializations: \\(0\\) for unconstrained parameters; \\(1\\) for parameters constrained to be positive; \\(0.5\\) for variables to constrained to lie between \\(0\\) and \\(1\\); a symmetric (uniform) vector for simplexes; unit matrices for both correlation and covariance matrices; and so on.\nfilepath - A data file in JSON or Rdump format containing initial parameters values for some or all of the model parameters. User specified initial values must satisfy the constraints declared in the model (i.e., they are on the constrained scale). Parameters which aren’t explicitly initialized will be initialized randomly over the range \\([-2, 2]\\).\n\n\n\n\nThe random-number generator’s behavior is determined by the unsigned seed (positive integer) it is started with. If a seed is not specified, or a seed of 0 or less is specified, the system time is used to generate a seed. The seed is recorded and included with Stan’s output regardless of whether it was specified or generated randomly from the system time.\nThe syntax for the random seed argument is:\nrandom seed=&lt;int&gt;\nThe keyword random must be followed directly by the keyword-value pair seed=&lt;int&gt;.\n\n\n\nThe chain identifier argument is used in conjunction with the random seed argument when running multiple Markov chains for sampling. The chain identifier is used to advance the random number generator a very large number of random variates so that two chains with the same seed and different identifiers draw from non-overlapping subsequences of the random-number sequence determined by the seed. Together, the seed and chain identifier determine the behavior of the random number generator.\nThe syntax for the random seed argument is:\nid=&lt;int&gt;\nThe default value is 1.\nWhen running a set of chains from the command line with a specified seed, this argument should be set to the chain index. E.g., when running 4 chains, the value should be 1,..,4, successively. When running multiple chains from a single command, Stan’s interfaces manage the chain identifier arguments automatically.\nFor complete reproducibility, every aspect of the environment needs to be locked down from the OS and version to the C++ compiler and version to the version of Stan and all dependent libraries. See the Stan Reference Manual Reproducibility chapter for further details.\n\n\n\nCmdStan provides a help and help-all mechanism that displays either the available top-level or keyword-specific key-value argument pairs. To display top-level help, call the CmdStan executable with keyword help:\n./bernoulli help\n\n\n\nCmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively. Some methods also generate warning messages when the algorithm detects potential problems with the inference. Depending on the method, these messages are sent to either standard out or standard error.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. To determine the signal number, subtract 128 from the return value, e.g. return code 139 results from termination signal 11 (segmentation violation).\n\nA non-zero return code or outputs sent to stderr indicate problems with the inference. However, a return code of zero and absence of error messages doesn’t necessarily mean that the inference is valid, it is still necessary to validate the inferences using all available summary and diagnostic techniques.",
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@@ -4230,7 +4180,7 @@
     "section": "",
     "text": "The values for all variables declared in the data block of the model are read in from an input data file in either JSON or Rdump format. The syntax for the input data argument is:\ndata file=&lt;filepath&gt;\nThe keyword data must be followed directly by the keyword-value pair file=&lt;filepath&gt;. If the model doesn’t declare any data variables, this argument is ignored.\nThe input data file must contain definitions for all data variables declared in the data block. If one or more data block variables are missing from the input data file, the program prints an error message to stderr and returns a non-zero return code. For example, the model bernoulli.stan defines two data variables N and y. If the input data file doesn’t include both variables, or if the data variable doesn’t match the declared type and dimensions, the program will exit with an error message at the point where it first encounters missing data.\nFor example if the input data file doesn’t include the definition for variable y, the executable exits with the following message:\nException: variable does not exist; processing stage=data initialization; variable name=y; base type=int (in 'examples/bernoulli/bernoulli.stan', line 3, column 2 to column 28)",
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@@ -4241,7 +4191,7 @@
     "section": "",
     "text": "The output keyword is used to specify non-default options for output files and messages written to the terminal window. The output keyword takes several keyword-value pair sub-arguments.\nThe keyword value pair file=&lt;filepath&gt; specifies the location of the Stan CSV output file. If unspecified, the output file is written to a file named output.csv in the current working directory.\nThe keyword value pair diagnostic_file=&lt;filepath&gt; specifies the location of the auxiliary output file. By default, no auxiliary output file is produced. This option is only valid for the iterative algorithms sample and variational.\nThe keyword value pair refresh=&lt;int&gt; specifies the number of iterations between progress messages written to the terminal window. The default value is 100 iterations.\nThe keyword value pair sig_figs=&lt;int&gt; specifies the number of significant digits for all numerical values in the output files. Allowable values are between 1 and 18, which is the maximum amount of precision available for 64-bit floating point arithmetic. The default value is 6.   Note: increasing sig_figs above the default will increase the size of the output CSV files accordingly.\nThe keyword value pair profile_file=&lt;filepath&gt; specifies the location of the output file for profiling data. If the model uses no profiling, the output profile file is not produced. If the model uses profiling and profile_file is unspecified, the profiling data is written to a file named profile.csv in the current working directory.\nThe keyword value pair save_cmdstan_config=&lt;int&gt; specifies whether to save the configuration options used to run the program to a file named &lt;output file&gt;_config.json alongside the other output files. The default value is 0, which means the configuration file is not saved. The contents of this file are similar to the comments in the Stan CSV file, but should be more portable across versions and easier to parse.",
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     "section": "",
     "text": "Initialization is only applied to parameters defined in the parameters block. By default, all parameters are initialized to random draws from a uniform distribution over the range \\([-2, 2]\\). These values are on the unconstrained scale, so must be inverse transformed back to satisfy the constraints declared for parameters. Because zero is chosen to be a reasonable default initial value for most parameters, the interval around zero provides a fairly diffuse starting point. For instance, unconstrained variables are initialized randomly in \\((-2, 2)\\), variables constrained to be positive are initialized roughly in \\((0.14, 7.4)\\), variables constrained to fall between 0 and 1 are initialized with values roughly in \\((0.12, 0.88)\\).\nThe initialization argument is specified as keyword-value pair with keyword init. The value can be one of the following:\n\npositive real number \\(x\\). All parameters will be initialized to random draws from a uniform distribution over the range \\([-x, x]\\).\n\\(0\\) - All parameters will be initialized to zero values on the unconstrained scale. The transforms are arranged in such a way that zero initialization provides reasonable variable initializations: \\(0\\) for unconstrained parameters; \\(1\\) for parameters constrained to be positive; \\(0.5\\) for variables to constrained to lie between \\(0\\) and \\(1\\); a symmetric (uniform) vector for simplexes; unit matrices for both correlation and covariance matrices; and so on.\nfilepath - A data file in JSON or Rdump format containing initial parameters values for some or all of the model parameters. User specified initial values must satisfy the constraints declared in the model (i.e., they are on the constrained scale). Parameters which aren’t explicitly initialized will be initialized randomly over the range \\([-2, 2]\\).",
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     "section": "",
     "text": "The random-number generator’s behavior is determined by the unsigned seed (positive integer) it is started with. If a seed is not specified, or a seed of 0 or less is specified, the system time is used to generate a seed. The seed is recorded and included with Stan’s output regardless of whether it was specified or generated randomly from the system time.\nThe syntax for the random seed argument is:\nrandom seed=&lt;int&gt;\nThe keyword random must be followed directly by the keyword-value pair seed=&lt;int&gt;.",
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     "section": "",
     "text": "The chain identifier argument is used in conjunction with the random seed argument when running multiple Markov chains for sampling. The chain identifier is used to advance the random number generator a very large number of random variates so that two chains with the same seed and different identifiers draw from non-overlapping subsequences of the random-number sequence determined by the seed. Together, the seed and chain identifier determine the behavior of the random number generator.\nThe syntax for the random seed argument is:\nid=&lt;int&gt;\nThe default value is 1.\nWhen running a set of chains from the command line with a specified seed, this argument should be set to the chain index. E.g., when running 4 chains, the value should be 1,..,4, successively. When running multiple chains from a single command, Stan’s interfaces manage the chain identifier arguments automatically.\nFor complete reproducibility, every aspect of the environment needs to be locked down from the OS and version to the C++ compiler and version to the version of Stan and all dependent libraries. See the Stan Reference Manual Reproducibility chapter for further details.",
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     "section": "",
     "text": "CmdStan provides a help and help-all mechanism that displays either the available top-level or keyword-specific key-value argument pairs. To display top-level help, call the CmdStan executable with keyword help:\n./bernoulli help",
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     "section": "",
     "text": "CmdStan executables and utility programs use streams standard output (stdout) and standard error (stderr) to report information and error messages, respectively. Some methods also generate warning messages when the algorithm detects potential problems with the inference. Depending on the method, these messages are sent to either standard out or standard error.\nAll program executables provide a return code between 0 and 255:\n\n0 - Program ran to termination as expected.\nvalue in range [1 : 125] - Method invoked could not run due to problems with model or data.\nvalue &gt; 128 - Fatal error during execution, process terminated by signal. To determine the signal number, subtract 128 from the return value, e.g. return code 139 results from termination signal 11 (segmentation violation).\n\nA non-zero return code or outputs sent to stderr indicate problems with the inference. However, a return code of zero and absence of error messages doesn’t necessarily mean that the inference is valid, it is still necessary to validate the inferences using all available summary and diagnostic techniques.",
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@@ -4321,7 +4271,7 @@
     "section": "",
     "text": "A Stan program must be in a file with extension .stan. The CmdStan makefile rules specify all necessary steps to translate files with suffix .stan to a CmdStan executable program. This is a two-stage process:\n\nfirst the Stan program is translated to C++ by the stanc compiler\nthen the C++ compiler compiles all C++ sources and links them together with the CmdStan interface program and the Stan and math libraries.\n\n\n\nTo compile Stan programs, you must invoke the Make program from the &lt;cmdstan-home&gt; directory. The Stan program can be in a different directory, but the directory path names cannot contain spaces - this limitation is imposed by Make.\n&gt; cd &lt;cmdstan_home&gt;\nIn the call to the Make program, the target is name of the CmdStan executable corresponding to the Stan program file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. To build the Bernoulli example, on Mac and Linux:\n&gt; make examples/bernoulli/bernoulli\nOn Windows, the command is the same with the addition of .exe at the end of the target (note the use of forward slashes):\n&gt; make examples/bernoulli/bernoulli.exe\nThe generated C++ code (bernoulli.hpp), object file (bernoulli.o) and the compiled executable will be placed in the same directory as the Stan program.\nThe compiled executable consists of the Stan model and the CmdStan command line interface which provides inference algorithms to do MCMC sampling, optimization, and variational inference. The following sections provide examples of doing inference using each method on the example model and data file.\n\n\n\nWhen executing a Make target, all its dependencies are checked to see if they are up to date, and if they are not, they are rebuilt. If the you call Make with target bernoulli twice in a row, without any editing bernoulli.stan or otherwise changing the system, on the second invocation, Make will determine that the executable is already newer than the Stan source file and will not recompile the program:\n&gt; make examples/bernoulli/bernoulli\nmake: `examples/bernoulli/bernoulli' is up to date.\nIf the file containing the Stan program is updated, the next call to make will rebuild the CmdStan executable.\n\n\n\nThe Stan probabilistic programming language is a programming language with a rich syntax, as such, it is often the case that a carefully written program contains errors.\nThe simplest class of errors are simple syntax errors such as forgetting the semi-colon statement termination marker at the end of a line, or typos such as a misspelled variable name. For example, if in the bernoulli.stan program, we introduce a typo on line \\(9\\) by writing thata instead of theta, the Make command fails with the following\n--- Translating Stan model to C++ code ---\nbin/stanc  --o=bernoulli.hpp bernoulli.stan\n\nSemantic error in 'bernoulli.stan', line 9, column 2 to column 7:\n   -------------------------------------------------\n     7:  }\n     8:  model {\n     9:    thata ~ beta(1, 1);  // uniform prior on interval 0, 1\n           ^\n    10:    y ~ bernoulli(theta);\n    11:  }\n   -------------------------------------------------\n\nIdentifier 'thata' not in scope.\n\nmake: *** [bernoulli.hpp] Error 1\nStan is a strongly-typed language; and the compiler will throw an error if statements or expressions violate the type rules. The following trivial program foo.stan contains an illegal assignment statement:\ndata {\n  real x;\n}\ntransformed data {\n  int y = x;\n}\nThe Make command fails with the following:\nSemantic error in 'foo.stan', line 5, column 2 to column 12:\n   -------------------------------------------------\n     3:  }\n     4:  transformed data {\n     5:    int y = x;\n           ^\n     6:  }\n   -------------------------------------------------\n\nIll-typed arguments supplied to assignment operator =:\n lhs has type int and rhs has type real\nThe Stan Reference Manual provides a complete specification of the Stan programming language. The Stan User’s Guide also contains a full description of the errors and warnings stanc can emit.\n\n\n\nIf the stanc compiler successfully translates a Stan program to C++, the resulting C++ code should be valid C++ which can be compiled into an executable. The stanc compiler is also a program, and while it has been extensively tested, it may still contain errors such that the generated C++ code fails to compile.\nThe Make command prints the following message to the terminal at the point when it compiles and links the C++ file:\n--- Compiling, linking C++ code ---\nIf the program fails to compile for any reason, the C++ compiler and linker will most likely print a long series of error messages to the console.\nIf this happens, please report the error, together with the Stan program on either the Stan Forums or on the Stan compiler GitHub issues tracker.\n\n\n\nUsers can set flags for the C++ compiler and linker and compiler to optimize their executables. We advise users to only do this once they are sure their basic setup of Cmdstan without flags works.\nThe CXXFLAGS and LDFLAGS makefile variables can be used to set compiler and linker flags respectively. We recommend setting these in the make/local file.\nFor example:\nCXXFLAGS = -O2\nA recommend a set of CXXFLAGS and LDFLAGS flags can be turned on by setting STAN_CPP_OPTIMS=true in the make/local file. These are tested compiler and link-time optimizations that can speed up execution of certain models. We have observed speedups up to 15 percent, but this depends on the model, operating system and hardware used. The use of these flags does considerably slow down compilation, so they are not used by default.\n\n\nWhen assigning or reading from with vectors, row_vectors, matrices or arrays using indexing, Stan checks that a supplied index is valid (not out of range), which avoids segmentation faults and other difficult-to-debug runtime errors.\nFor some models these checks can represent a significant part of the models execution time. By setting the STAN_NO_RANGE_CHECKS=true makefile flag in the make/local file the range checks can be removed. Use this flag with caution (only once the indexing has been validated). In case of any unexpected behavior remove the flag for easier debugging.",
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     "section": "",
     "text": "To compile Stan programs, you must invoke the Make program from the &lt;cmdstan-home&gt; directory. The Stan program can be in a different directory, but the directory path names cannot contain spaces - this limitation is imposed by Make.\n&gt; cd &lt;cmdstan_home&gt;\nIn the call to the Make program, the target is name of the CmdStan executable corresponding to the Stan program file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. To build the Bernoulli example, on Mac and Linux:\n&gt; make examples/bernoulli/bernoulli\nOn Windows, the command is the same with the addition of .exe at the end of the target (note the use of forward slashes):\n&gt; make examples/bernoulli/bernoulli.exe\nThe generated C++ code (bernoulli.hpp), object file (bernoulli.o) and the compiled executable will be placed in the same directory as the Stan program.\nThe compiled executable consists of the Stan model and the CmdStan command line interface which provides inference algorithms to do MCMC sampling, optimization, and variational inference. The following sections provide examples of doing inference using each method on the example model and data file.",
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     "section": "",
     "text": "When executing a Make target, all its dependencies are checked to see if they are up to date, and if they are not, they are rebuilt. If the you call Make with target bernoulli twice in a row, without any editing bernoulli.stan or otherwise changing the system, on the second invocation, Make will determine that the executable is already newer than the Stan source file and will not recompile the program:\n&gt; make examples/bernoulli/bernoulli\nmake: `examples/bernoulli/bernoulli' is up to date.\nIf the file containing the Stan program is updated, the next call to make will rebuild the CmdStan executable.",
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     "section": "",
     "text": "The Stan probabilistic programming language is a programming language with a rich syntax, as such, it is often the case that a carefully written program contains errors.\nThe simplest class of errors are simple syntax errors such as forgetting the semi-colon statement termination marker at the end of a line, or typos such as a misspelled variable name. For example, if in the bernoulli.stan program, we introduce a typo on line \\(9\\) by writing thata instead of theta, the Make command fails with the following\n--- Translating Stan model to C++ code ---\nbin/stanc  --o=bernoulli.hpp bernoulli.stan\n\nSemantic error in 'bernoulli.stan', line 9, column 2 to column 7:\n   -------------------------------------------------\n     7:  }\n     8:  model {\n     9:    thata ~ beta(1, 1);  // uniform prior on interval 0, 1\n           ^\n    10:    y ~ bernoulli(theta);\n    11:  }\n   -------------------------------------------------\n\nIdentifier 'thata' not in scope.\n\nmake: *** [bernoulli.hpp] Error 1\nStan is a strongly-typed language; and the compiler will throw an error if statements or expressions violate the type rules. The following trivial program foo.stan contains an illegal assignment statement:\ndata {\n  real x;\n}\ntransformed data {\n  int y = x;\n}\nThe Make command fails with the following:\nSemantic error in 'foo.stan', line 5, column 2 to column 12:\n   -------------------------------------------------\n     3:  }\n     4:  transformed data {\n     5:    int y = x;\n           ^\n     6:  }\n   -------------------------------------------------\n\nIll-typed arguments supplied to assignment operator =:\n lhs has type int and rhs has type real\nThe Stan Reference Manual provides a complete specification of the Stan programming language. The Stan User’s Guide also contains a full description of the errors and warnings stanc can emit.",
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     "text": "If the stanc compiler successfully translates a Stan program to C++, the resulting C++ code should be valid C++ which can be compiled into an executable. The stanc compiler is also a program, and while it has been extensively tested, it may still contain errors such that the generated C++ code fails to compile.\nThe Make command prints the following message to the terminal at the point when it compiles and links the C++ file:\n--- Compiling, linking C++ code ---\nIf the program fails to compile for any reason, the C++ compiler and linker will most likely print a long series of error messages to the console.\nIf this happens, please report the error, together with the Stan program on either the Stan Forums or on the Stan compiler GitHub issues tracker.",
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     "text": "Users can set flags for the C++ compiler and linker and compiler to optimize their executables. We advise users to only do this once they are sure their basic setup of Cmdstan without flags works.\nThe CXXFLAGS and LDFLAGS makefile variables can be used to set compiler and linker flags respectively. We recommend setting these in the make/local file.\nFor example:\nCXXFLAGS = -O2\nA recommend a set of CXXFLAGS and LDFLAGS flags can be turned on by setting STAN_CPP_OPTIMS=true in the make/local file. These are tested compiler and link-time optimizations that can speed up execution of certain models. We have observed speedups up to 15 percent, but this depends on the model, operating system and hardware used. The use of these flags does considerably slow down compilation, so they are not used by default.\n\n\nWhen assigning or reading from with vectors, row_vectors, matrices or arrays using indexing, Stan checks that a supplied index is valid (not out of range), which avoids segmentation faults and other difficult-to-debug runtime errors.\nFor some models these checks can represent a significant part of the models execution time. By setting the STAN_NO_RANGE_CHECKS=true makefile flag in the make/local file the range checks can be removed. Use this flag with caution (only once the indexing has been validated). In case of any unexpected behavior remove the flag for easier debugging.",
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-    "text": "The following is a simple, complete Stan program for a Bernoulli model of binary data.1 The model assumes the binary observed data y[1],...,y[N] are i.i.d. with Bernoulli chance-of-success theta.\ndata { \n  int&lt;lower=0&gt; N; \n  array[N] int&lt;lower=0, upper=1&gt; y;\n} \nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n} \nmodel {\n  theta ~ beta(1, 1);  // uniform prior on interval 0,1\n  y ~ bernoulli(theta);\n}\nThe input data file contains definitions for the two variables N and y which are specified in the data block of program bernoulli.stan (above).\nA data set of N=10 observations is included in the example Bernoulli model directory in both JSON notation and Rdump data format where 8 out of 10 trials had outcome 0 (failure) and 2 trials had outcome 1 (success). In JSON, this data is:\n{\n    \"N\" : 10,\n    \"y\" : [0,1,0,0,0,0,0,0,0,1]\n}",
+    "text": "The --allow-undefined flag can be passed to the call to stanc, which will allow undefined functions in the Stan language to be parsed without an error. We can then include a definition of the function in a C++ header file.\nThis requires specifying two makefile variables:\n\nSTANCFLAGS=--allow-undefined\nUSER_HEADER=&lt;header_file.hpp&gt;, where &lt;header_file.hpp&gt; is the name of a header file that defines a function with the same name and a compatible signature. This function can appear in the global namespace or in the model namespace, which is defined as the name of the model (either the file name, or the --name argument to stanc) followed by _namespace.\n\nThis is an advanced feature which is only recommended to users familiar with the internals of Stan’s Math library. Most existing C++ code will need to be modified to work with Stan, to varying degrees.\nAs an example, consider the following variant of the Bernoulli example\nfunctions {\n  real make_odds(data real theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  array[N] int&lt;lower=0, upper=1&gt; y;\n}\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n}\nmodel {\n  theta ~ beta(1, 1); // uniform prior on interval 0, 1\n  y ~ bernoulli(theta);\n}\ngenerated quantities {\n  real odds;\n  odds = make_odds(theta);\n}\nHere the make_odds function is declared but not defined, which would ordinarily result in a parser error. However, if you put STANCFLAGS = --allow-undefined into the make/local file or into the stanc call, then the stanc compiler will translate this program to C++, but the generated C++ code will not compile unless you write a file such as examples/bernoulli/make_odds.hpp with the following lines\n#include &lt;ostream&gt;\n\ndouble make_odds(const double& theta, std::ostream *pstream__) {\n  return theta / (1 - theta);\n}\nThe signature for this function needs to fulfill all the usages in the C++ class emitted by stanc. The pstream__ argument is mandatory in the signature but need not be used if your function does not print any output. Because make_odds was declared with a data argument and only used in generated quantites, a signature which accepts and returns double is acceptable. Functions which will have parameters passed as input in the transformed parameters or model blocks will require the ability to accept Stan’s autodiff types. If you wish to autodiff through this function, the simplest option is to make it a template, like\ntemplate &lt;typename T&gt;\nT make_odds(const T &theta, std::ostream *pstream__)\n{\n    return theta / (1 - theta);\n}\nGiven the above, the following make invocation should work\n&gt; make STANCFLAGS=--allow-undefined USER_HEADER=examples/bernoulli/make_odds.hpp examples/bernoulli/bernoulli # on Windows add .exe\nAlternatively, you could put STANCFLAGS and USER_HEADER into the make/local file instead of specifying them on the command-line.\nIf the function were more complicated and involved functions in the Stan Math Library, then you would need to add #include &lt;stan/model/model_header.hpp&gt; and prefix the function calls with stan::math::.\n\n\nExternal C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.\n\n\n\nSome functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
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-    "text": "Footnotes\n\n\nThe model is available with the CmdStan distribution at the path &lt;cmdstan-home&gt;/examples/bernoulli/bernoulli.stan.↩︎",
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+    "text": "External C++ functions are currently the only way to encode a function with a known analytic gradient outside the Stan Math Library. This is done very similarly to how a function would be added to the Math library with a reverse-mode specialization. The following code is adapted from the Stan Math documentation.\nSuppose you have the following (nonsensical) model which relies on a function called my_dot_self. We will implement this as a copy of the built-in dot_self function.\nfunctions {\n  // both overloads end up using the same C++ template\n  real my_dot_self(vector theta);\n  real my_dot_self(row_vector theta);\n}\ndata {\n  int&lt;lower=0&gt; N;\n  vector[N] input_data;\n}\ntransformed data {\n  // no autodiff for data - will call using doubles\n  real ds = my_dot_self(input_data);\n}\nparameters {\n  row_vector[N] thetas;\n}\nmodel {\n  thetas ~ normal(0,1);\n  // autodiff - will call using stan::math::var types\n  input_data ~ normal(thetas, my_dot_self(thetas));\n}\nIf you wanted to autodiff through this function, the following header would suffice1:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr&gt;\ninline stan::value_type_t&lt;EigVec&gt; my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    const auto &x_ref = stan::math::to_ref(x);\n    stan::value_type_t&lt;EigVec&gt; sum_x = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum_x += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum_x;\n}\nHowever, we know the derivative of this function directly. To leverage this, we could use a more complicated form which has two function templates that differentiate themselves based on whether or not derivatives are required:\n#include &lt;stan/model/model_header.hpp&gt;\n#include &lt;ostream&gt;\n\ntemplate &lt;typename EigVec, stan::require_eigen_vector_t&lt;EigVec&gt; * = nullptr,\n          stan::require_not_st_var&lt;EigVec&gt; * = nullptr&gt;\ninline double my_dot_self(const EigVec &x, std::ostream *pstream__)\n{\n    auto x_ref = stan::math::to_ref(x);\n    double sum = 0.0;\n    for (int i = 0; i &lt; x.size(); ++i)\n    {\n        sum += x_ref.coeff(i) * x_ref.coeff(i);\n    }\n    return sum;\n}\n\ntemplate &lt;typename EigVec, stan::require_eigen_vt&lt;stan::is_var, EigVec&gt; * = nullptr&gt;\ninline stan::math::var my_dot_self(const EigVec &v, std::ostream *pstream__)\n{\n    // (1) put v into our memory arena\n    stan::arena_t&lt;EigVec&gt; arena_v(v);\n    // (2) calculate forward pass using\n    // (3) the .val() method for matrices of var types\n    stan::math::var res = my_dot_self(arena_v.val(), pstream__);\n    // (4) Place a callback for the reverse pass on the callback stack.\n    stan::math::reverse_pass_callback(\n        [res, arena_v]() mutable\n        { arena_v.adj() += 2.0 * res.adj() * arena_v.val(); });\n    return res;\n}\nFor more details about how to write C++ code using the Stan Math Library, see the Math library documentation at https://mc-stan.org/math/ or the paper at https://arxiv.org/abs/1509.07164.",
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-    "text": "Standalone Generate Quantities\nThe generate_quantities method allows you to generate additional quantities of interest from a fitted model without re-running the sampler. For an overview of the uses of this feature, see the QuickStart Guide section and the Stan User’s Guide section on Stand-alone generated quantities and ongoing prediction.\nThis method requires sub-argument fitted_params which takes as its value an existing Stan CSV file that contains a sample from an equivalent model, i.e., a model with the same parameters, transformed parameters, and model blocks, conditioned on the same data.\nIf we run the bernoulli.stan program for a single chain to generate a sample in file bernoulli_fit.csv:\n&gt; ./bernoulli sample data file=bernoulli.data.json output file=bernoulli_fit.csv\nThen we can run the bernoulli_ppc.stan to carry out the posterior predictive checks:\n&gt; ./bernoulli_ppc generate_quantities fitted_params=bernoulli_fit.csv \\\n                  data file=bernoulli.data.json \\\n                  output file=bernoulli_ppc.csv\nThe fitted_params file must be a Stan CSV file; attempts to use a regular CSV file will result an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe fitted_params file must contain columns corresponding to legal values for all parameters defined in the model. If any parameters are missing, the program will exit with an error message of the form:\nError reading fitted param names from sample csv file &lt;filename.csv&gt;\nThe parameter values of the fitted_params are on the constrained scale and must obey all constraints. For example, if we modify the contencts of the first reported draw in bernoulli_fit.csv so that the value of theta is outside the declared bounds real&lt;lower=0, upper=1&gt;, the program will return the following error message:\nException: lub_free: Bounded variable is 1.21397, but must be in the interval [0, 1] (in 'bernoulli_ppc.stan', line 5, column 2 to column 30)\n\n\n\n\n Back to top",
+    "text": "Some functions have special meanings in Stan and place additional requirements on their signatures if used in external C++.\n\nRNGs must end with _rng. They will be passed a “base RNG object” as the second to last argument, before the pointer to the ostream. We recommend making this a template, since it may change, but at the moment it is always a boost::random::ecuyer1988 object.\nFunctions which edit the target directly must end with _lp and will be passed a reference to lp__ and a reference to a stan::math::accumulator object as the final parameters before the ostream pointer. They are also expected to have a boolean template parameter propto__ which controls whether or not constant terms can be dropped.\nProbability distributions must end with _lpdf or _lpmf and will be passed a boolean template parameter propto__ which controls whether or not constant terms can be dropped.",
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+    "text": "Footnotes\n\n\nDetails of programming in the Stan Math style are omitted from this section, it is presented only as an example↩︎",
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     "text": "CmdStan can return the computed log probability and the gradient with respect to a set of parameters.\nThis is similar to the diagnose subcommand, but the output format differs and the results here are not compared with those from finite differences.\nNote: Startup and data initialization costs mean that this method is not an efficient way to calculate these quantities. It is provided only for convenience and should not be used for serious computation.\n\n\nThis method takes 3 arguments:\n\njacobian - Whether or not the Jacobian adjustment for constrained parameters should be included in the gradient. Default value is 1 (include adjustment).\nconstrained_params - Input file of parameters values on the constrained scale. A single set of constrained parameters can be specified using JSON format. Alternatively, the input file can be set of draws in StanCSV format.\nunconstrained_params - Input file (JSON or R dump) of parameter values on unconstrained scale. These files should contain a single variable, called params_r, which is a flattened vector of all unconstrained parameters. If this object is two dimensional, each entry should be a vector of the same form and the output will feature multiple rows.\n\nOnly one of constrained_params and unconstrained_params can be specified.\nFor more on the differences between constrained and unconstrained parameters, see the Stan reference manual section on variable transforms.\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the log_prob method is reported at the beginning of the output file as CSV comments.\nColumn headers, the first column is labelled lp__, and the rest are named after parameters. These will be the unconstrained parameters, regardless of whether constrained or unconstrained parameters were supplied as input.\nValues which correspond to the value of the log density (column 1) and the gradient with respect to each parameter (remaining columns).\n\nFor example, if we have a file called params.json:\n{\n    \"theta\" : 0.1\n}\nWe can run the example model:\n/bernoulli log_prob constrained_params=params.json data file=bernoulli.data.json\nThis yields\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-11-17 20:46:06 UTC\n# method = log_prob\n#   log_prob\n#     unconstrained_params =  (Default)\n#     constrained_params = params.json\n#     jacobian = 1 (Default)\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2390820139 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 2.31.0 (Linux)\n# stancflags =\nlp_,theta\n-7.856,1.8",
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     "text": "This method takes 3 arguments:\n\njacobian - Whether or not the Jacobian adjustment for constrained parameters should be included in the gradient. Default value is 1 (include adjustment).\nconstrained_params - Input file of parameters values on the constrained scale. A single set of constrained parameters can be specified using JSON format. Alternatively, the input file can be set of draws in StanCSV format.\nunconstrained_params - Input file (JSON or R dump) of parameter values on unconstrained scale. These files should contain a single variable, called params_r, which is a flattened vector of all unconstrained parameters. If this object is two dimensional, each entry should be a vector of the same form and the output will feature multiple rows.\n\nOnly one of constrained_params and unconstrained_params can be specified.\nFor more on the differences between constrained and unconstrained parameters, see the Stan reference manual section on variable transforms.",
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     "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the log_prob method is reported at the beginning of the output file as CSV comments.\nColumn headers, the first column is labelled lp__, and the rest are named after parameters. These will be the unconstrained parameters, regardless of whether constrained or unconstrained parameters were supplied as input.\nValues which correspond to the value of the log density (column 1) and the gradient with respect to each parameter (remaining columns).\n\nFor example, if we have a file called params.json:\n{\n    \"theta\" : 0.1\n}\nWe can run the example model:\n/bernoulli log_prob constrained_params=params.json data file=bernoulli.data.json\nThis yields\n# stan_version_major = 2\n# stan_version_minor = 31\n# stan_version_patch = 0\n# model = bernoulli_model\n# start_datetime = 2022-11-17 20:46:06 UTC\n# method = log_prob\n#   log_prob\n#     unconstrained_params =  (Default)\n#     constrained_params = params.json\n#     jacobian = 1 (Default)\n# id = 1 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2390820139 (Default)\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\n#   sig_figs = -1 (Default)\n#   profile_file = profile.csv (Default)\n# num_threads = 1 (Default)\n# stanc_version = stanc3 2.31.0 (Linux)\n# stancflags =\nlp_,theta\n-7.856,1.8",
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-    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n....\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)\n\n\n\nA Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThere are two different ways of running multiple chains, with the num_chains argument using a single executable and by using the Unix and DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine.\nExample that will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially and will produce the sample in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS.\n\n\n\n\nEach execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n...\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)\n\n\n\nThe stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. 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-    "text": "To generate a sample from the posterior distribution of the model conditioned on the data, we run the executable program with the argument sample or method=sample together with the input data. The executable can be run from any directory. Here, we run it in the directory which contains the Stan program and input data, &lt;cmdstan-home&gt;/examples/bernoulli:\n&gt; cd examples/bernoulli\nTo execute sampling of the model under Linux or Mac, use:\n&gt; ./bernoulli sample data file=bernoulli.data.json\nIn Windows, the ./ prefix is not needed:\n&gt; bernoulli.exe sample data file=bernoulli.data.json\nThe output is the same across all supported platforms. First, the configuration of the program is echoed to the standard output:\nmethod = sample (Default)\n  sample\n    num_samples = 1000 (Default)\n    num_warmup = 1000 (Default)\n    save_warmup = 0 (Default)\n    thin = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      gamma = 0.050000000000000003 (Default)\n      delta = 0.80000000000000004 (Default)\n      kappa = 0.75 (Default)\n      t0 = 10 (Default)\n      init_buffer = 75 (Default)\n      term_buffer = 50 (Default)\n      window = 25 (Default)\n      save_metric = 0 (Default)\n    algorithm = hmc (Default)\n      hmc\n        engine = nuts (Default)\n          nuts\n            max_depth = 10 (Default)\n        metric = diag_e (Default)\n        metric_file =  (Default)\n        stepsize = 1 (Default)\n        stepsize_jitter = 0 (Default)\n    num_chains = 1 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3252652196 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, a short timing message is given.\nGradient evaluation took 1.2e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.\nAdjust your expectations accordingly!\nNext, the sampler reports the iteration number, reporting the percentage complete.\nIteration:    1 / 2000 [  0%]  (Warmup)\n....\nIteration: 2000 / 2000 [100%]  (Sampling)\nFinally, the sampler reports timing information:\n Elapsed Time: 0.007 seconds (Warm-up)\n               0.017 seconds (Sampling)\n               0.024 seconds (Total)",
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-    "text": "A Markov chain generates samples from the target distribution only after it has converged to equilibrium. In theory, convergence is only guaranteed asymptotically as the number of draws grows without bound. In practice, diagnostics must be applied to monitor convergence for the finite number of draws actually available. One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. For robust diagnostics, we recommend running 4 chains.\nThere are two different ways of running multiple chains, with the num_chains argument using a single executable and by using the Unix and DOS shell to run multiple executables.\n\n\nThe num_chains argument can be used for all of Stan’s samplers with the exception of the static HMC engine.\nExample that will run 4 chains:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv\nIf the model was not compiled with STAN_THREADS=true, the above command will run 4 chains sequentially and will produce the sample in output_1.csv, output_2.csv, output_3.csv, output_4.csv. A suffix with the chain id is appended to the provided output filename (output.csv in the above command).\nIf the model was compiled with STAN_THREADS=true, the chains can run in parallel, with the num_threads argument defining the maximum number of threads used to run the chains. If the model uses no within-chain parallelization (map_rect or reduce_sum calls), the below command will run 4 chains in parallel, provided there are cores available:\n./bernoulli sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=4\nIf the model uses within-chain parallelization (map_rect or reduce_sum calls), the threads are automatically scheduled to run the parallel parts of a single chain or run the sequential parts of another chains. The below call starts 4 chains that can use 16 threads. At a given moment a single chain may use all 16 threads, 1 thread, anything in between, or can wait for a thread to be available. The scheduling is left to the Threading Building Blocks scheduler.\n./bernoulli_par sample num_chains=4 data file=bernoulli.data.json output file=output.csv num_threads=16\n\n\n\nTo run multiple chains given a model and data, either sequentially or in parallel, we can also use the Unix or DOS shell for loop to set up index variables needed to identify each chain and its outputs.\nOn MacOS or Linux, the for-loop syntax for both the bash and zsh interpreters is:\nfor NAME [in LIST]; do COMMANDS; done\nThe list can be a simple sequence of numbers, or you can use the shell expansion syntax {1..N} which expands to the sequence from \\(1\\) to \\(N\\), e.g. {1..4} expands to 1 2 3 4. Note that the expression {1..N} cannot contain spaces.\nTo run 4 chains for the example bernoulli model on MacOS or Linux:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv\n    done\nThe backslash (\\) indicates a line continuation in Unix. The expression ${i} substitutes in the value of loop index variable i. To run chains in parallel, put an ampersand (&) at the end of the nested sampler command:\n&gt; for i in {1..4}\n    do\n      ./bernoulli sample data file=bernoulli.data.json \\\n      output file=output_${i}.csv &\n    done\nThis pushes each process into the background which allows the loop to continue without waiting for the current chain to finish.\nOn Windows, the DOS for-loop syntax is one of:\nfor %i in (SET) do COMMAND COMMAND-ARGUMENTS\nfor /l %i in (START, STEP, END) do COMMAND COMMAND-ARGUMENTS\nTo run 4 chains in parallel on Windows:\n&gt;for /l %i in (1, 1, 4) do start /b bernoulli.exe sample ^\n                                    data file=bernoulli.data.json my_data ^\n                                    output file=output_%i.csv\nThe caret (^) indicates a line continuation in DOS.",
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-    "text": "Each execution of the model results in draws from a single Markov chain being written to a file in comma-separated value (CSV) format. The default name of the output file is output.csv.\nThe first part of the output file records the version of the underlying Stan library and the configuration as comments (i.e., lines beginning with the pound sign (#)).\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 1000 (Default)\n#     num_warmup = 1000 (Default)\n...\n# output\n#   file = output.csv (Default)\n#   diagnostic_file =  (Default)\n#   refresh = 100 (Default)\nThis is followed by a CSV header indicating the names of the values sampled.\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nThe first output columns report the HMC sampler information:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause the above header is from the NUTS sampler, it has columns treedepth__, n_leapfrog__, and divergent__ and doesn’t have column int_time__. The remaining columns correspond to model parameters. For the Bernoulli model, it is just the final column, theta.\nThe header line is written to the output file before warmup begins. If option save_warmup is set to 1, the warmup draws are output directly after the header. The total number of warmup draws saved is num_warmup divided by thin, rounded up (i.e., ceiling).\nFollowing the warmup draws (if any), are comments which record the results of adaptation: the stepsize, and inverse mass metric used during sampling:\n# Adaptation terminated\n# Step size = 0.884484\n# Diagonal elements of inverse mass matrix:\n# 0.535006\nThe default sampler is NUTS with an adapted step size and a diagonal inverse mass matrix. For this example, the step size is 0.884484, and the inverse mass contains the single entry 0.535006 corresponding to the parameter theta.\nDraws from the posterior distribution are printed out next, each line containing a single draw with the columns corresponding to the header.\n-6.84097,0.974135,0.884484,1,3,0,6.89299,0.198853\n-6.91767,0.985167,0.884484,1,1,0,6.92236,0.182295\n-7.04879,0.976609,0.884484,1,1,0,7.05641,0.162299\n-6.88712,1,0.884484,1,1,0,7.02101,0.188229\n-7.22917,0.899446,0.884484,1,3,0,7.73663,0.383596\n...\nThe output ends with timing details:\n#  Elapsed Time: 0.007 seconds (Warm-up)\n#                0.017 seconds (Sampling)\n#                0.024 seconds (Total)",
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-    "text": "The stansummary utility processes one or more output files from a run or set of runs of Stan’s HMC sampler given a model and data. For all columns in the Stan CSV output file stansummary reports a set of statistics including mean, standard deviation, percentiles, effective number of samples, and \\(\\hat{R}\\) values.\nTo run stansummary on the output files generated by the for loop above, by the above run of the bernoulli model on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary output_*.csv\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe output_*.csv\nThe stansummary output consists of one row of statistics per column in the Stan CSV output file. Therefore, the first rows in the stansummary report statistics over the sampler state. The final row of output summarizes the estimates of the model variable theta:\nInference for Stan model: bernoulli_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.0070, 0.0070, 0.0070, 0.0070) seconds, 0.028 seconds total\nSampling took (0.020, 0.017, 0.021, 0.019) seconds, 0.077 seconds total\n\n                Mean     MCSE  StdDev     5%   50%   95%    N_Eff  N_Eff/s    R_hat\nlp__            -7.3  1.8e-02    0.75   -8.8  -7.0  -6.8  1.8e+03  2.4e+04  1.0e+00\naccept_stat__   0.89  2.7e-03    0.17   0.52  0.96   1.0  3.9e+03  5.1e+04  1.0e+00\nstepsize__       1.1  7.5e-02    0.11   0.93   1.2   1.2  2.0e+00  2.6e+01  2.5e+13\ntreedepth__      1.4  8.1e-03    0.49    1.0   1.0   2.0  3.6e+03  4.7e+04  1.0e+00\nn_leapfrog__     2.3  1.7e-02    0.98    1.0   3.0   3.0  3.3e+03  4.3e+04  1.0e+00\ndivergent__     0.00      nan    0.00   0.00  0.00  0.00      nan      nan      nan\nenergy__         7.8  2.6e-02     1.0    6.8   7.5   9.9  1.7e+03  2.2e+04  1.0e+00\ntheta           0.25  2.9e-03    0.12  0.079  0.23  0.46  1.7e+03  2.1e+04  1.0e+00\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nIn this example, we conditioned the model on a dataset consisting of the outcomes of 10 bernoulli trials, where only 2 trials reported success. The 5%, 50%, and 95% percentile values for theta reflect the uncertainty in our estimate, due to the small amount of data, given the prior of beta(1, 1)",
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-    "text": "The optimize method finds the mode of the posterior distribution, assuming that there is one. If the posterior is not convex, there is no guarantee Stan will be able to find the global mode as opposed to a local optimum of log probability. For optimization, the mode is calculated without the Jacobian adjustment for constrained variables, which shifts the mode due to the change of variables. Thus modes correspond to modes of the model as written.\nThe full set of configuration options available for the optimize method is reported at the beginning of the sampler output file as CSV comments. When the example model bernoulli.stan is run with method=optimize via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n#       lbfgs\n#         init_alpha = 0.001 (Default)\n#         tol_obj = 9.9999999999999998e-13 (Default)\n#         tol_rel_obj = 10000 (Default)\n#         tol_grad = 1e-08 (Default)\n#         tol_rel_grad = 10000000 (Default)\n#         tol_param = 1e-08 (Default)\n#         history_size = 5 (Default)\n#     jacobian = 0 (Default)\n#     iter = 2000 (Default)\n#     save_iterations = 0 (Default)\n\n\nThe jacobian argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms. Without the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\), the value which maximizes the likelihood of the data given the parameters, (including prior terms). Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\,p(\\theta)\\). By default this value is 0 (false), do not include the Jacobian adjustment.\n\n\n\nThe algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.\n\n\n\nFor both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. Being too big or too small just means that the first iteration will take longer (i.e., require more gradient evaluations) before the line search finds a good step length.\nIn addition to the above, the L-BFGS algorithm has argument history_size which controls the size of the history it uses to approximate the Hessian. The value should be less than the dimensionality of the parameter space and, in general, relatively small values (\\(5\\)-\\(10\\)) are sufficient; the default value is \\(5\\).\nIf L-BFGS performs poorly but BFGS performs well, consider increasing the history size. Increasing history size will increase the memory usage, although this is unlikely to be an issue for typical Stan models.\n\n\n\nThere are no configuration parameters for the Newton optimizer. It is not recommended because of the slow Hessian calculation involving finite differences.",
+    "text": "The CmdStan executable can run Stan’s optimization algorithms, which provide a deterministic method to find the posterior mode. If the posterior is not convex, there is no guarantee Stan will be able to find the global optimum as opposed to a local optimum of log probability.\nThe full set of configuration options available for the optimize method is available by using the optimize help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the optimizer console output and in the output CSV file’s comments.\nThe executable does not need to be recompiled in order to switch from sampling to optimization, and the data input format is the same. The following is a minimal call to Stan’s optimizer using defaults for everything but the location of the data file.\n&gt; ./bernoulli optimize data file=bernoulli.data.json\nExecuting this command prints both output to the console and to a CSV file.\nThe first part of the console output reports on the configuration used. The above command uses all default configurations, therefore the optimizer used is the L-BFGS optimizer and its default initial stepsize and tolerances for monitoring convergence:\n ./bernoulli optimize data file=bernoulli.data.json\nmethod = optimize\n  optimize\n    algorithm = lbfgs (Default)\n      lbfgs\n        init_alpha = 0.001 (Default)\n        tol_obj = 1e-12 (Default)\n        tol_rel_obj = 10000 (Default)\n        tol_grad = 1e-08 (Default)\n        tol_rel_grad = 10000000 (Default)\n        tol_param = 1e-08 (Default)\n        history_size = 5 (Default)\n    iter = 2000 (Default)\n    save_iterations = 0 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 87122538 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nThe second part of the output indicates how well the algorithm fared, here converging and terminating normally. The numbers reported indicate that it took 5 iterations and 8 gradient evaluations. This is, not surprisingly, far fewer iterations than required for sampling; even fewer iterations would be used with less stringent user-specified convergence tolerances. The alpha value is for step size used. In the final state the change in parameters was roughly \\(0.002\\) and the length of the gradient roughly 3e-05 (\\(0.00003\\)).\nInitial log joint probability = -6.85653\n    Iter      log prob        ||dx||      ||grad||       alpha      alpha0  # evals  Notes\n       5      -5.00402    0.00184936   3.35074e-05           1           1        8\nOptimization terminated normally:\n  Convergence detected: relative gradient magnitude is below tolerance\nThe output from optimization is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as comment lines:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = optimize\n#   optimize\n#     algorithm = lbfgs (Default)\n#       lbfgs\n#         init_alpha = 0.001 (Default)\n#         tol_obj = 9.9999999999999998e-13 (Default)\n#         tol_rel_obj = 10000 (Default)\n#         tol_grad = 1e-08 (Default)\n#         tol_rel_grad = 10000000 (Default)\n#         tol_param = 1e-08 (Default)\n#         history_size = 5 (Default)\n#     jacobian = 0 (Default)\n#     iter = 2000 (Default)\n#     save_iterations = 0 (Default)\n...\nFollowing the config information are two lines of output, the CSV headers and the recorded values:\nlp__,theta\n-5.00402,0.200003\nNote that everything is a comment other than a line for the header, and a line for the values. Here, the header indicates the unnormalized log probability with lp__ and the model parameter theta. The maximum log probability is -5.0 and the posterior mode for theta is 0.20. The mode exactly matches what we would expect from the data. Because the prior was uniform, the result 0.20 represents the maximum likelihood estimate (MLE) for the very simple Bernoulli model. Note that no uncertainty is reported.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file. Running the optimizer with save_iterations=1 writes both the initial log joint probability and values for all iterations to the output CSV file.\nRunning the example model with option save_iterations=1, i.e., the command\n&gt; ./bernoulli optimize save_iterations=1 data file=bernoulli.data.json\nproduces CSV file output rows:\nlp__,theta\n-6.85653,0.493689\n-6.10128,0.420936\n-5.02953,0.22956\n-5.00517,0.206107\n-5.00403,0.200299\n-5.00402,0.200003\n\n\nThe jacobian argument specifies whether or not the call to the model’s log probability function should include the log absolute Jacobian determinant of inverse parameter transforms.\nWithout the Jacobian adjustment, optimization returns the (regularized) maximum likelihood estimate (MLE), \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\), the value which maximizes the likelihood of the data given the parameters, (including prior terms).\nApplying the Jacobian adjustment produces the maximum a posteriori estimate (MAP), the maximum value of the posterior distribution, \\(\\mathrm{argmax}_{\\theta}\\ p(y | \\theta)\\,p(\\theta)\\).\nBy default this value is 0 (false), do not include the Jacobian adjustment.\n\n\n\nThe algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.\n\n\n\nFor both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. 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     "text": "The algorithm argument specifies the optimization algorithm. This argument takes one of the following three values:\n\nlbfgs A quasi-Newton optimizer. This is the default optimizer and also much faster than the other optimizers.\nbfgs A quasi-Newton optimizer.\nnewton A Newton optimizer. This is the least efficient optimization algorithm, but has the advantage of setting its own stepsize.\n\nSee the Stan Reference Manual’s Optimization chapter for a description of these algorithms.\nAll of the optimizers stream per-iteration intermediate approximations to the command line console. The sub-argument save_iterations specifies whether or not to save the intermediate iterations to the output file. Allowed values are \\(0\\) or \\(1\\), corresponding to False and True respectively. The default value is \\(0\\), i.e., intermediate iterations are not saved to the output file.",
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     "text": "For both BFGS and L-BFGS optimizers, convergence monitoring is controlled by a number of tolerance values, any one of which being satisfied causes the algorithm to terminate with a solution. See the BFGS and L-BFGS configuration section for details on the convergence tests.\nBoth BFGS and L-BFGS have the following configuration arguments:\n\ninit_alpha - The initial step size parameter. Must be a positive real number. Default value is \\(0.001\\)\ntol_obj - Convergence tolerance on changes in objective function value. Must be a positive real number. Default value is \\(1^{-12}\\).\ntol_rel_obj - Convergence tolerance on relative changes in objective function value. Must be a positive real number. Default value is \\(1^{4}\\).\ntol_grad - Convergence tolerance on the norm of the gradient. Must be a positive real number. Default value is \\(1^{-8}\\).\ntol_rel_grad - Convergence tolerance on the relative norm of the gradient. Must be a positive real number. Default value is \\(1^{7}\\).\ntol_param - Convergence tolerance on changes in parameter value. Must be a positive real number. Default value is \\(1^{-8}\\).\n\nThe init_alpha argument specifies the first step size to try on the initial iteration. If the first iteration takes a long time (and requires a lot of function evaluations), set init_alpha to be the roughly equal to the alpha used in that first iteration. The default value is very small, which is reasonable for many problems but might be too large or too small depending on the objective function and initialization. Being too big or too small just means that the first iteration will take longer (i.e., require more gradient evaluations) before the line search finds a good step length.\nIn addition to the above, the L-BFGS algorithm has argument history_size which controls the size of the history it uses to approximate the Hessian. The value should be less than the dimensionality of the parameter space and, in general, relatively small values (\\(5\\)-\\(10\\)) are sufficient; the default value is \\(5\\).\nIf L-BFGS performs poorly but BFGS performs well, consider increasing the history size. Increasing history size will increase the memory usage, although this is unlikely to be an issue for typical Stan models.",
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     "text": "There are no configuration parameters for the Newton optimizer. It is not recommended because of the slow Hessian calculation involving finite differences.",
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-    "text": "The Pathfinder algorithm is described in section Pathfinder overview.\nThe pathfinder method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument num_paths specifies the number of single-path Pathfinders, the default is \\(4\\). If num_paths is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the pathfinder output file as CSV comments. When the example model bernoulli.stan is run with method=pathfinder via the command line with all default arguments, the resulting Stan CSV file header comments show the complete set of default configuration options:\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\n\n\n\nnum_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\n\n\n\n\nArguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.\n\n\n\nBy default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\n\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the sampler output file as CSV comments.\nThe CSV header row consists of columns lp_approx__, lp__, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.\nThe data rows contain the draws from the single- or multi-path run.\nFinal comments containing timing information.\n\n\n\n\nThe boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
+    "text": "The CmdStan method pathfinder uses the Pathfinder algorithm of Zhang et al. (2022), which is further described in the Stan Reference Manual.\nA single run of the Pathfinder algorithm generates a set of approximate draws. Inference is improved by running multiple Pathfinder instances and using Pareto-smoothed importance resampling (PSIS) of the resulting sets of draws. This better matches non-normal target densities and also eliminates minor modes.\nThe pathfinder method runs multi-path Pathfinder by default, which returns a PSIS sample over the draws from several individual (“single-path”) Pathfinder runs. Argument num_paths specifies the number of single-path Pathfinders, the default is \\(4\\). If num_paths is set to 1, then only one individual Pathfinder is run without the PSIS reweighting of the sample.\nThe full set of configuration options available for the pathfinder method is available by using the pathfinder help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.\nThe following is a minimal call the Pathfinder algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli pathfinder data file=bernoulli.data.R\nExecuting this command prints both output to the console and csv files.\nThe first part of the console output reports on the configuration used.\nmethod = pathfinder\n  pathfinder\n    init_alpha = 0.001 (Default)\n    tol_obj = 9.9999999999999998e-13 (Default)\n    tol_rel_obj = 10000 (Default)\n    tol_grad = 1e-08 (Default)\n    tol_rel_grad = 10000000 (Default)\n    tol_param = 1e-08 (Default)\n    history_size = 5 (Default)\n    num_psis_draws = 1000 (Default)\n    num_paths = 4 (Default)\n    psis_resample = 1 (Default)\n    calculate_lp = 1 (Default)\n    save_single_paths = 0 (Default)\n    max_lbfgs_iters = 1000 (Default)\n    num_draws = 1000 (Default)\n    num_elbo_draws = 25 (Default)\nid = 1 (Default)\ndata\n  file = examples/bernoulli/bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 1995513073 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\n  sig_figs = -1 (Default)\n  profile_file = profile.csv (Default)\nnum_threads = 1 (Default)\nThe rest of the output describes the progression of the algorithm.\nBy default, the Pathfinder algorithm runs 4 single-path Pathfinders in parallel, then uses importance resampling on the set of returned draws to produce the specified number of draws.\nPath [1] :Initial log joint density = -11.543343\nPath [1] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      1.070e-03   1.707e-05    1.000e+00  1.000e+00       126 -6.220e+00 -6.220e+00\nPath [1] :Best Iter: [5] ELBO (-6.219833) evaluations: (126)\nPath [2] :Initial log joint density = -7.443345\nPath [2] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      9.936e-05   3.738e-07    1.000e+00  1.000e+00       126 -6.164e+00 -6.164e+00\nPath [2] :Best Iter: [5] ELBO (-6.164015) evaluations: (126)\nPath [3] :Initial log joint density = -18.986308\nPath [3] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.996e-04   4.018e-06    1.000e+00  1.000e+00       126 -6.201e+00 -6.201e+00\nPath [3] :Best Iter: [5] ELBO (-6.200559) evaluations: (126)\nPath [4] :Initial log joint density = -8.304453\nPath [4] : Iter      log prob        ||dx||      ||grad||     alpha      alpha0      # evals       ELBO    Best ELBO        Notes\n              5      -6.748e+00      2.814e-04   2.034e-06    1.000e+00  1.000e+00       126 -6.221e+00 -6.221e+00\nPath [4] :Best Iter: [3] ELBO (-6.161276) evaluations: (126)\nTotal log probability function evaluations:8404\n\n\n\nnum_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\n\n\n\n\nArguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.\n\n\n\nBy default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\nThe initial CSV comment rows contain the complete set of CmdStan configuration options.\n...\n# method = pathfinder\n#   pathfinder\n#     init_alpha = 0.001 (Default)\n#     tol_obj = 9.9999999999999998e-13 (Default)\n#     tol_rel_obj = 10000 (Default)\n#     tol_grad = 1e-08 (Default)\n#     tol_rel_grad = 10000000 (Default)\n#     tol_param = 1e-08 (Default)\n#     history_size = 5 (Default)\n#     num_psis_draws = 1000 (Default)\n#     num_paths = 4 (Default)\n#     psis_resample = 1 (Default)\n#     calculate_lp = 1 (Default)\n#     save_single_paths = 0 (Default)\n#     max_lbfgs_iters = 1000 (Default)\n#     num_draws = 1000 (Default)\n#     num_elbo_draws = 25 (Default)\n...\nNext is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs.\n\n\n\nThe boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
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@@ -4672,7 +4578,7 @@
     "section": "",
     "text": "num_psis_draws - Final number of draws from multi-path pathfinder. Must be a positive integer. Default value is \\(1000\\).\nnum_paths - Number of single pathfinders. Must be a positive integer. Default value is \\(4\\).\nsave_single_paths - When True (\\(1\\)), save outputs from single pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(0\\) (False).\nmax_lbfgs_iters - Maximum number of L-BFGS iterations. Must be a positive integer. Default value is \\(1000\\).\nnum_draws - Number of approximate posterior draws for each single pathfinder. Must be a positive integer. Default value is \\(1000\\). Can differ from num_psis_draws.\nnum_elbo_draws - Number of Monte Carlo draws to evaluate ELBO. Must be a positive integer. Default value is \\(25\\).\npsis_resample - If True (\\(1\\)), perform psis resampling on samples returned from individual pathfinders. If False (\\(0\\)), returns all num_paths * num_draws samples draws from the individual pathfinders. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).\ncalculate_lp - If True (\\(1\\)), log probabilities of the approximate draws are calculated and returned with the output. If False (\\(0\\)), each pathfinder will only calculate the lp values needed for the ELBO calculation. If False, psis resampling cannot be performed and the algorithm returns num_paths * num_draws samples. The output will still contain any lp values used when calculating ELBO scores within L-BFGS iterations. Valid values: \\(\\{0, 1\\}\\). Default is \\(1\\) (True).",
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@@ -4683,7 +4589,7 @@
     "section": "",
     "text": "Arguments init_alpha through history_size are the full set of arguments to the L-BFGS optimizer and have the same defaults for optimization.",
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@@ -4692,9 +4598,9 @@
     "href": "cmdstan-guide/pathfinder_config.html#pathfinder_csv",
     "title": "Pathfinder Method for Approximate Bayesian Inference",
     "section": "",
-    "text": "By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\n\nThe full set of configuration options available for the pathfinder method is reported at the beginning of the sampler output file as CSV comments.\nThe CSV header row consists of columns lp_approx__, lp__, and the Stan model parameters, transformed parameters, and generated quantities in the order in which they are declared in the Stan program.\nThe data rows contain the draws from the single- or multi-path run.\nFinal comments containing timing information.",
+    "text": "By default, the pathfinder method uses 4 independent Pathfinder runs, each of which produces 1000 approximate draws, which are then importance resampled down to 1000 final draws. The importance resampled draws are output as a StanCSV file.\nThe CSV files have the following structure:\nThe initial CSV comment rows contain the complete set of CmdStan configuration options.\n...\n# method = pathfinder\n#   pathfinder\n#     init_alpha = 0.001 (Default)\n#     tol_obj = 9.9999999999999998e-13 (Default)\n#     tol_rel_obj = 10000 (Default)\n#     tol_grad = 1e-08 (Default)\n#     tol_rel_grad = 10000000 (Default)\n#     tol_param = 1e-08 (Default)\n#     history_size = 5 (Default)\n#     num_psis_draws = 1000 (Default)\n#     num_paths = 4 (Default)\n#     psis_resample = 1 (Default)\n#     calculate_lp = 1 (Default)\n#     save_single_paths = 0 (Default)\n#     max_lbfgs_iters = 1000 (Default)\n#     num_draws = 1000 (Default)\n#     num_elbo_draws = 25 (Default)\n...\nNext is the column header line, followed the set of approximate draws. The Pathfinder algorithm first outputs lp_approx__, the log density in the approximating distribution, and lp__, the log density in the target distribution, followed by estimates of the model parameters, transformed parameters, and generated quantities.\nlp_approx__,lp__,theta\n-2.4973, -8.2951, 0.0811852\n-0.87445, -7.06526, 0.160207\n-0.812285, -7.07124, 0.35819\n...\nThe final lines are comment lines which give timing information.\n# Elapsed Time: 0.016000 seconds (Pathfinders)\n#               0.003000 seconds (PSIS)\n#               0.019000 seconds (Total)\nPathfinder provides option save_single_paths which will save output from the single-path Pathfinder runs.",
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@@ -4705,150 +4611,205 @@
     "section": "",
     "text": "The boolean option save_single_paths is used to save both the draws and the ELBO iterations from the individual Pathfinder runs. When save_single_paths is \\(1\\), the draws from each are saved to StanCSV files with the same format as the PSIS sample and the ELBO evaluations along the L-BFGS trajectory for each are saved as JSON. Given an output file name, CmdStan adds suffixes to the base filename to distinguish between the output files. For the default output file name output.csv and default number of runs (4), the resulting CSV files are\noutput.csv\noutput_path_1.csv\noutput_path_1.json\noutput_path_2.csv\noutput_path_2.json\noutput_path_3.csv\noutput_path_3.json\noutput_path_4.csv\noutput_path_4.json\nThe individual sample CSV files have the same structure as the PSIS sample CSV file. The JSON files contain information from each ELBO iteration.\nTo see how this works, we run Pathfinder on the centered-parameterization of the eight-schools model, where the posterior distribution has a funnel shape:\n&gt; eight_schools pathfinder save_single_paths=1 data file=eight_schools.data.json\nEach JSON file records the approximations to the target density at each point along the trajectory of the L-BFGS optimization algorithms.\n{\n  \"0\": {\n    \"iter\": 0,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445]\n  },\n  \"1\": {\n    \"iter\": 1,\n    \"unconstrained_parameters\": [1.00595, -0.503687, 1.79367, 0.99083, 0.498077, -0.65816, 1.49176, -1.22647, 1.62911, 0.767445],\n    \"grads\": [-0.868919, 0.45198, -0.107675, -0.0123304, 0.163172, 0.354362, -0.108746, 0.673306, -0.102268, -4.51445],\n    \"history_size\": 1,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [0.126047, -0.065048, 1.55708, 0.958509, 0.628075, -0.217041, 1.32032, -0.561338, 1.42988, 1.23213],\n    \"logDetCholHk\": -2.6839,\n    \"L_approx\": [[-0.0630456, -0.0187959], [0, 1.08328]],\n    \"Qk\": [[-0.361073, 0.5624], [0.183922, -0.279474], [-0.0708175, 0.15715], [-0.00917823, 0.0215802], [0.0606019, -0.0814513], [0.164071, -0.285769], [-0.057723, 0.112428], [0.276376, -0.424348], [-0.0620524, 0.131786], [-0.846488, -0.531094]],\n    \"alpha\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n    \"full\": false,\n    \"lbfgs_note\": \"\"\n  },\n  ...,\n  \"171\": {\n    \"iter\": 171,\n    \"unconstrained_parameters\": [1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, 1.60479, -35.7821],\n    \"grads\": [2.66927e+15, -0.117312, -0.0639521, -2.66927e+15, -0.0445885, 0.0321579, 0.00499827, -0.163952, -0.032084, 6.4073],\n    \"history_size\": 5,\n    \"lbfgs_success\": true,\n    \"pathfinder_success\": true,\n    \"x_center\": [5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, 5.58876e+15, -2.02979e+17],\n    \"logDetCholHk\": 299.023,\n    \"L_approx\": [[4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, 4.6852e+06, -1.70162e+08], [0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 2.19511e+13, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 2.19511e+13, -7.97244e+14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2.89552e+16]],\n    \"Qk\": [],\n    \"alpha\": [1.11027e-12, 2.24669e-12, 2.05603e-12, 3.71177e-12, 5.7855e-12, 1.80169e-12, 3.40291e-12, 2.29699e-12, 3.43423e-12, 1.25815e-08],\n    \"full\": true,\n    \"lbfgs_note\": \"\"\n  },\n  \"172\": {\n    \"iter\": 172,\n    \"unconstrained_parameters\": [1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, 1.60531, -35.801],\n    \"grads\": [-0, -0.11731, -0.0639469, 0.0179895, -0.0445842, 0.0321643, 0.00500256, -0.163947, -0.0320824, 7],\n    \"history_size\": 5,\n    \"lbfgs_success\": false,\n    \"pathfinder_success\": false,\n    \"lbfgs_note\": \"\"\n  }\n}\nOption num_paths=1 runs one single-path Pathfinder and output CSV file contains the draws from that run without PSIS reweighting. The combination of arguments num_paths=1 save_single_paths=1 creates just two output files, the CSV sample and the set of ELBO iterations. In this case, the default output file name is “output.csv” and the default diagnostic file name is “output.json”.",
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   {
-    "objectID": "cmdstan-guide/print.html",
-    "href": "cmdstan-guide/print.html",
-    "title": "print (deprecated): MCMC Output Analysis",
+    "objectID": "cmdstan-guide/rdump_apdx.html",
+    "href": "cmdstan-guide/rdump_apdx.html",
+    "title": "RDump Format for CmdStan",
     "section": "",
-    "text": "print (deprecated): MCMC Output Analysis\nThe print utility is deprecated, but is still available until CmdStan v3.0. It has been replaced by the stansummary utility.\n\n\n\n\n Back to top",
+    "text": "NOTE: Although the RDump format is still supported, I/O with JSON is faster and recommended. See the chapter on JSON for more details.\nRDump format can be used to represent values for Stan variables. This format was introduced in SPLUS and is used in R, JAGS, and in BUGS (but with a different ordering).\nA dump file is structured as a sequence of variable definitions. Each variable is defined in terms of its dimensionality and its values. There are three kinds of variable declarations: - scalars - sequences - general arrays\n\n\nDump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.\n\n\n\nA simple scalar value can be thought of as having an empty list of dimensions. Its declaration in the dump format follows the SPLUS assignment syntax. For example, the following would constitute a valid dump file defining a single scalar variable y with value \\(17.2\\):\ny &lt;- 17.2\n\n\n\nOne-dimensional arrays may be specified directly using the SPLUS sequence notation. The following example defines an integer-value and a real-valued sequence.\nn &lt;- c(1,2,3) y &lt;- c(2.0,3.0,9.7)\nArrays are provided without a declaration of dimensionality because the reader just counts the number of entries to determine the size of the array.\nSequence variables may alternatively be represented with R’s colon-based notation. For instance, the first example above could equivalently be written as\nn &lt;- 1:3\nThe sequence denoted by 1:3 is of length \\(3\\), running from \\(1\\) to \\(3\\) inclusive. The colon notation allows sequences going from high to low. The following are equivalent:\nn &lt;- 2:-2\nn &lt;- c(2,1,0,-1,-2)\nAs a special case, a sequence of zeros can also be represented in the dump format by integer(x) and double(x), for type int and double, respectively. Here x is a non-negative integer to specify the length. If x is \\(0\\), it can be omitted. The following are some examples.\nx1 &lt;- integer()\nx2 &lt;- integer(0)\nx3 &lt;- integer(2)\ny1 &lt;- double()\ny2 &lt;- double(0)\ny3 &lt;- double(2)\n\n\n\nFor more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))\n\n\n\nThe dump format for matrices and vectors, including arrays of matrices and vectors, is the same as that for arrays of the same shape.\n\n\nThe following three declarations have the same dump format for their data.\narray[K] real a;\nvector[K] b;\nrow_vector[K] c;\n\n\n\nThe following declarations have the same dump format.\narray[M, N] real a;\nmatrix[M, N] b;\n\n\n\nThe key to understanding arrays is that the array indexing comes before any of the container indexing. That is, an array of vectors is just that: each array element is a vector. See the chapter on array and matrix types in the user’s guide section of the language manual for more information.\nFor the dump data format, the following declarations have the same arrangement.\narray[M, N] real a;\nmatrix[M, N] b;\narray[M] vector[N] c;\narray[M] row_vector[N] d;\nSimilarly, the following also have the same dump format.\narray[P, M, N] real a;\narray[P] matrix[M, N] b;\narray[P, M] vector[N] c;\narray[P, M] row_vector[N] d;\n\n\n\n\nAt this time, there is no support for complex number input through the R dump format. As an alternative, the JSON input format supports complex numbers.\n\n\n\nThere is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.\n\n\n\n\nIn order to support JAGS data files, variables may be double quoted. For instance, the following definition is legal in a dump file.\n\"y\" &lt;- c(1,2,3) \\end{Verbatim}\n\n\n\nThe line breaks in a dump file are required to be consistent with the way R reads in data. Both of the following declarations are legal.\ny &lt;- 2\ny &lt;-\n3\nAlso following R, breaking before the assignment arrow are not allowed, so the following is invalid.\ny\n&lt;- 2 # Syntax Error\nLines may also be broken in the middle of sequences declared using the c(...) notation., as well as between the comma following a sequence definition and the dimensionality declaration. 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-    "text": "The output from all CmdStan methods is in CSV format. A Stan CSV file is a data table where the columns are the method and model parameters and quantities of interest. Each row contains one record’s worth of data in plain-text format using the comma character (‘,’) as the field delimiter (hence the name).\nFor the Stan CSV files, data is strictly numerical, however, possible values include both positive and negative infinity and “Not-a-Number” which are represented as the strings NaN, inf, +inf, -inf. All other values are written in decimal notation with at most 6 digits of precision.\nStan CSV files have a header row containing the column names. They also make extensive use of CSV comments, i.e., lines which begin with the # character. In addition to initial and final comment rows, some methods also put comment rows in the middle of the data table, which makes it difficult to use many of the commonly used CSV parser packages.\n\n\nThe data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3\n\n\n\nThe sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the Quickstart Guide chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. It is possible that some profile statements have only one entry (if they were only executed by one thread) and others have multiple entries (if they were executed by multiple threads).\nThe columns are as follows:\n\nname, The name of the profile statement that is being timed\nthread_id, The thread that executed the profile statement\ntotal_time, The combined time spent executing statements inside the profile which includes calculation with and without automatic differentiation\nforward_time, The time spent in the profile statement during the forward pass of a reverse mode automatic differentiation calculation or during a calculation without automatic differentiation\nreverse_time, The time spent in the profile statement during the reverse (backward) pass of reverse mode automatic differentiation\nchain_stack, The number of objects allocated on the chaining automatic differentiation stack. There is a function call for each of these objects in the reverse pass\nno_chain_stack, The number of objects allocated on the non-chaining automatic differentiation stack\nautodiff_calls, The total number of times the profile statement was executed with automatic differentiation\nno_autodiff_calls - The total number of times the profile statement was executed without automatic differentiation\n\n\n\n\n\n\nConfig as comments\nHeader row\nPenalized maximum likelihood estimate\n\n\n\n\n\nConfig as comments\nHeader row\nAdaptation as comments\nVariational estimate\nSample draws from estimate of the posterior\n\n\n\n\n\nHeader row\nQuantities of interest\n\n\n\n\n\nHeader row\nGradients",
+    "text": "Dump files can be created from R using RStan, via the rstan package function stan_rdump. Stan RDump files must be created via stan_rdump and not by R’s native dump function because R’s dump function uses a richer syntax than is supported by the underlying Stan i/o libraries.",
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-    "text": "The data table is laid out with zero or more method-specific columns followed by the Stan program variables declared in the parameter block, then the variables in the transformed parameters block, finally variables declared in the generated quantities, in declaration order.\nStan provides three types of container objects: arrays, vectors, and matrices. In order to output all elements of a container object, it is necessary to choose an indexing notation and a serialization order. The Stan CSV file indexing notation is\n\nThe column name consists of the variable name followed by the element indices.\nIndices are delimited by periods (‘.’).\nIndexing is 1-based, i.e., given a dimension of size \\(N\\), the first element index is \\(1\\) and the last element index is \\(N\\).\nTuples are laid out element-by-element, with each tuple slot being delimited by a colon (‘:’).\n\nContainer variables are serialized in column major order, a.k.a. “Fortran” order. In column major-order, all elements of column 1 are listed in ascending order, followed by all elements of column 2, thus the first index changes the slowest and the last index changes the fastest.\nTo see how this works, consider a 3-dimensional variable with dimension sizes 2, 3, and 4, e.g., an array of matrices, a 2-D array of vectors or row_vectors, or a 3-D array of scalars. Given a Stan program with model parameter variable:\n array[2, 3, 4] real foo;\nThe Stan CSV file will require 24 columns to output the elements of foo. The first 6 columns will be labeled:\nfoo.1.1.1, foo.1.1.2, foo.1.1.3, foo.1.1.4, foo.1.2.1, foo.1.2.2\nThe final 6 columns will be labeled:\nfoo.2.2.3, foo.2.2.4, foo.2.3.1, foo.2.3.2, foo.2.3.3, foo.2.3.4\nTo see how a tuple would be laid out, consider the following variable:\ntuple(real, array[3] real) bar;\nThis will correspond to 4 columns in the CSV file, which are labeled\nbar:1,bar:2.1,bar:2.2,bar:2.3",
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-    "text": "The sample method produces both a Stan CSV output file and a diagnostic file which contains the sampler parameters together with the gradients on the unconstrained scale and log probabilities for all parameters in the model.\nTo see how this works, we show snippets of the output file resulting from the following command:\n./bernoulli sample save_warmup=1 num_warmup=200 num_samples=100 \\\n            data file=bernoulli.data.json \\\n            output file=bernoulli_samples.csv\n\n\nThe sampler output file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nComment rows for adaptation listing step size and metric used for sampling\nSampling draws\nComment rows giving timing information\n\nInitial comments rows: argument configuration\nAll configuration arguments are listed, one per line, indented according to CmdStan’s hierarchy of arguments and sub-arguments. Arguments not overtly specified on the command line are annotated as (Default).\nIn the above example the num_samples, num_warmup, and save_warmup arguments were specified, whereas subargument thin is left at its default value, as seen in the initial comment rows:\n# stan_version_major = 2\n# stan_version_minor = 24\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = sample (Default)\n#   sample\n#     num_samples = 100\n#     num_warmup = 200\n#     save_warmup = 1\n#     thin = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       gamma = 0.050000000000000003 (Default)\n#       delta = 0.80000000000000004 (Default)\n#       kappa = 0.75 (Default)\n#       t0 = 10 (Default)\n#       init_buffer = 75 (Default)\n#       term_buffer = 50 (Default)\n#       window = 25 (Default)\n#     algorithm = hmc (Default)\n#       hmc\n#         engine = nuts (Default)\n#           nuts\n#             max_depth = 10 (Default)\n#         metric = diag_e (Default)\n#         metric_file =  (Default)\n#         stepsize = 1 (Default)\n#         stepsize_jitter = 0 (Default)\n# id = 0 (Default)\n# data\n#   file = bernoulli.data.json\n# init = 2 (Default)\n# random\n#   seed = 2991989946 (Default)\n# output\n#   file = bernoulli_samples.csv\n#   diagnostic_file = bernoulli_diagnostics.csv\n#   refresh = 100 (Default)\nNote that when running multi-threaded programs which use reduce_sum for high-level parallelization, the number of threads used will also be included in this initial comment header.\nColumn headers\nThe CSV header row lists all sampler parameters, model parameters, transformed parameters, and quantities of interest. The sampler parameters are described in detail in the output file section of the Quickstart Guide chapter on MCMC Sampling. The example model bernoulli.stan only contains one parameter theta, therefore the CSV file data table consists of 7 sampler parameter columns and one column for the model parameter:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta\nAs a second example, we show the output of the eight_schools.stan model on run on example dataset. This model has 3 parameters: mu, theta a vector whose length is dependent on the input data, here N = 8, and tau. The initial columns are for the 7 sampler parameters, as before. The column headers for the model parameters are:\nmu,theta.1,theta.2,theta.3,theta.4,theta.5,theta.6,theta.7,theta.8,tau\nData rows containing warmup draws\nWhen run with option save_warmup=1, the thinned warmup draws are written to the CSV output file directly after the CSV header line. Since the default option is save_warmup=0, this section is usually not present in the output file.\nHere we specified num_warmup=200 and left thin at the default value \\(1\\), therefore the next 200 lines are data rows containing the sampler and model parameter values for each warmup draw.\n-6.74827,1,1,1,1,0,6.75348,0.247195\n-6.74827,4.1311e-103,14.3855,1,1,0,6.95087,0.247195\n-6.74827,1.74545e-21,2.43117,1,1,0,7.67546,0.247195\n-6.77655,0.99873,0.239791,2,7,0,6.81982,0.280619\n-6.7552,0.999392,0.323158,1,3,0,6.79175,0.26517\nComment rows for adaptation\nDuring warmup, the sampler adjusts the stepsize and the metric. At the end warmup, the sampler outputs this information as comments.\n# Adaptation terminated\n# Step size = 0.813694\n# Diagonal elements of inverse mass matrix:\n# 0.592879\nAs the example bernoulli model only contains a single parameter, and as the default metric is diag_e, the inverse mass matrix is a \\(1 \\times 1\\) matrix, and the length of the diagonal vector is also \\(1\\).\nIn contrast, if we run the eight schools example model with metric dense_e, the adaptation comments section lists both the stepsize and the full \\(10 \\times 10\\) inverse mass matrix:\n# Adaptation terminated\n# Step size = 0.211252\n# Elements of inverse mass matrix:\n# 25.6389, 17.3379, 13.9455, 15.9036, 15.1953, 8.73729, 16.9486, 14.4231, 17.4969, 0.518757\n# 17.3379, 79.8719, 12.2989, -1.28006, 9.92895, -3.51622, 10.073, 22.0196, 19.8151, 4.71028\n# 13.9455, 12.2989, 36.1572, 12.8734, 11.9446, 9.09582, 9.74519, 10.9539, 12.1204, 0.211353\n# 15.9036, -1.28006, 12.8734, 59.9998, 10.245, 8.03461, 16.9754, 3.13443, 9.68292, -1.36097\n# 15.1953, 9.92895, 11.9446, 10.245, 43.548, 15.3403, 13.0537, 7.69818, 10.1093, 0.155245\n# 8.73729, -3.51622, 9.09582, 8.03461, 15.3403, 39.981, 12.7695, 1.16248, 6.13749, -2.08507\n# 16.9486, 10.073, 9.74519, 16.9754, 13.0537, 12.7695, 45.8884, 11.6074, 8.96413, -1.15946\n# 14.4231, 22.0196, 10.9539, 3.13443, 7.69818, 1.16248, 11.6074, 49.4083, 18.9169, 3.15661\n# 17.4969, 19.8151, 12.1204, 9.68292, 10.1093, 6.13749, 8.96413, 18.9169, 68.0228, 1.74104\n# 0.518757, 4.71028, 0.211353, -1.36097, 0.155245, -2.08507, -1.15946, 3.15661, 1.74104, 1.50433\nNote that when the sampler is run with arguments algorithm=fixed_param, this section will be missing.\nData rows containing sampling draws\nThe output file contains the values for the thinned set draws during sampling. Here we specified num_sampling=100 and left thin at the default value \\(1\\), therefore the next 100 lines are data rows containing the sampler and model parameter values for each sampling iteration.\n-8.76921,0.796814,0.813694,1,1,0,9.75854,0.535093\n-6.79143,0.979604,0.813694,1,3,0,9.13092,0.214431\n-6.79451,0.955359,0.813694,2,3,0,7.19149,0.289341\nTiming information\nUpon successful completion, the sampler writes timing information to the output CSV file as a series of final comment lines:\n#\n#  Elapsed Time: 0.005 seconds (Warm-up)\n#                0.002 seconds (Sampling)\n#                0.007 seconds (Total)\n#\n\n\n\nThe diagnostic file contains the following:\n\nInitial comment rows listing full CmdStan argument configuration.\nHeader row\nData rows containing warmup draws, if run with option save_warmup=1\nSampling draws\nComment rows giving timing information\n\nThe columns in this file contain, in order:\n\nall sampler parameters\nall model parameter estimates (on the unconstrained scale)\nthe latent Hamiltonian for each parameter\nthe gradient for each parameters\n\nThe labels for the latent Hamiltonian columns are the parameter column label with prefix p_ and the labels for the gradient columns are the parameter column label with prefix g_.\nThese are the column labels from the file bernoulli_diagnostic.csv:\nlp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,divergent__,energy__,theta,p_theta,g_theta\n\n\n\nThe profiling information is stored in a plain CSV format with no meta information in the comments.\nEach row represents timing information collected in a profile statement for a given thread. 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+    "text": "For more than one dimension, the dump format uses a dimensionality specification. For example, the following defines a \\(2 \\times 3\\) array:\ny &lt;- structure(c(1,2,3,4,5,6), .Dim = c(2,3))\nData is stored column-major, thus the values for y will be:\ny[1, 1] = 1\ny[1, 2] = 3\ny[1, 3] = 5\ny[2, 1] = 2\ny[2, 2] = 4\ny[2, 3] = 6\nThe structure keyword just wraps a sequence of values and a dimensionality declaration, which is itself just a sequence of non-negative integer values. The product of the dimensions must equal the length of the array.\nIf the values happen to form a contiguous sequence of integers, they may be written with colon notation. Thus the example above is equivalent to the following.\ny &lt;- structure(1:6, .Dim = c(2,3))\nSequence notation can be used within any call to the generic c() function in R. In the above example, c(2,3) could be written as c(2:3).\nThe generalization of column-major indexing is last-index major indexing. Arrays of more than two dimensions are written in a last-index major form. For example,\nz &lt;- structure(1:24, .Dim = c(2,3,4))\nproduces a three-dimensional int (assignable to real) array z with values:\nz[1, 1, 1] = 1\nz[2, 1, 1] = 2\nz[1, 2, 1] = 3\nz[2, 2, 1] = 4\nz[1, 3, 1] = 5\nz[2, 3, 1] = 6\nz[1, 1, 2] = 7\nz[2, 1, 2] = 8\nz[1, 2, 2] = 9\nz[2, 2, 2] = 10\nz[1, 3, 2] = 11\nz[2, 3, 2] = 12\nz[1, 1, 3] = 13\nz[2, 1, 3] = 14\nz[1, 2, 3] = 15\nz[2, 2, 3] = 16\nz[1, 3, 3] = 17\nz[2, 3, 3] = 18\nz[1, 1, 4] = 19\nz[2, 1, 4] = 20\nz[1, 2, 4] = 21\nz[2, 2, 4] = 22\nz[1, 3, 4] = 23\nz[2, 3, 4] = 24\nIf the underlying 3-D array is stored as a 1-D array in last-index major format, the innermost array elements will be contiguous.\nThe sequence of values inside structure can also be integer(x) or double(x). In particular, if one or more dimensions is zero, integer() can be put inside structure. For instance, the following example is supported by the dump format.\ny &lt;- structure(integer(), .Dim = c(2,0))",
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+    "text": "There is no declaration in a dump file that distinguishes integer versus continuous values. If a value in a dump file’s definition of a variable contains a decimal point (e.g., \\(132.3\\)) or uses scientific notation (e.g., \\(1.323e2\\)), Stan assumes that the values are real.\nFor a single value, if there is no decimal point, it may be assigned to an int or real variable in Stan. An array value may only be assigned to an int array if there is no decimal point or scientific notation in any of the values. This convention is compatible with the way R writes data.\nThe following dump file declares an integer value for y.\ny &lt;- 2\nThis definition can be used for a Stan variable y declared as real or as int. Assigning an integer value to a real variable automatically promotes the integer value to a real value.\nInteger values may optionally be followed by L or l, denoting long integer values. The following example, where the type is explicit, is equivalent to the above.\ny &lt;- 2L\nThe following dump file provides a real value for y.\ny &lt;- 2.0\nEven though this is a round value, the occurrence of the decimal point in the value, \\(2.0\\), causes Stan to infer that y is real valued. This dump file may only be used for variables y declared as real in Stan.\n\n\nNumbers written in scientific notation may only be used for real values in Stan. R will write out the integer one million as \\(1e+06\\).\n\n\n\nStan’s reader supports infinite and not-a-number values for scalar quantities (see the section of the reference manual section of the language manual for more information on Stan’s numerical data types). Both infinite and not-a-number values are supported by Stan’s dump-format readers.\n\n\n\nValue\nPreferred Form\nAlternative Forms\n\n\n\n\npositive infinity\nInf\nInfinity, infinity\n\n\nnegative infinity\n-Inf\n-Infinity, -infinity\n\n\nnot a number\nNaN\n\n\n\n\nThese strings are not case sensitive, so inf may also be used for positive infinity, or NAN for not-a-number.",
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-    "text": "The CmdStan stansummary program reports statistics for one or more sampler chains over all sampler and model parameters and quantities of interest. The statistics reported include both summary statistics of the estimates and diagnostic statistics on the sampler chains, reported in the following order:\n\nMean - sample mean\nMCSE - Monte Carlo Standard Error, a measure of the amount of noise in the sample\nStdDev - sample standard deviation\nQuantiles - default 5%, 50%, 95%\nN_eff - effective sample size - the number of independent draws in the sample\nN_eff/S - the number of independent draws per second\nR_hat - \\(\\hat{R}\\) statistic, a measure of chain equilibrium, must be within \\(0.05\\) of \\(1.0\\).\n\nWhen reviewing the stansummary output, it is important to check the final three output columns first - these are the diagnostic statistics on chain convergence and number of independent draws in the sample. A \\(\\hat{R}\\) statistic of greater than \\(1.05\\) indicates that the chain has not converged and therefore the sample is not drawn from the posterior, thus the estimates of the mean and all other summary statistics are invalid.\nEstimation by sampling produces an approximate value for the model parameters; the MCSE statistic indicates the amount of noise in the estimate. Therefore MCSE column is placed next to the sample mean column, in order to make it easy to compare this sample with others.\nFor more information, see the Posterior Analysis chapter of the Stan Reference Manual which describes both the theory and practice of MCMC estimation techniques. The summary statistics and the algorithms used to compute them are described in sections Notation for samples and Effective Sample Size.\n\n\nThe CmdStan makefile task build compiles the stansummary utility into the bin directory. It can be compiled directly using the makefile as follows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; make bin/stansummary\n\n\n\nThe stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].\n\n\n\n\nThe stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
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+    "text": "A more precise definition of the dump data format is provided by the following (mildly templated) Backus-Naur form grammar.\ndefinition ::= name &lt;- value optional_semicolon\n\nname ::= char*     | ''' char* '''     | '\"' char* '\"'\n\nvalue ::= value&lt;int&gt; | value&lt;double&gt;\n\nvalue&lt;T&gt; ::= T       | seq&lt;T&gt;       | zero_array&lt;T&gt;       |\n'structure' '(' seq&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'       | 'structure'\n'(' zero_array&lt;T&gt; ',' \".Dim\" '=' seq&lt;int&gt; ')'\n\nseq&lt;int&gt; ::= int ':' int       | cseq&lt;int&gt;\n\nzero_array&lt;int&gt; ::= \"integer\" '(' &lt;non-negative int&gt;? ')'\n\nzero_array&lt;real&gt; ::= \"double\" '(' &lt;non-negative int&gt;? ')'\n\nseq&lt;real&gt; ::= cseq&lt;real&gt;\n\ncseq&lt;T&gt; ::= 'c' '(' vseq&lt;T&gt; ')'\n\nvseq&lt;T&gt; ::= T      | T ',' vseq&lt;T&gt;\nThe template parameters T will be set to either int or real. Because Stan allows promotion of integer values to real values, an integer sequence specification in the dump data format may be assigned to either an integer- or real-based variable in Stan.",
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+    "text": "CmdStan translates Stan programs to C++ using the Stan compiler program which is included in the CmdStan release bin directory as program stanc. One can view the complete stanc documentation in the Stan User’s Guide.\nAs of release 2.22, the CmdStan Stan to C++ compiler is written in OCaml. This compiler is called “stanc3” and has has its own repository https://github.com/stan-dev/stanc3, from which pre-built binaries for Linux, Mac, and Windows can be downloaded.\n\n\nBefore the Stan compiler can be used, the binary stanc must be created. This can be done using the makefile as follows. For Mac and Linux:\nmake bin/stanc\nFor Windows:\nmake bin/stanc.exe\nThis is also done as part of the make build command.\n\n\n\nThe Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
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-    "text": "The stansummary utility processes one or more output files from a set of chains from one run of the HMC sampler. To run stansummary on the output file or files generated by a run of the sampler, on Mac or Linux:\n&lt;cmdstan-home&gt;/bin/stansummary &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nOn Windows, use backslashes to call the stansummary.exe.\n&lt;cmdstan-home&gt;\\bin\\stansummary.exe &lt;file_1.csv&gt; ... &lt;file_N.csv&gt;\nFor example, after running 4 chains to fit the example model eight_schools.stan to the supplied example data file, we run stansummary on the resulting Stan CSV output files to get the following report:\n&gt; bin/stansummary eight_*.csv\nInput files: eight_1.csv, eight_2.csv, eight_3.csv, eight_4.csv\nInference for Stan model: eight_schools_model\n4 chains: each with iter=(1000,1000,1000,1000); warmup=(0,0,0,0); thin=(1,1,1,1); 4000 iterations saved.\n\nWarmup took (0.048, 0.060, 0.047, 0.045) seconds, 0.20 seconds total\nSampling took (0.057, 0.058, 0.061, 0.066) seconds, 0.24 seconds total\n\n                 Mean     MCSE  StdDev     5%   50%   95%  N_Eff  N_Eff/s    R_hat\n\nlp__              -18     0.33     5.1    -26   -19  -9.1    233      963      1.0\naccept_stat__    0.88  1.6e-02    0.23   0.21  0.98  1.00    203      838  1.0e+00\nstepsize__       0.18  2.2e-02   0.031   0.14  0.20  0.22    2.0      8.3  3.9e+13\ntreedepth__       3.8  5.9e-02    0.78    2.0   4.0   5.0    175      724  1.0e+00\nn_leapfrog__       18  1.3e+00     9.4    7.0    15    31     51      212  1.0e+00\ndivergent__     0.015  4.1e-03    0.12   0.00  0.00  0.00    865     3576  1.0e+00\nenergy__           23  3.4e-01     5.5     13    23    32    258     1066  1.0e+00\n\nmu                7.9     0.16     5.1  -0.23   7.9    16   1021     4221      1.0\ntheta[1]           12     0.30     8.6  -0.48    11    28    837     3459      1.0\ntheta[2]          7.8     0.15     6.4   -2.7   7.7    18   1717     7096     1.00\ntheta[3]          6.1     0.19     7.7   -6.5   6.5    18   1684     6958      1.0\ntheta[4]          7.5     0.15     6.7   -3.1   7.4    18   2026     8373      1.0\ntheta[5]          4.7     0.17     6.4   -6.7   5.3    15   1391     5747     1.00\ntheta[6]          5.9     0.16     6.7   -5.8   6.2    16   1673     6915     1.00\ntheta[7]           11     0.22     7.0  0.057    10    23   1069     4419      1.0\ntheta[8]          8.3     0.20     7.9   -4.2   8.0    22   1503     6209     1.00\ntau               7.2     0.26     5.2    1.5   5.9    17    401     1657      1.0\n\nSamples were drawn using hmc with nuts.\nFor each parameter, N_Eff is a crude measure of effective sample size,\nand R_hat is the potential scale reduction factor on split chains (at\nconvergence, R_hat=1).\nThe console output information consists of\n\nModel, chains, and timing summaries\nSampler parameter statistics\nModel parameter statistics\nSampling algorithm - either nuts (shown here) or static HMC.\n\nThere is one row per parameter and the row order in the summary report corresponds to the column order in the Stan CSV output file.\n\n\nThe initial Stan CSV columns provide information on the sampler state for each draw:\n\nlp__ - the total log probability density (up to an additive constant) at each sample\naccept_stat__ - the average Metropolis acceptance probability over each simulated Hamiltonian trajectory\nstepsize__ - integrator step size\ntreedepth__ - depth of tree used by NUTS (NUTS sampler)\nn_leapfrog__ - number of leapfrog calculations (NUTS sampler)\ndivergent__ - has value 1 if trajectory diverged, otherwise 0. (NUTS sampler)\nenergy__ - value of the Hamiltonian\nint_time__ - total integration time (static HMC sampler)\n\nBecause we ran the NUTS sampler, the above summary reports sampler parameters treedepth__, n_leapfrog__, and divergent__; the static HMC sampler would report int_time__ instead.\n\n\n\nThe remaining Stan CSV columns report the values of all parameters, transformed parameters, and generated quantities in the order in which these variables are declared in the Stan program. For container variables, i.e., vector, row_vector, matrix, and array variables, the statistics for each element are reported separately, in row-major order. The eight_schools.stan program parameters block contains the following parameter variable declarations:\n  real mu;\n  array[J] real theta;\n  real&lt;lower=0&gt; tau;\nIn the example data, J is \\(8\\); therefore the stansummary listing reports on theta[1] through theta[8].",
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-    "text": "The stansummary command syntax provides a set of flags to customize the output which must precede the list of filenames. When invoked with no arguments or with the -h or --help option, the program prints the usage message to the console and exits.\nReport statistics for one or more Stan CSV files from a HMC sampler run.\nExample:  stansummary model_chain_1.csv model_chain_2.csv\nOptions:\n  -a, --autocorr [n]          Display the chain autocorrelation for the n-th\n                              input file, in addition to statistics.\n  -c, --csv_filename [file]   Write statistics to a CSV file.\n  -h, --help                  Produce help message, then exit.\n  -p, --percentiles [values]  Percentiles to report as ordered set of\n                              comma-separated numbers from (0.1,99.9), inclusive.\n                              Default is 5,50,95.\n  -s, --sig_figs [n]          Significant figures reported. Default is 2.\n                              Must be an integer from (1, 18), inclusive.\n  -i, --include_param [name]  Include the named parameter in the summary output.\n                              By default, all parameters in the file are summarized,\n                              passing this argument one or more times will filter\n                              the output down to just the requested arguments.\nBoth short an long option names are allowed. Short names are specified as -&lt;o&gt; &lt;value&gt;; long option names can be specified either as --&lt;option&gt;=&lt;value&gt; or --&lt;option&gt; &lt;value&gt;.\nThe --percentiles argument can also be passed an empty string \"\", which results in no percentiles being displayed in the output of the command.\nThe amount of precision in the sampler output limits the amount of real precision in the summary report. CmdStan’s command line interface also has output argument sig_figs. The default sampler output precision is 6. The --sig_figs argument to the stansummary program should not exceed the sig_figs argument to the sampler.",
+    "text": "The Stan compiler program stanc converts Stan programs to C++ concepts. If the compiler encounters syntax errors in the program, it will provide an error message indicating the location in the input where the failure occurred and reason for the failure. The following example illustrates a fully qualified call to stanc to generate the C++ translation of the example model bernoulli.stan. For Linux and Mac:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nFor Windows:\n&gt; cd &lt;cmdstan-home&gt;\n&gt; bin/stanc.exe --o=bernoulli.hpp examples/bernoulli/bernoulli.stan\nThe base name of the Stan program file determines the name of the C++ model class. Because this name is the name of a C++ class, it must start with an alphabetic character (a--z or A--Z) and contain only alphanumeric characters (a--z, A--Z, and 0--9) and underscores (_) and should not conflict with any C++ reserved keyword.\nThe C++ code implementing the class is written to the file bernoulli.hpp in the current directory. The final argument, bernoulli.stan, is the file from which to read the Stan program.\nIn practice, stanc is invoked indirectly, via the GNU Make utility, which contains rules that compile a Stan program to its corresponding executable. To build the simple Bernoulli model via make, we specify the name of the target executable file. On Mac and Linux, this is the name of the Stan program with the .stan omitted. On Windows, replace .stan with .exe, and make sure that the path is given with slashes and not backslashes. For Linux and Mac:\n&gt; make examples/bernoulli/bernoulli\nFor Windows:\n&gt; make examples/bernoulli/bernoulli.exe\nThe makefile rules first invoke the stanc compiler to translate the Stan model to C++ , then compiles and links the C++ code to a binary executable. The makefile variable STANCFLAGS can be used to to override the default arguments to stanc, e.g.,\n&gt; make STANCFLAGS=\"--include-paths=~/foo\" examples/bernoulli/bernoulli",
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+    "text": "Stan implements an automatic variational inference algorithm, called Automatic Differentiation Variational Inference (ADVI) Kucukelbir et al. (2017). ADVI uses Monte Carlo integration to approximate the variational objective function, the ELBO (evidence lower bound). ADVI optimizes the ELBO in the real-coordinate space using stochastic gradient ascent. The measures of convergence are similar to the relative tolerance scheme of Stan’s optimization algorithms.\nThe algorithm progression consists of an adaptation phase followed by a sampling phase. The adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. The algorithm runs until the mean change in ELBO drops below the specified tolerance.\nThe full set of configuration options available for the variational method is available by using the variational help-all subcommand. The arguments with their requested values or defaults are also reported at the beginning of the algorithm’s console output and in the output CSV file’s comments.\nThe following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli variational data file=bernoulli.data.R\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used: the default option algorithm=meanfield and the default tolerances for monitoring the algorithm’s convergence.\nmethod = variational\n  variational\n    algorithm = meanfield (Default)\n      meanfield\n    iter = 10000 (Default)\n    grad_samples = 1 (Default)\n    elbo_samples = 100 (Default)\n    eta = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      iter = 50 (Default)\n    tol_rel_obj = 0.01 (Default)\n    eval_elbo = 100 (Default)\n    output_samples = 1000 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3323783840 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, informational and timing messages are output:\n------------------------------------------------------------\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n------------------------------------------------------------\n\nGradient evaluation took 2.1e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.\nAdjust your expectations accordingly!\nThe rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in \\(\\sim\\) 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.\nBegin eta adaptation.\nIteration:   1 / 250 [  0%]  (Adaptation)\nIteration:  50 / 250 [ 20%]  (Adaptation)\nIteration: 100 / 250 [ 40%]  (Adaptation)\nIteration: 150 / 250 [ 60%]  (Adaptation)\nIteration: 200 / 250 [ 80%]  (Adaptation)\nSuccess! Found best value [eta = 1] earlier than expected.\n\nBegin stochastic gradient ascent.\n  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes\n   100           -6.131             1.000            1.000\n   200           -6.458             0.525            1.000\n   300           -6.300             0.359            0.051\n   400           -6.137             0.276            0.051\n   500           -6.243             0.224            0.027\n   600           -6.305             0.188            0.027\n   700           -6.289             0.162            0.025\n   800           -6.402             0.144            0.025\n   900           -6.103             0.133            0.025\n  1000           -6.314             0.123            0.027\n  1100           -6.348             0.024            0.025\n  1200           -6.244             0.020            0.018\n  1300           -6.293             0.019            0.017\n  1400           -6.250             0.017            0.017\n  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED\n\nDrawing a sample of size 1000 from the approximate posterior...\nCOMPLETED.\n\n\nStan implements two variational algorithms. They differ in the approximating distribution used in the unconstrained variable space. By default, ADVI uses option algorithm=meanfield. The algorithm argument specifies the variational algorithm.\n\nalgorithm=meanfield - Use a fully factorized Gaussian for the approximation. This is the default algorithm.\nalgorithm=fullrank Use a Gaussian with a full-rank covariance matrix for the approximation.\n\n\n\n\n\niter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.\n\n\n\n\nThe output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv.\nThe output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext, the column header row:\nlp__,log_p__,log_g__,theta\nAdditional comments provide stepsize adaptation information:\n# Stepsize adaptation complete.\n# eta = 1\nFollowed by the data rows. The first line is special — it is the mean of the variational approximation.\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.",
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+    "text": "iter=&lt;int&gt; Maximum number of iterations. Must be \\(&gt; 0\\). Default is \\(10000\\).\ngrad_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of gradients. Must be \\(&gt; 0\\). Default is \\(1\\).\nelbo_samples=&lt;int&gt; Number of samples for Monte Carlo estimate of ELBO (objective function). Must be \\(&gt; 0\\). Default is \\(100\\).\neta=&lt;double&gt; Stepsize weighting parameter for adaptive stepsize sequence. Must be \\(&gt; 0\\). Default is \\(1.0\\).\nadapt Warmup Adaptation keyword, takes sub-arguments:\n\nengaged=&lt;boolean&gt; Adaptation engaged? Valid values: \\((0, 1)\\). Default is \\(1\\).\niter=&lt;int&gt; Maximum number of adaptation iterations. Must be \\(&gt; 0\\). Default is \\(50\\).\n\ntol_rel_obj=&lt;double&gt; Convergence tolerance on the relative norm of the objective. Must be \\(&gt; 0\\). Default is \\(0.01\\).\neval_elbo=&lt;int&gt; Evaluate ELBO every Nth iteration. Must be \\(&gt; 0\\). Default is 100.\noutput_samples=&lt;int&gt; Number of posterior samples to draw and save. Must be \\(&gt; 0\\). Default is 1000.",
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-    "text": "Variational Inference using ADVI\nThe CmdStan method variational uses the Automatic Differentiation Variational Inference (ADVI) algorithm of Kucukelbir et al. (2017) to provide an approximate posterior distribution of the model conditioned on the data. The approximating distribution it uses is a Gaussian in the unconstrained variable space, either a fully factorized Gaussian approximation, specified by argument algorithm=meanfield option, or a Gaussian approximation using a full-rank covariance matrix, specified by argument algorithm=fullrank. By default, ADVI uses option algorithm=meanfield.\nThe following is a minimal call to Stan’s variational inference algorithm using defaults for everything but the location of the data file.\n&gt; ./bernoulli variational data file=bernoulli.data.R\nExecuting this command prints both output to the console and to a csv file.\nThe first part of the console output reports on the configuration used: the default option algorithm=meanfield and the default tolerances for monitoring the algorithm’s convergence.\nmethod = variational\n  variational\n    algorithm = meanfield (Default)\n      meanfield\n    iter = 10000 (Default)\n    grad_samples = 1 (Default)\n    elbo_samples = 100 (Default)\n    eta = 1 (Default)\n    adapt\n      engaged = 1 (Default)\n      iter = 50 (Default)\n    tol_rel_obj = 0.01 (Default)\n    eval_elbo = 100 (Default)\n    output_samples = 1000 (Default)\nid = 0 (Default)\ndata\n  file = bernoulli.data.json\ninit = 2 (Default)\nrandom\n  seed = 3323783840 (Default)\noutput\n  file = output.csv (Default)\n  diagnostic_file =  (Default)\n  refresh = 100 (Default)\nAfter the configuration has been displayed, informational and timing messages are output:\n------------------------------------------------------------\nEXPERIMENTAL ALGORITHM:\n  This procedure has not been thoroughly tested and may be unstable\n  or buggy. The interface is subject to change.\n------------------------------------------------------------\n\nGradient evaluation took 2.1e-05 seconds\n1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.\nAdjust your expectations accordingly!\nThe rest of the output describes the progression of the algorithm. An adaptation phase finds a good value for the step size scaling parameter eta. The evidence lower bound (ELBO) is the variational objective function and is evaluated based on a Monte Carlo estimate. The variational inference algorithm in Stan is stochastic, which makes it challenging to assess convergence. That is, while the algorithm appears to have converged in \\(\\sim\\) 250 iterations, the algorithm runs for another few thousand iterations until mean change in ELBO drops below the default tolerance of 0.01.\nBegin eta adaptation.\nIteration:   1 / 250 [  0%]  (Adaptation)\nIteration:  50 / 250 [ 20%]  (Adaptation)\nIteration: 100 / 250 [ 40%]  (Adaptation)\nIteration: 150 / 250 [ 60%]  (Adaptation)\nIteration: 200 / 250 [ 80%]  (Adaptation)\nSuccess! Found best value [eta = 1] earlier than expected.\n\nBegin stochastic gradient ascent.\n  iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes \n   100           -6.131             1.000            1.000\n   200           -6.458             0.525            1.000\n   300           -6.300             0.359            0.051\n   400           -6.137             0.276            0.051\n   500           -6.243             0.224            0.027\n   600           -6.305             0.188            0.027\n   700           -6.289             0.162            0.025\n   800           -6.402             0.144            0.025\n   900           -6.103             0.133            0.025\n  1000           -6.314             0.123            0.027\n  1100           -6.348             0.024            0.025\n  1200           -6.244             0.020            0.018\n  1300           -6.293             0.019            0.017\n  1400           -6.250             0.017            0.017\n  1500           -6.241             0.015            0.010   MEDIAN ELBO CONVERGED\n\nDrawing a sample of size 1000 from the approximate posterior... \nCOMPLETED.\nThe output from variational is written into the file output.csv by default. The output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext is the column header line, followed more CSV comments reporting the adapted value for the stepsize, followed by the values. The first line is special: it is the mean of the variational approximation. The rest of the output contains output_samples number of samples drawn from the variational approximation.\nlp__,log_p__,log_g__,theta\n# Stepsize adaptation complete.\n# eta = 1\n0,0,0,0.236261\n0,-6.82318,-0.0929121,0.300415\n0,-6.89701,-0.158687,0.321982\n0,-6.99391,-0.23916,0.343643\n0,-7.35801,-0.51787,0.401554\n0,-7.4668,-0.539473,0.123081\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.\n\n\n\n\n\n Back to topReferences\n\nKucukelbir, Alp, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David M Blei. 2017. “Automatic Differentiation Variational Inference.” Journal of Machine Learning Research.",
+    "text": "The output file consists of the following pieces of information:\n\nThe full set of configuration options available for the variational method is reported at the beginning of the sampler output file as CSV comments.\nThe first three output columns are labelled lp__, log_p__, log_g__, the rest are the model parameters.\nThe stepsize adaptation information is output as CSV comments following column header row.\nThe following line contains the mean of the variational approximation.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\n\nTo illustrate, we call Stan’s variational inference on the example model and data:\n&gt; ./bernoulli variational data file=bernoulli.data.R\nBy default, the output file is output.csv.\nThe output follows the same pattern as the output for sampling, first dumping the entire set of parameters used as CSV comments:\n# stan_version_major = 2\n# stan_version_minor = 23\n# stan_version_patch = 0\n# model = bernoulli_model\n# method = variational\n#   variational\n#     algorithm = meanfield (Default)\n#       meanfield\n#     iter = 10000 (Default)\n#     grad_samples = 1 (Default)\n#     elbo_samples = 100 (Default)\n#     eta = 1 (Default)\n#     adapt\n#       engaged = 1 (Default)\n#       iter = 50 (Default)\n#     tol_rel_obj = 0.01 (Default)\n#     eval_elbo = 100 (Default)\n#     output_samples = 1000 (Default)\n...\nNext, the column header row:\nlp__,log_p__,log_g__,theta\nAdditional comments provide stepsize adaptation information:\n# Stepsize adaptation complete.\n# eta = 1\nFollowed by the data rows. The first line is special — it is the mean of the variational approximation.\n0,0,0,0.214911\nThat is, the estimate for theta given the data is 0.2.\nThe rest of the output contains output_samples number of samples drawn from the variational approximation.\nThe following is a sample based on this approximation:\n0,-14.0252,-5.21718,0.770397\n0,-7.05063,-0.10025,0.162061\n0,-6.75031,-0.0191099,0.241606\n...\nThe header indicates the unnormalized log probability with lp__. This is a legacy feature that we do not use for variational inference. The ELBO is not stored unless a diagnostic option is given.",
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-    "text": "Stan’s diagnostic mode runs a Stan program with data, initializing parameters either randomly or with user-specified initial values, and then evaluates the log probability and its gradients. The gradients computed by the Stan program are compared to values calculated by finite differences.\nDiagnostic mode may be configured with two parameters.\nDiagnostic Mode Configuration Table. The diagnostic model configuration parameters, constraints, and default values.\n\n\n\n\n\n\n\n\n\nparameter\ndescription\nconstraints\ndefault\n\n\n\n\nepsilon\nfinite difference size\n(0, infty)\n1e–6\n\n\nerror\nerror threshold for matching\n(0, infty)\n1e–6\n\n\n\nIf the difference between the Stan program’s gradient value and that calculated by finite difference is higher than the specified threshold, the argument will be flagged.\n\n\nDiagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.\n\n\n\n\nThe general configuration options for diagnostics are the same as those for MCMC. Initial values may be specified, or they may be drawn at random. Setting the random number generator will only have an effect if a random initialization is specified.\n\n\n\nDue to the application of finite differences, the computation time grows linearly with the number of parameters. This can be require a very long time, especially in models with latent parameters that grow with the data size. It can be helpful to diagnose a model with smaller data sizes in such cases.",
+    "text": "Stan’s diagnostic mode runs a Stan program with data, initializing parameters either randomly or with user-specified initial values, and then evaluates the log probability and its gradients. The gradients computed by the Stan program are compared to values calculated by finite differences.\nDiagnostic mode may be configured with two parameters.\nDiagnostic Mode Configuration Table. The diagnostic model configuration parameters, constraints, and default values.\n\n\n\n\n\n\n\n\n\nparameter\ndescription\nconstraints\ndefault\n\n\n\n\nepsilon\nfinite difference size\n(0, infty)\n1e–6\n\n\nerror\nerror threshold for matching\n(0, infty)\n1e–6\n\n\n\nIf the difference between the Stan program’s gradient value and that calculated by finite difference is higher than the specified threshold, the argument will be flagged.\n\n\nDiagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.\n\n\n\n\nThe general configuration options for diagnostics are the same as those for MCMC. Initial values may be specified, or they may be drawn at random. Setting the random number generator will only have an effect if a random initialization is specified.\n\n\n\nDue to the application of finite differences, the computation time grows linearly with the number of parameters. This can be require a very long time, especially in models with latent parameters that grow with the data size. It can be helpful to diagnose a model with smaller data sizes in such cases.",
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-    "text": "Diagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms will be turned off if optimization is used in practice, but there is as of yet no way to turn it off in diagnostic mode.",
+    "text": "Diagnostic mode prints the log posterior density (up to a proportion) calculated by the Stan program for the specified initial values. For each parameter, it prints the gradient at the initial parameter values calculated by Stan’s program and by finite differences over Stan’s program for the log probability.\n\n\nThe output is for the variable values and their gradients are on the unconstrained scale, which means each variable is a vector of size corresponding to the number of unconstrained variables required to define it. For example, an \\(N \\times N\\) correlation matrix, requires \\(\\binom{N}{2}\\) unconstrained parameters. The transformations from constrained to unconstrained parameters are based on the constraints in the parameter declarations and described in the reference manual chapter on transforms.\n\n\n\nThe log density includes the Jacobian adjustment implied by the constraints declared on variables. The Jacobian adjustment for constrained parameter transforms may be turned off for optimization, but there is as of yet no way to turn it off in diagnostic mode.",
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     "title": "Program Execution",
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-    "text": "This chapter provides a sketch of how a compiled Stan model is executed using sampling. Optimization shares the same data reading and initialization steps, but then does optimization rather than sampling.\nThis sketch is elaborated in the following chapters of this part, which cover variable declarations, expressions, statements, and blocks in more detail.\n\n\nThe reading and transforming data steps are the same for sampling, optimization and diagnostics.\n\n\nThe first step of execution is to read data into memory. Data may be read in through file (in CmdStan) or through memory (RStan and PyStan); see their respective manuals for details.1\nAll of the variables declared in the data block will be read. If a variable cannot be read, the program will halt with a message indicating which data variable is missing.\nAfter each variable is read, if it has a declared constraint, the constraint is validated. For example, if a variable N is declared as int&lt;lower=0&gt;, after N is read, it will be tested to make sure it is greater than or equal to zero. If a variable violates its declared constraint, the program will halt with a warning message indicating which variable contains an illegal value, the value that was read, and the constraint that was declared.\n\n\n\nAfter data is read into the model, the transformed data variable statements are executed in order to define the transformed data variables. As the statements execute, declared constraints on variables are not enforced.\nTransformed data variables are initialized with real values set to NaN and integer values set to the smallest integer (large absolute value negative number).\nAfter the statements are executed, all declared constraints on transformed data variables are validated. If the validation fails, execution halts and the variable’s name, value and constraints are displayed.\n\n\n\n\nInitialization is the same for sampling, optimization, and diagnosis\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.\n\n\n\n\nSampling is based on simulating the Hamiltonian of a particle with a starting position equal to the current parameter values and an initial momentum (kinetic energy) generated randomly. The potential energy at work on the particle is taken to be the negative log (unnormalized) total probability function defined by the model. In the usual approach to implementing HMC, the Hamiltonian dynamics of the particle is simulated using the leapfrog integrator, which discretizes the smooth path of the particle into a number of small time steps called leapfrog steps.\n\n\nFor each leapfrog step, the negative log probability function and its gradient need to be evaluated at the position corresponding to the current parameter values (a more detailed sketch is provided in the next section). These are used to update the momentum based on the gradient and the position based on the momentum.\nFor simple models, only a few leapfrog steps with large step sizes are needed. For models with complex posterior geometries, many small leapfrog steps may be needed to accurately model the path of the parameters.\nIf the user specifies the number of leapfrog steps (i.e., chooses to use standard HMC), that number of leapfrog steps are simulated. If the user has not specified the number of leapfrog steps, the No-U-Turn sampler (NUTS) will determine the number of leapfrog steps adaptively (Hoffman and Gelman 2014).\n\n\n\nDuring each leapfrog step, the log probability function and its gradient must be calculated. This is where most of the time in the Stan algorithm is spent. This log probability function, which is used by the sampling algorithm, is defined over the unconstrained parameters.\nThe first step of the calculation requires the inverse transform of the unconstrained parameter values back to the constrained parameters in terms of which the model is defined. There is no error checking required because the inverse transform is a total function on every point in whose range satisfies the constraints.\nBecause the probability statements in the model are defined in terms of constrained parameters, the log Jacobian of the inverse transform must be added to the accumulated log probability.\nNext, the transformed parameter statements are executed. After they complete, any constraints declared for the transformed parameters are checked. If the constraints are violated, the model will halt with a diagnostic error message.\nThe final step in the log probability function calculation is to execute the statements defined in the model block.\nAs the log probability function executes, it accumulates an in-memory representation of the expression tree used to calculate the log probability. This includes all of the transformed parameter operations and all of the Jacobian adjustments. This tree is then used to evaluate the gradients by propagating partial derivatives backward along the expression graph. The gradient calculations account for the majority of the cycles consumed by a Stan program.\n\n\n\nA standard Metropolis accept/reject step is required to retain detailed balance and ensure draws are marginally distributed according to the probability function defined by the model. This Metropolis adjustment is based on comparing log probabilities, here defined by the Hamiltonian, which is the sum of the potential (negative log probability) and kinetic (squared momentum) energies. In theory, the Hamiltonian is invariant over the path of the particle and rejection should never occur. In practice, the probability of rejection is determined by the accuracy of the leapfrog approximation to the true trajectory of the parameters.\nIf step sizes are small, very few updates will be rejected, but many steps will be required to move the same distance. If step sizes are large, more updates will be rejected, but fewer steps will be required to move the same distance. Thus a balance between effort and rejection rate is required. If the user has not specified a step size, Stan will tune the step size during warmup sampling to achieve a desired rejection rate (thus balancing rejection versus number of steps).\nIf the proposal is accepted, the parameters are updated to their new values. Otherwise, the sample is the current set of parameter values.\n\n\n\n\nOptimization runs very much like sampling in that it starts by reading the data and then initializing parameters. Unlike sampling, it produces a deterministic output which requires no further analysis other than to verify that the optimizer itself converged to a posterior mode. The output for optimization is also similar to that for sampling.\n\n\n\nVariational inference also runs similar to sampling. It begins by reading the data and initializing the algorithm. The initial variational approximation is a random draw from the standard normal distribution in the unconstrained (real-coordinate) space. Again, similar to sampling, it outputs draws from the approximate posterior once the algorithm has decided that it has converged. Thus, the tools we use for analyzing the result of Stan’s sampling routines can also be used for variational inference.\n\n\n\nModel diagnostics are like sampling and optimization in that they depend on a model’s data being read and its parameters being initialized. The user’s guides for the interfaces (RStan, PyStan, CmdStan) provide more details on the diagnostics available; as of Stan 2.0, that’s just gradients on the unconstrained scale and log probabilities.\n\n\n\nFor each final draw (not counting draws during warmup or draws that are thinned), there is an output stage of writing the draw.\n\n\nBefore generating any output, the statements in the generated quantities block are executed. This can be used for any forward simulation based on parameters of the model. Or it may be used to transform parameters to an appropriate form for output.\nAfter the generated quantities statements execute, the constraints declared on generated quantities variables are validated. If these constraints are violated, the program will terminate with a diagnostic message.\n\n\n\nThe final step is to write the actual values. The values of all variables declared as parameters, transformed parameters, or generated quantities are written. Local variables are not written, nor is the data or transformed data. All values are written in their constrained forms, that is the form that is used in the model definitions.\nIn the executable form of a Stan models, parameters, transformed parameters, and generated quantities are written to a file in comma-separated value (CSV) notation with a header defining the names of the parameters (including indices for multivariate parameters).2",
+    "text": "This chapter provides a sketch of how a compiled Stan model is executed using sampling. Optimization shares the same data reading and initialization steps, but then does optimization rather than sampling.\nThis sketch is elaborated in the following chapters of this part, which cover variable declarations, expressions, statements, and blocks in more detail.\n\n\nThe reading and transforming data steps are the same for sampling, optimization and diagnostics.\n\n\nThe first step of execution is to read data into memory. Data may be read in through file (in CmdStan) or through memory (RStan and PyStan); see their respective manuals for details.1\nAll of the variables declared in the data block will be read. If a variable cannot be read, the program will halt with a message indicating which data variable is missing.\nAfter each variable is read, if it has a declared constraint, the constraint is validated. For example, if a variable N is declared as int&lt;lower=0&gt;, after N is read, it will be tested to make sure it is greater than or equal to zero. If a variable violates its declared constraint, the program will halt with a warning message indicating which variable contains an illegal value, the value that was read, and the constraint that was declared.\n\n\n\nAfter data is read into the model, the transformed data variable statements are executed in order to define the transformed data variables. As the statements execute, declared constraints on variables are not enforced.\nTransformed data variables are initialized with real values set to NaN and integer values set to the smallest integer (large absolute value negative number).\nAfter the statements are executed, all declared constraints on transformed data variables are validated. If the validation fails, execution halts and the variable’s name, value and constraints are displayed.\n\n\n\n\nInitialization is the same for all of Stan’s algorithms.\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.\n\n\n\n\nSampling is based on simulating the Hamiltonian of a particle with a starting position equal to the current parameter values and an initial momentum (kinetic energy) generated randomly. The potential energy at work on the particle is taken to be the negative log (unnormalized) total probability function defined by the model. In the usual approach to implementing HMC, the Hamiltonian dynamics of the particle is simulated using the leapfrog integrator, which discretizes the smooth path of the particle into a number of small time steps called leapfrog steps.\n\n\nFor each leapfrog step, the negative log probability function and its gradient need to be evaluated at the position corresponding to the current parameter values (a more detailed sketch is provided in the next section). These are used to update the momentum based on the gradient and the position based on the momentum.\nFor simple models, only a few leapfrog steps with large step sizes are needed. For models with complex posterior geometries, many small leapfrog steps may be needed to accurately model the path of the parameters.\nIf the user specifies the number of leapfrog steps (i.e., chooses to use standard HMC), that number of leapfrog steps are simulated. If the user has not specified the number of leapfrog steps, the No-U-Turn sampler (NUTS) will determine the number of leapfrog steps adaptively (Hoffman and Gelman 2014).\n\n\n\nDuring each leapfrog step, the log probability function and its gradient must be calculated. This is where most of the time in the Stan algorithm is spent. This log probability function, which is used by the sampling algorithm, is defined over the unconstrained parameters.\nThe first step of the calculation requires the inverse transform of the unconstrained parameter values back to the constrained parameters in terms of which the model is defined. There is no error checking required because the inverse transform is a total function on every point in whose range satisfies the constraints.\nBecause the probability statements in the model are defined in terms of constrained parameters, the log Jacobian of the inverse transform must be added to the accumulated log probability.\nNext, the transformed parameter statements are executed. After they complete, any constraints declared for the transformed parameters are checked. If the constraints are violated, the model will halt with a diagnostic error message.\nThe final step in the log probability function calculation is to execute the statements defined in the model block.\nAs the log probability function executes, it accumulates an in-memory representation of the expression tree used to calculate the log probability. This includes all of the transformed parameter operations and all of the Jacobian adjustments. This tree is then used to evaluate the gradients by propagating partial derivatives backward along the expression graph. The gradient calculations account for the majority of the cycles consumed by a Stan program.\n\n\n\nA standard Metropolis accept/reject step is required to retain detailed balance and ensure draws are marginally distributed according to the probability function defined by the model. This Metropolis adjustment is based on comparing log probabilities, here defined by the Hamiltonian, which is the sum of the potential (negative log probability) and kinetic (squared momentum) energies. In theory, the Hamiltonian is invariant over the path of the particle and rejection should never occur. In practice, the probability of rejection is determined by the accuracy of the leapfrog approximation to the true trajectory of the parameters.\nIf step sizes are small, very few updates will be rejected, but many steps will be required to move the same distance. If step sizes are large, more updates will be rejected, but fewer steps will be required to move the same distance. Thus a balance between effort and rejection rate is required. If the user has not specified a step size, Stan will tune the step size during warmup sampling to achieve a desired rejection rate (thus balancing rejection versus number of steps).\nIf the proposal is accepted, the parameters are updated to their new values. Otherwise, the sample is the current set of parameter values.\n\n\n\n\nOptimization runs very much like sampling in that it starts by reading the data and then initializing parameters. Unlike sampling, it produces a deterministic output which requires no further analysis other than to verify that the optimizer itself converged to a posterior mode. The output for optimization is also similar to that for sampling.\n\n\n\nVariational inference also runs similar to sampling. It begins by reading the data and initializing the algorithm. The initial variational approximation is a random draw from the standard normal distribution in the unconstrained (real-coordinate) space. Again, similar to sampling, it outputs draws from the approximate posterior once the algorithm has decided that it has converged. Thus, the tools we use for analyzing the result of Stan’s sampling routines can also be used for variational inference.\n\n\n\nModel diagnostics are like sampling and optimization in that they depend on a model’s data being read and its parameters being initialized. The user’s guides for the interfaces (RStan, PyStan, CmdStan) provide more details on the diagnostics available; as of Stan 2.0, that’s just gradients on the unconstrained scale and log probabilities.\n\n\n\nFor each final draw (not counting draws during warmup or draws that are thinned), there is an output stage of writing the draw.\n\n\nBefore generating any output, the statements in the generated quantities block are executed. This can be used for any forward simulation based on parameters of the model. Or it may be used to transform parameters to an appropriate form for output.\nAfter the generated quantities statements execute, the constraints declared on generated quantities variables are validated. If these constraints are violated, the program will terminate with a diagnostic message.\n\n\n\nThe final step is to write the actual values. The values of all variables declared as parameters, transformed parameters, or generated quantities are written. Local variables are not written, nor is the data or transformed data. All values are written in their constrained forms, that is the form that is used in the model definitions.\nIn the executable form of a Stan models, parameters, transformed parameters, and generated quantities are written to a file in comma-separated value (CSV) notation with a header defining the names of the parameters (including indices for multivariate parameters).2",
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-    "text": "Initialization is the same for sampling, optimization, and diagnosis\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.",
+    "text": "Initialization is the same for all of Stan’s algorithms.\n\n\nIf there are user-supplied initial values for parameters, these are read using the same input mechanism and same file format as data reads. Any constraints declared on the parameters are validated for the initial values. If a variable’s value violates its declared constraint, the program halts and a diagnostic message is printed.\nAfter being read, initial values are transformed to unconstrained values that will be used to initialize the sampler.\n\n\nBecause of the way Stan defines its transforms from the constrained to the unconstrained space, initializing parameters on the boundaries of their constraints is usually problematic. For instance, with a constraint\nparameters {\n  real&lt;lower=0, upper=1&gt; theta;\n  // ...\n}\nan initial value of 0 for theta leads to an unconstrained value of \\(-\\infty\\), whereas a value of 1 leads to an unconstrained value of \\(+\\infty\\). While this will be inverse transformed back correctly given the behavior of floating point arithmetic, the Jacobian will be infinite and the log probability function will fail and raise an exception.\n\n\n\n\nIf there are no user-supplied initial values, the default initialization strategy is to initialize the unconstrained parameters directly with values drawn uniformly from the interval \\((-2,2)\\). The bounds of this initialization can be changed but it is always symmetric around 0. The value of 0 is special in that it represents the median of the initialization. An unconstrained value of 0 corresponds to different parameter values depending on the constraints declared on the parameters.\nAn unconstrained real does not involve any transform, so an initial value of 0 for the unconstrained parameters is also a value of 0 for the constrained parameters.\nFor parameters that are bounded below at 0, the initial value of 0 on the unconstrained scale corresponds to \\(\\exp(0) = 1\\) on the constrained scale. A value of -2 corresponds to \\(\\exp(-2) = .13\\) and a value of 2 corresponds to \\(\\exp(2) = 7.4\\).\nFor parameters bounded above and below, the initial value of 0 on the unconstrained scale corresponds to a value at the midpoint of the constraint interval. For probability parameters, bounded below by 0 and above by 1, the transform is the inverse logit, so that an initial unconstrained value of 0 corresponds to a constrained value of 0.5, -2 corresponds to 0.12 and 2 to 0.88. Bounds other than 0 and 1 are just scaled and translated.\nSimplexes with initial values of 0 on the unconstrained basis correspond to symmetric values on the constrained values (i.e., each value is \\(1/K\\) in a \\(K\\)-simplex).\nCholesky factors for positive-definite matrices are initialized to 1 on the diagonal and 0 elsewhere; this is because the diagonal is log transformed and the below-diagonal values are unconstrained.\nThe initial values for other parameters can be determined from the transform that is applied. The transforms are all described in full detail in the chapter on variable transforms.\n\n\n\nThe initial values may all be set to 0 on the unconstrained scale. This can be helpful for diagnosis, and may also be a good starting point for sampling. Once a model is running, multiple chains with more diffuse starting points can help diagnose problems with convergence; see the user’s guide for more information on convergence monitoring.",
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-    "text": "Stan allows one file to be included within another file using a syntax similar to that from C++. For example, suppose the file my-std-normal.stan defines the standard normal log probability density function (up to an additive constant).\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nSuppose we also have a file containing a Stan program with an include statement.\n#include my-std-normal.stan\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThis Stan program behaves as if the contents of the file my-std-normal.stan replace the line with the #include statement, behaving as if a single Stan program were provided.\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThere are no restrictions on where include statements may be placed within a file or what the contents are of the replaced file.\n\n\nIt is possible to use includes on a line non-initially. For example, the previous example could’ve included space before the # in the include line:\n    #include my-std-normal.stan\nparameters {\n// ...\nIf there is initial space before an include, it will be discarded.\n\n\n\nIt is also possible to include line-based comments after the include. For example, the previous example can be coded as:\n#include my-std-normal.stan  // definition of standard normal\nparameters {\n// ...\nLine comments are discarded when the entire line is replaced with the contents of the included file.\n\n\n\nRecursive includes will be ignored. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThe result of processing this file will be empty, because a.stan will include b.stan, from which the include of a.stan is ignored and a warning printed.\n\n\n\nThe Stan interfaces may provide a mechanism for specifying a sequence of system paths in which to search for include files. The file included is the first one that is found in the sequence.\n\n\nIf there is not a final / or \\ in the path, a / will be appended between the path and the included file name.",
+    "text": "Stan allows one file to be included within another file using a syntax similar to that from C++. For example, suppose the file my-std-normal.stan defines the standard normal log probability density function (up to an additive constant).\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nSuppose we also have a file containing a Stan program with an include statement.\n#include my-std-normal.stan\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThis Stan program behaves as if the contents of the file my-std-normal.stan replace the line with the #include statement, behaving as if a single Stan program were provided.\nfunctions {\n  real my_std_normal_lpdf(vector y) {\n    return -0.5 * y' * y;\n  }\n}\nparameters {\n  real y;\n}\nmodel {\n  y ~ my_std_normal();\n}\nThere are no restrictions on where include statements may be placed within a file or what the contents are of the replaced file.\n\n\nIt is possible to use includes on a line non-initially. For example, the previous example could’ve included space before the # in the include line:\n    #include my-std-normal.stan\nparameters {\n// ...\nIf there is initial space before an include, it will be discarded.\n\n\n\nIt is also possible to include line-based comments after the include. For example, the previous example can be coded as:\n#include my-std-normal.stan  // definition of standard normal\nparameters {\n// ...\nLine comments are discarded when the entire line is replaced with the contents of the included file.\n\n\n\nRecursive includes will lead to a compiler error. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThis will result in an error explaining the circular dependency:\nSyntax error in './b.stan', line 1, column 0, included from\n'./a.stan', line 1, column 0, included from\n'./b.stan', line 1, column 0, included from\n'a.stan', line 1, column 0, include error:\n   -------------------------------------------------\n     1:  #include a.stan\n         ^\n   -------------------------------------------------\n\nFile a.stan recursively included itself.\n\n\n\nThe Stan interfaces may provide a mechanism for specifying a sequence of system paths in which to search for include files. The file included is the first one that is found in the sequence.\n\n\nIf there is not a final / or \\ in the path, a / will be appended between the path and the included file name.",
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-    "text": "Recursive includes will be ignored. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThe result of processing this file will be empty, because a.stan will include b.stan, from which the include of a.stan is ignored and a warning printed.",
+    "text": "Recursive includes will lead to a compiler error. For example, suppose a.stan contains\n#include b.stan\nand b.stan contains\n#include a.stan\nThis will result in an error explaining the circular dependency:\nSyntax error in './b.stan', line 1, column 0, included from\n'./a.stan', line 1, column 0, included from\n'./b.stan', line 1, column 0, included from\n'a.stan', line 1, column 0, include error:\n   -------------------------------------------------\n     1:  #include a.stan\n         ^\n   -------------------------------------------------\n\nFile a.stan recursively included itself.",
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-    "text": "The blocks of a Stan program are made up of variable declarations and statements; see the blocks chapter for details. Unlike programs in BUGS, the declarations and statements making up a Stan program are executed in the order in which they are written. Variables must be defined to have some value (as well as declared to have some type) before they are used — if they do not, the behavior is undefined.\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters. Log probability function can be constructed by using assignment statements. Statements may be grouped into sequences and into for-each loops. In addition, Stan allows local variables to be declared in blocks and also allows an empty statement consisting only of a semicolon.\n\n\nThe data and parameters blocks do not allow statements of any kind because these blocks are solely used to declare the data variables for input and the parameter variables for sampling. All other blocks allow statements. In these blocks, both variable declarations and statements are allowed. All top-level variables in a block are considered block variables. See the blocks chapter for more information about the block structure of Stan programs.\n\n\n\nAn assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,\n\na variable, or\na variable with one or more indices.\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y\n\n\n\n\n\n\n\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.1\ntarget += -0.5 * y * y;\nThe keyword target here is actually not a variable, and may not be accessed as such (though see below on how to access the value of target through a special function).\nIn this example, the unnormalized log probability of a unit normal variable \\(y\\) is added to the total log probability. In the general case, the argument can be any expression.2\nAn entire Stan model can be implemented this way. For instance, the following model will draw a single variable according to a unit normal probability.\nparameters {\n  real y;\n}\nmodel {\n  target += -0.5 * y * y;\n}\nThis model defines a log probability function\n\\[\n\\log p(y) = - \\, \\frac{y^2}{2} - \\log Z\n\\]\nwhere \\(Z\\) is a normalizing constant that does not depend on \\(y\\). The constant \\(Z\\) is conventionally written this way because on the linear scale, \\[\np(y) = \\frac{1}{Z} \\exp\\left(-\\frac{y^2}{2}\\right).\n\\] which is typically written without reference to \\(Z\\) as \\[\np(y) \\propto \\exp\\left(-\\frac{y^2}{2}\\right).\n\\]\nStan only requires models to be defined up to a constant that does not depend on the parameters. This is convenient because often the normalizing constant \\(Z\\) is either time-consuming to compute or intractable to evaluate.\n\n\nThe built in distribution functions in Stan are all available in normalized and unnormalized form. The normalized forms include all of the terms in the log density, and the unnormalized forms drop terms which are not directly or indirectly a function of the model parameters.\nFor instance, the normal_lpdf function returns the log density of a normal distribution:\n\\[\n\\textsf{normal\\_lpdf}(x | \\mu, \\sigma) =\n-\\log \\left( \\sigma \\sqrt{2 \\pi} \\right)\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nThe normal_lupdf function returns the log density of an unnormalized distribution. With the unnormalized version of the function, Stan does not define what the normalization constant will be, though usually as many terms as possible are dropped to make the calculation fast. Dropping a constant sigma term, normal_lupdf would be equivalent to:\n\\[\n\\textsf{normal\\_lupdf}(x | \\mu, \\sigma) =\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nAll functions ending in _lpdf have a corresponding _lupdf version which evaluates and returns the unnormalized density. The same is true for _lpmf and _lupmf.\n\n\n\nThe increment log density statement looks syntactically like compound addition and assignment (see the compound arithmetic/assignment section, it is treated as a primitive statement because target is not itself a variable. So, even though\ntarget += lp;\nis a legal statement, the corresponding long form is not legal.\ntarget = target + lp;  // BAD, target is not a variable\n\n\n\nThe target += ... statement accepts an argument in place of ... for any expression type, including integers, reals, vectors, row vectors, matrices, and arrays of any dimensionality, including arrays of vectors and matrices. For container arguments, their sum will be added to the total log density.\n\n\n\nTo access accumulated log density up to the current execution point, the function target() may be used.\n\n\n\n\nStan supports writing probability statements also in sampling notation, such as\ny ~ normal(mu, sigma);\nThe name “sampling statement” is meant to be suggestive, not interpreted literally. Conceptually, the variable y, which may be an unknown parameter or known, modeled data, is being declared to have the distribution indicated by the right-hand side of the sampling statement. The above statement can read as “y is distributed as normal distribution with parameters mu and sigma.”. The symbol ~ is called tilde.\nExecuting such a statement does not perform any sampling. In Stan, a sampling statement is merely a notational convenience following the typical notation used to present models in the literature. The above sampling statement could be expressed as a direct increment on the total log probability density as\ntarget += normal_lpdf(y | mu, sigma);\nIn general, a sampling statement of the form\ny ~ dist(theta1, ..., thetaN);\ninvolving subexpressions y and theta1 through thetaN (including the case where N is zero) will be well formed if and only if the corresponding assignment statement is well-formed. For densities allowing real y values, the log probability density function is used,\ntarget += dist_lpdf(y | theta1, ..., thetaN);\nFor those restricted to integer y values, the log probability mass function is used,\ntarget += dist_lpmf(y | theta1, ..., thetaN);\nThis will be well formed if and only if dist_lpdf(y | theta1,   ..., thetaN) or dist_lpmf(y | theta1, ..., thetaN) is a well-formed expression of type real.\n\n\nAlthough both lead to the same sampling behavior in Stan, there is one critical difference between using the sampling statement, as in\ny ~ normal(mu, sigma);\nand explicitly incrementing the log probability function, as in\ntarget += normal_lpdf(y | mu, sigma);\nThe sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions, y, mu, or sigma, involve only constants, data variables, and transformed data variables.\n\n\n\nThe left-hand side of a sampling statement may be a complex expression. For instance, it is legal syntactically to write\nparameters {\n  real&lt;lower=0&gt; beta;\n}\n// ...\nmodel {\n  log(beta) ~ normal(mu, sigma);\n}\nUnfortunately, this is not enough to properly model beta as having a lognormal distribution. Whenever a nonlinear transform is applied to a parameter, such as the logarithm function being applied to beta here, and then used on the left-hand side of a sampling statement or on the left of a vertical bar in a log pdf function, an adjustment must be made to account for the differential change in scale and ensure beta gets the correct distribution. The correction required is to add the log Jacobian of the transform to the target log density; see the change of variables section for full definitions. For the case above, the following adjustment will account for the log transform.3\ntarget += - log(abs(y));\n\n\n\nStan supports truncating distributions with lower bounds, upper bounds, or both.\n\n\nA probability density function \\(p(x)\\) for a continuous distribution may be truncated to an interval \\([a, b]\\) to define a new density \\(p_{[a, b]}(x)\\) with support \\([a, b]\\) by setting\n\\[\np_{[a, b]}(x)\n= \\frac{p(x)}\n       {\\int_a^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) for a discrete distribution may be truncated to the closed interval \\([a, b]\\) by\n\\[\np_{[a, b]}(x) = \\frac{p(x)}\n                  {\\sum_{u = a}^b p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x)\n= \\frac{p(x)}\n       {\\int_a^{\\infty} p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x) = \\frac{p(x)}\n                  {\\sum_{a &lt;= u} p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty, b]}(x)\n= \\frac{p(x)}\n       {\\int_{-\\infty}^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty,b]}(x) = \\frac{p(x)}\n                  {\\sum_{u &lt;= b} p(u)}.\n\\]\n\n\n\nGiven a probability function \\(p_X(x)\\) for a random variable \\(X\\), its cumulative distribution function (cdf) \\(F_X(x)\\) is defined to be the probability that \\(X \\leq x\\),\n\\[\nF_X(x) = \\Pr[X \\leq x].\n\\]\nThe upper-case variable \\(X\\) is the random variable whereas the lower-case variable \\(x\\) is just an ordinary bound variable. For continuous random variables, the definition of the cdf works out to\n\\[\nF_X(x) \\ = \\ \\int_{-\\infty}^{x} p_X(u) \\, du,\n\\]\nFor discrete variables, the cdf is defined to include the upper bound given by the argument,\n\\[\nF_X(x) = \\sum_{u \\leq x} p_X(u).\n\\]\n\n\n\nThe complementary cumulative distribution function (ccdf) in both the continuous and discrete cases is given by\n\\[\nF^C_X(x)\n\\ = \\ \\Pr[X &gt; x]\n\\ = \\ 1 - F_X(x).\n\\]\nUnlike the cdf, the ccdf is exclusive of the bound, hence the event \\(X &gt; x\\) rather than the cdf’s event \\(X \\leq x\\).\nFor continuous distributions, the ccdf works out to\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\int_{-\\infty}^x p_X(u) \\, du\n\\ = \\ \\int_x^{\\infty} p_X(u) \\, du.\n\\]\nThe lower boundary can be included in the integration bounds because it is a single point on a line and hence has no probability mass. For the discrete case, the lower bound must be excluded in the summation explicitly by summing over \\(u &gt; x\\),\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\sum_{u \\leq x} p_X(u)\n\\ = \\ \\sum_{u &gt; x} p_X(u).\n\\]\nCumulative distribution functions provide the necessary integral calculations to define truncated distributions. For truncation with lower and upper bounds, the denominator is defined by \\[\n\\int_a^b p(u) \\, du = F_X(b) - F_X(a).\n\\] This allows truncated distributions to be defined as \\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a)}.\n\\]\nFor discrete distributions, a slightly more complicated form is required to explicitly insert the lower truncation point, which is otherwise excluded from \\(F_X(b) - F_X(a)\\),\n\\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a) + p_X(a)}.\n\\]\n\n\n\nStan allows probability functions to be truncated. For example, a truncated unit normal distributions restricted to \\([-0.5, 2.1]\\) can be coded with the following sampling statement.\ny ~ normal(0, 1) T[-0.5, 2.1];\nTruncated distributions are translated as an additional term in the accumulated log density function plus error checking to make sure the variate in the sampling statement is within the bounds of the truncation.\nIn general, the truncation bounds and parameters may be parameters or local variables.\nBecause the example above involves a continuous distribution, it behaves the same way as the following more verbose form.\ny ~ normal(0, 1);\nif (y &lt; -0.5 || y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -log_diff_exp(normal_lcdf(2.1 | 0, 1),\n}\n                          normal_lcdf(-0.5 | 0, 1));\nBecause a Stan program defines a log density function, all calculations are on the log scale. The function normal_lcdf is the log of the cumulative normal distribution function and the function log_diff_exp(a, b) is a more arithmetically stable form of log(exp(a) - exp(b)).\nFor a discrete distribution, another term is necessary in the denominator to account for the excluded boundary. The truncated discrete distribution\ny ~ poisson(3.7) T[2, 10];\nbehaves in the same way as the following code.\ny ~ poisson(3.7);\nif (y &lt; 2 || y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         log_diff_exp(poisson_lcdf(10 | 3.7),\n                                      poisson_lcdf(2 | 3.7)));\n}\nRecall that log_sum_exp(a, b) is just the arithmetically stable form of log(exp(a) + exp(b)).\n\n\n\nFor truncating with only a lower bound, the upper limit is left blank.\ny ~ normal(0, 1) T[-0.5, ];\nThis truncated sampling statement has the same behavior as the following code.\ny ~ normal(0, 1);\nif (y &lt; -0.5) {\n  target += negative_infinity();\n} else {\n  target += -normal_lccdf(-0.5 | 0, 1);\n}\nThe normal_lccdf function is the normal complementary cumulative distribution function.\nAs with lower and upper truncation, the discrete case requires a more complicated denominator to add back in the probability mass for the lower bound. Thus\ny ~ poisson(3.7) T[2, ];\nbehaves the same way as\ny ~ poisson(3.7);\nif (y &lt; 2) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         poisson_lccdf(2 | 3.7));\n}\n\n\n\nTo truncate with only an upper bound, the lower bound is left blank. The upper truncated sampling statement\ny ~ normal(0, 1) T[ , 2.1];\nproduces the same result as the following code.\ntarget += normal_lpdf(y | 0, 1);\nif (y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -normal_lcdf(2.1 | 0, 1);\n}\nWith only an upper bound, the discrete case does not need a boundary adjustment. The upper-truncated sampling statement\ny ~ poisson(3.7) T[ , 10];\nbehaves the same way as the following code.\ny ~ poisson(3.7);\nif (y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -poisson_lcdf(10 | 3.7);\n}\n\n\n\nIn all cases, the truncation is only well formed if the appropriate log density or mass function and necessary log cumulative distribution functions are defined. Not every distribution built into Stan has log cdf and log ccdfs defined, nor will every user-defined distribution. The discrete probability function documentations describes the available discrete and continuous cumulative distribution functions; most univariate distributions have log cdf and log ccdf functions.\n\n\n\nFor continuous distributions, truncation points must be expressions of type int or real. For discrete distributions, truncation points must be expressions of type int.\n\n\n\nFor a truncated sampling statement, if the value sampled is not within the bounds specified by the truncation expression, the result is zero probability and the entire statement adds \\(-\\infty\\) to the total log probability, which in turn results in the sample being rejected.\n\n\n\nVectorization of distribution functions with truncation is available if the underlying distribution, lcdf, and lccdf functions meet the required signatures.\nThe equivalent code for a vectorized truncation depends on which of the variables are non-scalars (arrays, vectors, etc.):\n\nIf the variate y is the only non-scalar, the result is the same as described in the above sections, but the lcdf/lccdf calculation is multiplied by size(y).\nIf the other arguments to the distribution are non-scalars, then the vectorized version of the lcdf/lccdf is used. These functions return the sum of their terms, so no multiplication by the size is needed.\nThe exception to the above is when a non-variate is a vector and both a lower and upper bound are specified in the truncation. In this case, a for loop is generated over the elements of the non-scalar arguments. This is required since the log_diff_exp of two sums is not the same as the sum of the pairwise log_diff_exp operations.\n\nNote that while a lower-and-upper truncated distribution may generate a for-loop internally as part of translating the truncation statement, this is still preferable to manually constructing a loop, since the distribution function itself can still be evaluated in a vectorized manner.\n\n\n\n\n\nSuppose N is a variable of type int, y is a one-dimensional array of type array[] real, and mu and sigma are variables of type real. Furthermore, suppose that n has not been defined as a variable. Then the following is a well-formed for-loop statement.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nThe loop variable is n, the loop bounds are the values in the range 1:N, and the body is the statement following the loop bounds.\n\n\nThe type of the loop variable is int. Unlike in C++ and similarly to R, this variable must not be declared explicitly.\nThe bounds in a for loop must be integers. Unlike in R, the loop is always interpreted as an upward counting loop. The range L:H will cause the loop to execute the loop with the loop variable taking on all integer values greater than or equal to L and less than or equal to H. For example, the loop for (n in 2:5) will cause the body of the for loop to be executed with n equal to 2, 3, 4, and 5, in order. The variable and bound for (n in   5:2) will not execute anything because there are no integers greater than or equal to 5 and less than or equal to 2.\nThe scope of the loop variable is limited to the body of the loop.\n\n\n\nUnlike in BUGS, Stan allows variables to be reassigned. For example, the variable theta in the following program is reassigned in each iteration of the loop.\nfor (n in 1:N) {\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nSuch reassignment is not permitted in BUGS. In BUGS, for loops are declarative, defining plates in directed graphical model notation, which can be thought of as repeated substructures in the graphical model. Therefore, it is illegal in BUGS or JAGS to have a for loop that repeatedly reassigns a value to a variable.4\nIn Stan, assignments are executed in the order they are encountered. As a consequence, the following Stan program has a very different interpretation than the previous one.\nfor (n in 1:N) {\n  y[n] ~ bernoulli(theta);\n  theta = inv_logit(alpha + x[n] * beta);\n}\nIn this program, theta is assigned after it is used in the probability statement. This presupposes it was defined before the first loop iteration (otherwise behavior is undefined), and then each loop uses the assignment from the previous iteration.\nStan loops may be used to accumulate values. Thus it is possible to sum the values of an array directly using code such as the following.\ntotal = 0.0;\nfor (n in 1:N) {\n  total = total + x[n];\n}\nAfter the for loop is executed, the variable total will hold the sum of the elements in the array x. This example was purely pedagogical; it is easier and more efficient to write\ntotal = sum(x);\nA variable inside (or outside) a loop may even be reassigned multiple times, as in the following legal code.\nfor (n in 1:100) {\n  y += y * epsilon;\n  epsilon = 0.5 * epsilon;\n  y += y * epsilon;\n}\n\n\n\n\nA second form of for loops allows iteration over elements of containers. If ys is an expression denoting a container (vector, row vector, matrix, or array) with elements of type T, then the following is a well-formed foreach statement.\nfor (y in ys) {\n  // ... do something with y ...\n}\nThe order in which elements of ys are visited is defined for container types as follows.\n\nvector, row_vector: elements visited in order, y is of type double\nmatrix: elements visited in column-major order, y is of type double\narray[] T: elements visited in order, y is of type T.\n\nConsequently, if ys is a two dimensional array array[,] real, y will be a one-dimensional array of real values (type array[] real). If ’ysis a matrix, thenywill be a real value (typereal`). To loop over all values of a two-dimensional array using foreach statements would require a doubly-nested loop,\narray[2, 3] real yss;\nfor (ys in yss) {\n  for (y in ys) {\n    // ... do something with y ...\n  }\n}\nwhereas a matrix can be looped over in one foreach statement\nmatrix[2, 3] yss;\nfor (y in yss) {\n   // ... do something with y...\n}\nIn both cases, the loop variable y is of type real.  The elements of the matrix are visited in column-major order (e.g.,y[1, 1],y[2, 1],y[1, 2], ...,y[2, 3]), whereas the elements of the two-dimensional array are visited in row-major order (e.g.,y[1, 1],y[1, 2],y[1, 3],y[2, 1], ...,y[2, 3]`).\n\n\n\nStan supports full conditional statements using the same if-then-else syntax as C++. The general format is\nif (condition1)\n  statement1\nelse if (condition2)\n  statement2\n// ...\nelse if (conditionN-1)\n  statementN-1\nelse\n  statementN\nThere must be a single leading if clause, which may be followed by any number of else if clauses, all of which may be optionally followed by an else clause. Each condition must be an integer value, with non-zero values interpreted as true and the zero value as false.\nThe entire sequence of if-then-else clauses forms a single conditional statement for evaluation. The conditions are evaluated in order until one of the conditions evaluates to a non-zero value, at which point its corresponding statement is executed and the conditional statement finishes execution. If none of the conditions evaluate to a non-zero value and there is a final else clause, its statement is executed.\n\n\n\nStan supports standard while loops using the same syntax as C++. The general format is as follows.\nwhile (condition)\n  body\nThe condition must be an integer expression and the body can be any statement (or sequence of statements in curly braces).\nEvaluation of a while loop starts by evaluating the condition. If the condition evaluates to a false (zero) value, the execution of the loop terminates and control moves to the position after the loop. If the loop’s condition evaluates to a true (non-zero) value, the body statement is executed, then the whole loop is executed again. Thus the loop is continually executed as long as the condition evaluates to a true value.\nThe rest of the body of a while loop may be skipped using a continue. The loop will be exited with a break statement. See the section on continue and break statements for more details.\n\n\n\nJust as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are\nint, real, vector[K], row_vector[K], matrix[M, N].\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.\n\n\n\n\nThe one-token statements continue and break may be used within loops to alter control flow; continue causes the next iteration of the loop to run immediately, whereas break terminates the loop and causes execution to resume after the loop. Both control structures must appear in loops. Both break and continue scope to the most deeply nested loop, but pass through non-loop statements.\nAlthough these control statements may seem undesirable because of their goto-like behavior, their judicious use can greatly improve readability by reducing the level of nesting or eliminating bookkeeping inside loops.\n\n\nWhen a break statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:\nwhile (1) {\n  if (n &lt; 0) {\n    break;\n  }\n  foo(n);\n  n = n - 1;\n}\nThe while~(1) loop is a “forever” loop, because 1 is the true value, so the test always succeeds. Within the loop, if the value of n is less than 0, the loop terminates, otherwise it executes foo(n) and then decrements n. The statement above does exactly the same thing as\nwhile (n &gt;= 0) {\n  foo(n);\n  n = n - 1;\n}\nThis case is simply illustrative of the behavior; it is not a case where a break simplifies the loop.\n\n\n\nThe continue statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array x,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &lt;= 0) {\n    continue;\n  }\n  sum += x[n];\n}\nWhen the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an x[n] less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &gt; 0) {\n    sum += x[n];\n  }\n}\nWhile the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.\n\n\n\nIf there is a loop nested within a loop, a break or continue statement only breaks out of the inner loop. So\nwhile (cond1) {\n  // ...\n  while (cond2) {\n    // ...\n    if (cond3) {\n      break;\n    }\n    // ...\n  }\n  // execution continues here after break\n  // ...\n}\nIf the break is triggered by cond3 being true, execution will continue after the nested loop.\nAs with break statements, continue statements go back to the top of the most deeply nested loop in which the continue appears.\nAlthough break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.\n\n\n\n\nStan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\n\n\n\n\nThe Stan reject statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.\nLike the print statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.\nReject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable x’s value is negative.\nif (x &lt; 0) {\n  reject(\"x must not be negative; found x=\", x);\n}\n\n\nReject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the normal_lpdf function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.\nIn all cases of rejection, the interface accessing the Stan program should print the arguments to the reject statement.\n\n\nRejections in user-defined functions are just passed to the calling function or program block. Reject statements can be used in functions to validate the function arguments, allowing user-defined functions to fully emulate built-in function behavior. It is better to find out earlier rather than later when there is a problem.\n\n\n\nIn both the transformed data block and generated quantities block, rejections are fatal. This is because if initialization fails or if generating output fails, there is no way to recover values.\nReject statements placed in the transformed data block can be used to validate both the data and transformed data (if any). This allows more complicated constraints to be enforced that can be specified with Stan’s constrained variable declarations.\n\n\n\nRejections in the transformed parameters and model blocks are not in and of themselves instantly fatal. The result has the same effect as assigning a \\(-\\infty\\) log probability, which causes rejection of the current proposal in MCMC samplers and adjustment of search parameters in optimization.\nIf the log probability function results in a rejection every time it is called, the containing application (MCMC sampler or optimization) should diagnose this problem and terminate with an appropriate error message. To aid in diagnosing problems, the message for each reject statement will be printed as a result of executing it.\n\n\n\n\nRejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;\n  real&lt;lower=a, upper=b&gt; theta;\n  // ...\n}\nmodel {\n  // **wrong** needs explicit truncation\n  theta ~ normal(0, 1);\n  // ...\n}\nThis program is wrong because its truncation bounds on theta depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.\n  theta ~ normal(0, 1) T[a, b];\nThe conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over \\([a ,b]\\); in these simple univariate cases, we can overcome that with the T[ , ] notation, which essentially divides by whatever the prior integrates to over \\([a, b]\\).\nThis problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.\n  if (theta &lt; a || theta &gt; b) {\n    reject(\"theta not in (a, b)\");\n  }\n  // still **wrong**, needs T[a,b]\n  theta ~ normal(0, 1);\nIn this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.\nEven if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.",
+    "text": "The blocks of a Stan program are made up of variable declarations and statements; see the blocks chapter for details. Unlike programs in BUGS, the declarations and statements making up a Stan program are executed in the order in which they are written. Variables must be defined to have some value (as well as declared to have some type) before they are used — if they do not, the behavior is undefined.\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters. Log probability function can be constructed by using assignment statements. Statements may be grouped into sequences and into for-each loops. In addition, Stan allows local variables to be declared in blocks and also allows an empty statement consisting only of a semicolon.\n\n\nThe data and parameters blocks do not allow statements of any kind because these blocks are solely used to declare the data variables for input and the parameter variables for sampling. All other blocks allow statements. In these blocks, both variable declarations and statements are allowed. All top-level variables in a block are considered block variables. See the blocks chapter for more information about the block structure of Stan programs.\n\n\n\nAn assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,\n\na variable, or\na variable with one or more indices, or\na comma separated list of lvalues surrounded by ( and )\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y\n\n\n\n\n\n\n\nThe basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.1\ntarget += -0.5 * y * y;\nThe keyword target here is actually not a variable, and may not be accessed as such (though see below on how to access the value of target through a special function).\nIn this example, the unnormalized log probability of a unit normal variable \\(y\\) is added to the total log probability. In the general case, the argument can be any expression.2\nAn entire Stan model can be implemented this way. For instance, the following model will draw a single variable according to a unit normal probability.\nparameters {\n  real y;\n}\nmodel {\n  target += -0.5 * y * y;\n}\nThis model defines a log probability function\n\\[\n\\log p(y) = - \\, \\frac{y^2}{2} - \\log Z\n\\]\nwhere \\(Z\\) is a normalizing constant that does not depend on \\(y\\). The constant \\(Z\\) is conventionally written this way because on the linear scale, \\[\np(y) = \\frac{1}{Z} \\exp\\left(-\\frac{y^2}{2}\\right).\n\\] which is typically written without reference to \\(Z\\) as \\[\np(y) \\propto \\exp\\left(-\\frac{y^2}{2}\\right).\n\\]\nStan only requires models to be defined up to a constant that does not depend on the parameters. This is convenient because often the normalizing constant \\(Z\\) is either time-consuming to compute or intractable to evaluate.\n\n\nThe built in distribution functions in Stan are all available in normalized and unnormalized form. The normalized forms include all of the terms in the log density, and the unnormalized forms drop terms which are not directly or indirectly a function of the model parameters.\nFor instance, the normal_lpdf function returns the log density of a normal distribution:\n\\[\n\\textsf{normal\\_lpdf}(x | \\mu, \\sigma) =\n-\\log \\left( \\sigma \\sqrt{2 \\pi} \\right)\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nThe normal_lupdf function returns the log density of an unnormalized distribution. With the unnormalized version of the function, Stan does not define what the normalization constant will be, though usually as many terms as possible are dropped to make the calculation fast. Dropping a constant sigma term, normal_lupdf would be equivalent to:\n\\[\n\\textsf{normal\\_lupdf}(x | \\mu, \\sigma) =\n-\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\n\\]\nAll functions ending in _lpdf have a corresponding _lupdf version which evaluates and returns the unnormalized density. The same is true for _lpmf and _lupmf.\n\n\n\nThe increment log density statement looks syntactically like compound addition and assignment (see the compound arithmetic/assignment section, it is treated as a primitive statement because target is not itself a variable. So, even though\ntarget += lp;\nis a legal statement, the corresponding long form is not legal.\ntarget = target + lp;  // BAD, target is not a variable\n\n\n\nThe target += ... statement accepts an argument in place of ... for any expression type, including integers, reals, vectors, row vectors, matrices, and arrays of any dimensionality, including arrays of vectors and matrices. For container arguments, their sum will be added to the total log density.\n\n\n\nTo access accumulated log density up to the current execution point, the function target() may be used.\n\n\n\n\nStan supports writing probability statements also in sampling notation, such as\ny ~ normal(mu, sigma);\nThe name “sampling statement” is meant to be suggestive, not interpreted literally. Conceptually, the variable y, which may be an unknown parameter or known, modeled data, is being declared to have the distribution indicated by the right-hand side of the sampling statement. The above statement can read as “y is distributed as normal distribution with parameters mu and sigma.”. The symbol ~ is called tilde.\nExecuting such a statement does not perform any sampling. In Stan, a sampling statement is merely a notational convenience following the typical notation used to present models in the literature. The above sampling statement could be expressed as a direct increment on the total log probability density as\ntarget += normal_lpdf(y | mu, sigma);\nIn general, a sampling statement of the form\ny ~ dist(theta1, ..., thetaN);\ninvolving subexpressions y and theta1 through thetaN (including the case where N is zero) will be well formed if and only if the corresponding assignment statement is well-formed. For densities allowing real y values, the log probability density function is used,\ntarget += dist_lpdf(y | theta1, ..., thetaN);\nFor those restricted to integer y values, the log probability mass function is used,\ntarget += dist_lpmf(y | theta1, ..., thetaN);\nThis will be well formed if and only if dist_lpdf(y | theta1,   ..., thetaN) or dist_lpmf(y | theta1, ..., thetaN) is a well-formed expression of type real.\n\n\nAlthough both lead to the same sampling behavior in Stan, there is one critical difference between using the sampling statement, as in\ny ~ normal(mu, sigma);\nand explicitly incrementing the log probability function, as in\ntarget += normal_lpdf(y | mu, sigma);\nThe sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions, y, mu, or sigma, involve only constants, data variables, and transformed data variables.\n\n\n\nThe left-hand side of a sampling statement may be a complex expression. For instance, it is legal syntactically to write\nparameters {\n  real&lt;lower=0&gt; beta;\n}\n// ...\nmodel {\n  log(beta) ~ normal(mu, sigma);\n}\nUnfortunately, this is not enough to properly model beta as having a lognormal distribution. Whenever a nonlinear transform is applied to a parameter, such as the logarithm function being applied to beta here, and then used on the left-hand side of a sampling statement or on the left of a vertical bar in a log pdf function, an adjustment must be made to account for the differential change in scale and ensure beta gets the correct distribution. The correction required is to add the log Jacobian of the transform to the target log density; see the change of variables section for full definitions. For the case above, the following adjustment will account for the log transform.3\ntarget += - log(abs(y));\n\n\n\nStan supports truncating distributions with lower bounds, upper bounds, or both.\n\n\nA probability density function \\(p(x)\\) for a continuous distribution may be truncated to an interval \\([a, b]\\) to define a new density \\(p_{[a, b]}(x)\\) with support \\([a, b]\\) by setting\n\\[\np_{[a, b]}(x)\n= \\frac{p(x)}\n       {\\int_a^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) for a discrete distribution may be truncated to the closed interval \\([a, b]\\) by\n\\[\np_{[a, b]}(x) = \\frac{p(x)}\n                  {\\sum_{u = a}^b p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x)\n= \\frac{p(x)}\n       {\\int_a^{\\infty} p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([a, \\infty]\\) by defining\n\\[\np_{[a, \\infty]}(x) = \\frac{p(x)}\n                  {\\sum_{a &lt;= u} p(u)}.\n\\]\n\n\n\nA probability density function \\(p(x)\\) can be truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty, b]}(x)\n= \\frac{p(x)}\n       {\\int_{-\\infty}^b p(u) \\, du}.\n\\]\nA probability mass function \\(p(x)\\) is truncated to \\([-\\infty, b]\\) by defining\n\\[\np_{[-\\infty,b]}(x) = \\frac{p(x)}\n                  {\\sum_{u &lt;= b} p(u)}.\n\\]\n\n\n\nGiven a probability function \\(p_X(x)\\) for a random variable \\(X\\), its cumulative distribution function (cdf) \\(F_X(x)\\) is defined to be the probability that \\(X \\leq x\\),\n\\[\nF_X(x) = \\Pr[X \\leq x].\n\\]\nThe upper-case variable \\(X\\) is the random variable whereas the lower-case variable \\(x\\) is just an ordinary bound variable. For continuous random variables, the definition of the cdf works out to\n\\[\nF_X(x) \\ = \\ \\int_{-\\infty}^{x} p_X(u) \\, du,\n\\]\nFor discrete variables, the cdf is defined to include the upper bound given by the argument,\n\\[\nF_X(x) = \\sum_{u \\leq x} p_X(u).\n\\]\n\n\n\nThe complementary cumulative distribution function (ccdf) in both the continuous and discrete cases is given by\n\\[\nF^C_X(x)\n\\ = \\ \\Pr[X &gt; x]\n\\ = \\ 1 - F_X(x).\n\\]\nUnlike the cdf, the ccdf is exclusive of the bound, hence the event \\(X &gt; x\\) rather than the cdf’s event \\(X \\leq x\\).\nFor continuous distributions, the ccdf works out to\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\int_{-\\infty}^x p_X(u) \\, du\n\\ = \\ \\int_x^{\\infty} p_X(u) \\, du.\n\\]\nThe lower boundary can be included in the integration bounds because it is a single point on a line and hence has no probability mass. For the discrete case, the lower bound must be excluded in the summation explicitly by summing over \\(u &gt; x\\),\n\\[\nF^C_X(x)\n\\ = \\ 1 - \\sum_{u \\leq x} p_X(u)\n\\ = \\ \\sum_{u &gt; x} p_X(u).\n\\]\nCumulative distribution functions provide the necessary integral calculations to define truncated distributions. For truncation with lower and upper bounds, the denominator is defined by \\[\n\\int_a^b p(u) \\, du = F_X(b) - F_X(a).\n\\] This allows truncated distributions to be defined as \\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a)}.\n\\]\nFor discrete distributions, a slightly more complicated form is required to explicitly insert the lower truncation point, which is otherwise excluded from \\(F_X(b) - F_X(a)\\),\n\\[\np_{[a,b]}(x) = \\frac{p_X(x)}\n                  {F_X(b) - F_X(a) + p_X(a)}.\n\\]\n\n\n\nStan allows probability functions to be truncated. For example, a truncated unit normal distributions restricted to \\([-0.5, 2.1]\\) can be coded with the following sampling statement.\ny ~ normal(0, 1) T[-0.5, 2.1];\nTruncated distributions are translated as an additional term in the accumulated log density function plus error checking to make sure the variate in the sampling statement is within the bounds of the truncation.\nIn general, the truncation bounds and parameters may be parameters or local variables.\nBecause the example above involves a continuous distribution, it behaves the same way as the following more verbose form.\ny ~ normal(0, 1);\nif (y &lt; -0.5 || y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -log_diff_exp(normal_lcdf(2.1 | 0, 1),\n}\n                          normal_lcdf(-0.5 | 0, 1));\nBecause a Stan program defines a log density function, all calculations are on the log scale. The function normal_lcdf is the log of the cumulative normal distribution function and the function log_diff_exp(a, b) is a more arithmetically stable form of log(exp(a) - exp(b)).\nFor a discrete distribution, another term is necessary in the denominator to account for the excluded boundary. The truncated discrete distribution\ny ~ poisson(3.7) T[2, 10];\nbehaves in the same way as the following code.\ny ~ poisson(3.7);\nif (y &lt; 2 || y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         log_diff_exp(poisson_lcdf(10 | 3.7),\n                                      poisson_lcdf(2 | 3.7)));\n}\nRecall that log_sum_exp(a, b) is just the arithmetically stable form of log(exp(a) + exp(b)).\n\n\n\nFor truncating with only a lower bound, the upper limit is left blank.\ny ~ normal(0, 1) T[-0.5, ];\nThis truncated sampling statement has the same behavior as the following code.\ny ~ normal(0, 1);\nif (y &lt; -0.5) {\n  target += negative_infinity();\n} else {\n  target += -normal_lccdf(-0.5 | 0, 1);\n}\nThe normal_lccdf function is the normal complementary cumulative distribution function.\nAs with lower and upper truncation, the discrete case requires a more complicated denominator to add back in the probability mass for the lower bound. Thus\ny ~ poisson(3.7) T[2, ];\nbehaves the same way as\ny ~ poisson(3.7);\nif (y &lt; 2) {\n  target += negative_infinity();\n} else {\n  target += -log_sum_exp(poisson_lpmf(2 | 3.7),\n                         poisson_lccdf(2 | 3.7));\n}\n\n\n\nTo truncate with only an upper bound, the lower bound is left blank. The upper truncated sampling statement\ny ~ normal(0, 1) T[ , 2.1];\nproduces the same result as the following code.\ntarget += normal_lpdf(y | 0, 1);\nif (y &gt; 2.1) {\n  target += negative_infinity();\n} else {\n  target += -normal_lcdf(2.1 | 0, 1);\n}\nWith only an upper bound, the discrete case does not need a boundary adjustment. The upper-truncated sampling statement\ny ~ poisson(3.7) T[ , 10];\nbehaves the same way as the following code.\ny ~ poisson(3.7);\nif (y &gt; 10) {\n  target += negative_infinity();\n} else {\n  target += -poisson_lcdf(10 | 3.7);\n}\n\n\n\nIn all cases, the truncation is only well formed if the appropriate log density or mass function and necessary log cumulative distribution functions are defined. Not every distribution built into Stan has log cdf and log ccdfs defined, nor will every user-defined distribution. The discrete probability function documentations describes the available discrete and continuous cumulative distribution functions; most univariate distributions have log cdf and log ccdf functions.\n\n\n\nFor continuous distributions, truncation points must be expressions of type int or real. For discrete distributions, truncation points must be expressions of type int.\n\n\n\nFor a truncated sampling statement, if the value sampled is not within the bounds specified by the truncation expression, the result is zero probability and the entire statement adds \\(-\\infty\\) to the total log probability, which in turn results in the sample being rejected.\n\n\n\nVectorization of distribution functions with truncation is available if the underlying distribution, lcdf, and lccdf functions meet the required signatures.\nThe equivalent code for a vectorized truncation depends on which of the variables are non-scalars (arrays, vectors, etc.):\n\nIf the variate y is the only non-scalar, the result is the same as described in the above sections, but the lcdf/lccdf calculation is multiplied by size(y).\nIf the other arguments to the distribution are non-scalars, then the vectorized version of the lcdf/lccdf is used. These functions return the sum of their terms, so no multiplication by the size is needed.\nThe exception to the above is when a non-variate is a vector and both a lower and upper bound are specified in the truncation. In this case, a for loop is generated over the elements of the non-scalar arguments. This is required since the log_diff_exp of two sums is not the same as the sum of the pairwise log_diff_exp operations.\n\nNote that while a lower-and-upper truncated distribution may generate a for-loop internally as part of translating the truncation statement, this is still preferable to manually constructing a loop, since the distribution function itself can still be evaluated in a vectorized manner.\n\n\n\n\n\nSuppose N is a variable of type int, y is a one-dimensional array of type array[] real, and mu and sigma are variables of type real. Furthermore, suppose that n has not been defined as a variable. Then the following is a well-formed for-loop statement.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nThe loop variable is n, the loop bounds are the values in the range 1:N, and the body is the statement following the loop bounds.\n\n\nThe type of the loop variable is int. Unlike in C++ and similarly to R, this variable must not be declared explicitly.\nThe bounds in a for loop must be integers. Unlike in R, the loop is always interpreted as an upward counting loop. The range L:H will cause the loop to execute the loop with the loop variable taking on all integer values greater than or equal to L and less than or equal to H. For example, the loop for (n in 2:5) will cause the body of the for loop to be executed with n equal to 2, 3, 4, and 5, in order. The variable and bound for (n in   5:2) will not execute anything because there are no integers greater than or equal to 5 and less than or equal to 2.\nThe scope of the loop variable is limited to the body of the loop.\n\n\n\nUnlike in BUGS, Stan allows variables to be reassigned. For example, the variable theta in the following program is reassigned in each iteration of the loop.\nfor (n in 1:N) {\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nSuch reassignment is not permitted in BUGS. In BUGS, for loops are declarative, defining plates in directed graphical model notation, which can be thought of as repeated substructures in the graphical model. Therefore, it is illegal in BUGS or JAGS to have a for loop that repeatedly reassigns a value to a variable.4\nIn Stan, assignments are executed in the order they are encountered. As a consequence, the following Stan program has a very different interpretation than the previous one.\nfor (n in 1:N) {\n  y[n] ~ bernoulli(theta);\n  theta = inv_logit(alpha + x[n] * beta);\n}\nIn this program, theta is assigned after it is used in the probability statement. This presupposes it was defined before the first loop iteration (otherwise behavior is undefined), and then each loop uses the assignment from the previous iteration.\nStan loops may be used to accumulate values. Thus it is possible to sum the values of an array directly using code such as the following.\ntotal = 0.0;\nfor (n in 1:N) {\n  total = total + x[n];\n}\nAfter the for loop is executed, the variable total will hold the sum of the elements in the array x. This example was purely pedagogical; it is easier and more efficient to write\ntotal = sum(x);\nA variable inside (or outside) a loop may even be reassigned multiple times, as in the following legal code.\nfor (n in 1:100) {\n  y += y * epsilon;\n  epsilon = 0.5 * epsilon;\n  y += y * epsilon;\n}\n\n\n\n\nA second form of for loops allows iteration over elements of containers. If ys is an expression denoting a container (vector, row vector, matrix, or array) with elements of type T, then the following is a well-formed foreach statement.\nfor (y in ys) {\n  // ... do something with y ...\n}\nThe order in which elements of ys are visited is defined for container types as follows.\n\nvector, row_vector: elements visited in order, y is of type double\nmatrix: elements visited in column-major order, y is of type double\narray[] T: elements visited in order, y is of type T.\n\nConsequently, if ys is a two dimensional array array[,] real, y will be a one-dimensional array of real values (type array[] real). If ’ysis a matrix, thenywill be a real value (typereal`). To loop over all values of a two-dimensional array using foreach statements would require a doubly-nested loop,\narray[2, 3] real yss;\nfor (ys in yss) {\n  for (y in ys) {\n    // ... do something with y ...\n  }\n}\nwhereas a matrix can be looped over in one foreach statement\nmatrix[2, 3] yss;\nfor (y in yss) {\n   // ... do something with y...\n}\nIn both cases, the loop variable y is of type real.  The elements of the matrix are visited in column-major order (e.g.,y[1, 1],y[2, 1],y[1, 2], ...,y[2, 3]), whereas the elements of the two-dimensional array are visited in row-major order (e.g.,y[1, 1],y[1, 2],y[1, 3],y[2, 1], ...,y[2, 3]`).\n\n\n\nStan supports full conditional statements using the same if-then-else syntax as C++. The general format is\nif (condition1)\n  statement1\nelse if (condition2)\n  statement2\n// ...\nelse if (conditionN-1)\n  statementN-1\nelse\n  statementN\nThere must be a single leading if clause, which may be followed by any number of else if clauses, all of which may be optionally followed by an else clause. Each condition must be an integer value, with non-zero values interpreted as true and the zero value as false.\nThe entire sequence of if-then-else clauses forms a single conditional statement for evaluation. The conditions are evaluated in order until one of the conditions evaluates to a non-zero value, at which point its corresponding statement is executed and the conditional statement finishes execution. If none of the conditions evaluate to a non-zero value and there is a final else clause, its statement is executed.\n\n\n\nStan supports standard while loops using the same syntax as C++. The general format is as follows.\nwhile (condition)\n  body\nThe condition must be an integer expression and the body can be any statement (or sequence of statements in curly braces).\nEvaluation of a while loop starts by evaluating the condition. If the condition evaluates to a false (zero) value, the execution of the loop terminates and control moves to the position after the loop. If the loop’s condition evaluates to a true (non-zero) value, the body statement is executed, then the whole loop is executed again. Thus the loop is continually executed as long as the condition evaluates to a true value.\nThe rest of the body of a while loop may be skipped using a continue. The loop will be exited with a break statement. See the section on continue and break statements for more details.\n\n\n\nJust as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are listed in the types table under “local”.\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.\n\n\n\n\nThe one-token statements continue and break may be used within loops to alter control flow; continue causes the next iteration of the loop to run immediately, whereas break terminates the loop and causes execution to resume after the loop. Both control structures must appear in loops. Both break and continue scope to the most deeply nested loop, but pass through non-loop statements.\nAlthough these control statements may seem undesirable because of their goto-like behavior, their judicious use can greatly improve readability by reducing the level of nesting or eliminating bookkeeping inside loops.\n\n\nWhen a break statement is executed, the most deeply nested loop currently being executed is ended and execution picks up with the next statement after the loop. For example, consider the following program:\nwhile (1) {\n  if (n &lt; 0) {\n    break;\n  }\n  foo(n);\n  n = n - 1;\n}\nThe while~(1) loop is a “forever” loop, because 1 is the true value, so the test always succeeds. Within the loop, if the value of n is less than 0, the loop terminates, otherwise it executes foo(n) and then decrements n. The statement above does exactly the same thing as\nwhile (n &gt;= 0) {\n  foo(n);\n  n = n - 1;\n}\nThis case is simply illustrative of the behavior; it is not a case where a break simplifies the loop.\n\n\n\nThe continue statement ends the current operation of the loop and returns to the condition at the top of the loop. Such loops are typically used to exclude some values from calculations. For example, we could use the following loop to sum the positive values in the array x,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &lt;= 0) {\n    continue;\n  }\n  sum += x[n];\n}\nWhen the continue statement is executed, control jumps back to the conditional part of the loop. With while and for loops, this causes control to return to the conditional of the loop. With for loops, this advances the loop variable, so the the above program will not go into an infinite loop when faced with an x[n] less than zero. Thus the above program could be rewritten with deeper nesting by reversing the conditional,\nreal sum;\nsum = 0;\nfor (n in 1:size(x)) {\n  if (x[n] &gt; 0) {\n    sum += x[n];\n  }\n}\nWhile the latter form may seem more readable in this simple case, the former has the main line of execution nested one level less deep. Instead, the conditional at the top finds cases to exclude and doesn’t require the same level of nesting for code that’s not excluded. When there are several such exclusion conditions, the break or continue versions tend to be much easier to read.\n\n\n\nIf there is a loop nested within a loop, a break or continue statement only breaks out of the inner loop. So\nwhile (cond1) {\n  // ...\n  while (cond2) {\n    // ...\n    if (cond3) {\n      break;\n    }\n    // ...\n  }\n  // execution continues here after break\n  // ...\n}\nIf the break is triggered by cond3 being true, execution will continue after the nested loop.\nAs with break statements, continue statements go back to the top of the most deeply nested loop in which the continue appears.\nAlthough break and continue must appear within loops, they may appear in nested statements within loops, such as within the conditionals shown above or within nested statements. The break and continue statements jump past any control structure other than while-loops and for-loops.\n\n\n\n\nStan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\nNote that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.\n\n\n\n\nThe Stan reject statement provides a mechanism to report errors or problematic values encountered during program execution and either halt processing or reject iterations.\nLike the print statement, the reject statement accepts any number of quoted string literals or Stan expressions as arguments.\nReject statements are typically embedded in a conditional statement in order to detect variables in illegal states. For example, the following code handles the case where a variable x’s value is negative.\nif (x &lt; 0) {\n  reject(\"x must not be negative; found x=\", x);\n}\n\n\nReject statements have the same behavior as exceptions thrown by built-in Stan functions. For example, the normal_lpdf function raises an exception if the input scale is not positive and finite. The effect of a reject statement depends on the program block in which the rejection occurs.\nIn all cases of rejection, the interface accessing the Stan program should print the arguments to the reject statement.\n\n\nRejections in user-defined functions are just passed to the calling function or program block. Reject statements can be used in functions to validate the function arguments, allowing user-defined functions to fully emulate built-in function behavior. It is better to find out earlier rather than later when there is a problem.\n\n\n\nIn both the transformed data block and generated quantities block, rejections are fatal. This is because if initialization fails or if generating output fails, there is no way to recover values.\nReject statements placed in the transformed data block can be used to validate both the data and transformed data (if any). This allows more complicated constraints to be enforced that can be specified with Stan’s constrained variable declarations.\n\n\n\nRejections in the transformed parameters and model blocks are not in and of themselves instantly fatal. The result has the same effect as assigning a \\(-\\infty\\) log probability, which causes rejection of the current proposal in MCMC samplers and adjustment of search parameters in optimization.\nIf the log probability function results in a rejection every time it is called, the containing application (MCMC sampler or optimization) should diagnose this problem and terminate with an appropriate error message. To aid in diagnosing problems, the message for each reject statement will be printed as a result of executing it.\n\n\n\n\nRejection should be used for error handling, not defining arbitrary constraints. Consider the following errorful Stan program.\nparameters {\n  real a;\n  real&lt;lower=a&gt; b;\n  real&lt;lower=a, upper=b&gt; theta;\n  // ...\n}\nmodel {\n  // **wrong** needs explicit truncation\n  theta ~ normal(0, 1);\n  // ...\n}\nThis program is wrong because its truncation bounds on theta depend on parameters, and thus need to be accounted for using an explicit truncation on the distribution. This is the right way to do it.\n  theta ~ normal(0, 1) T[a, b];\nThe conceptual issue is that the prior does not integrate to one over the admissible parameter space; it integrates to one over all real numbers and integrates to something less than one over \\([a ,b]\\); in these simple univariate cases, we can overcome that with the T[ , ] notation, which essentially divides by whatever the prior integrates to over \\([a, b]\\).\nThis problem is exactly the same problem as you would get using reject statements to enforce complicated inequalities on multivariate functions. In this case, it is wrong to try to deal with truncation through constraints.\n  if (theta &lt; a || theta &gt; b) {\n    reject(\"theta not in (a, b)\");\n  }\n  // still **wrong**, needs T[a,b]\n  theta ~ normal(0, 1);\nIn this case, the prior integrates to something less than one over the region of the parameter space where the complicated inequalities are satisfied. But we don’t generally know what value the prior integrates to, so we can’t increment the log probability function to compensate.\nEven if this adjustment to a proper probability model may seem minor in particular models where the amount of truncated posterior density is negligible or constant, we can’t sample from that truncated posterior efficiently. Programs need to use one-to-one mappings that guarantee the constraints are satisfied and only use reject statements to raise errors or help with debugging.",
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-    "text": "An assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are only two kinds of legal lvalues,\n\na variable, or\na variable with one or more indices.\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y",
+    "text": "An assignment statement consists of a variable (possibly multivariate with indexing information) and an expression. Executing an assignment statement evaluates the expression on the right-hand side and assigns it to the (indexed) variable on the left-hand side. An example of a simple assignment is as follows.\nn = 0;\nExecuting this statement assigns the value of the expression 0, which is the integer zero, to the variable n. For an assignment to be well formed, the type of the expression on the right-hand side should be compatible with the type of the (indexed) variable on the left-hand side. For the above example, because 0 is an expression of type int, the variable n must be declared as being of type int or of type real. If the variable is of type real, the integer zero is promoted to a floating-point zero and assigned to the variable. After the assignment statement executes, the variable n will have the value zero (either as an integer or a floating-point value, depending on its type).\nSyntactically, every assignment statement must be followed by a semicolon. Otherwise, whitespace between the tokens does not matter (the tokens here being the left-hand-side (indexed) variable, the assignment operator, the right-hand-side expression and the semicolon).\nBecause the right-hand side is evaluated first, it is possible to increment a variable in Stan just as in C++ and other programming languages by writing\nn = n + 1;\nSuch self assignments are not allowed in BUGS, because they induce a cycle into the directed graphical model.\nThe left-hand side of an assignment may contain indices for array, matrix, or vector data structures. For instance, if Sigma is of type matrix, then\nSigma[1, 1] = 1.0;\nsets the value in the first column of the first row of Sigma to one.\nAssignments to subcomponents of larger multi-variate data structures are supported by Stan. For example, a is an array of type array[,] real and b is an array of type array[] real, then the following two statements are both well-formed.\na[3] = b;\nb = a[4];\nSimilarly, if x is a variable declared to have type row_vector and Y is a variable declared as type matrix, then the following sequence of statements to swap the first two rows of Y is well formed.\nx = Y[1];\nY[1] = Y[2];\nY[2] = x;\n\n\nStan allows assignment of lower types to higher types, but not vice-versa. That is, we can assign an expression of type int to an lvalue of type real, and we can assign an expression of type real to an lvalue of type complex. Furthermore, promotion is transitive, so that we can assign an expression of type int to an lvalue of type complex.\nPromotion extends to containers, so that arrays of int can be promoted to arrays of real during assignment, and arrays of real can be assigned to an lvalue of type array of complex. Similarly, an expression of type vector may be assigned to an lvalue of type complex_vector, and similarly for row vectors and matrices.\n\n\n\nThe expressions that are legal left-hand sides of assignment statements are known as “lvalues.” In Stan, there are three kinds of legal lvalues,\n\na variable, or\na variable with one or more indices, or\na comma separated list of lvalues surrounded by ( and )\n\nTo be used as an lvalue, an indexed variable must have at least as many dimensions as the number of indices provided. An array of real or integer types has as many dimensions as it is declared for. A matrix has two dimensions and a vector or row vector one dimension; this also holds for the constrained types, covariance and correlation matrices and their Cholesky factors and ordered, positive ordered, and simplex vectors. An array of matrices has two more dimensions than the array and an array of vectors or row vectors has one more dimension than the array. Note that the number of indices can be less than the number of dimensions of the variable, meaning that the right hand side must itself be multidimensional to match the remaining dimensions.\n\n\n\nMultiple indexes, as described in the multi-indexing section, are also permitted on the left-hand side of assignments. Indexing on the left side works exactly as it does for expressions, with multiple indexes preserving index positions and single indexes reducing them. The type on the left side must still match the type on the right side.\n\n\nAll assignment is carried out as if the right-hand side is copied before the assignment. This resolves any potential aliasing issues arising from he right-hand side changing in the middle of an assignment statement’s execution.\n\n\n\n\nStan’s arithmetic operators may be used in compound arithmetic and assignment operations. For example, consider the following example of compound addition and assignment.\nreal x = 5;\nx += 7;  // value of x is now 12\nThe compound arithmetic and assignment statement above is equivalent to the following long form.\nx = x + 7;\nIn general, the compound form\nx op= y\nwill be equivalent to\nx = x op y;\nThe compound statement will be legal whenever the long form is legal. This requires that the operation x op y must itself be well formed and that the result of the operation be assignable to x. For the expression x to be assignable, it must be an indexed variable where the variable is defined in the current block. For example, the following compound addition and assignment statement will increment a single element of a vector by two.\nvector[N] x;\nx[3] += 2;\nAs a further example, consider\nmatrix[M, M] x;\nvector[M] y;\nreal z;\nx *= x;  // OK, (x * x) is a matrix\nx *= z;  // OK, (x * z) is a matrix\nx *= y;  // BAD, (x * y) is a vector\nThe supported compound arithmetic and assignment operations are listed in the compound arithmetic/assignment table; they are also listed in the index prefaced by operator, e.g., operator+=.\nCompound Arithmetic/Assignment Table.  Stan allows compound arithmetic and assignment statements of the forms listed in the table. The compound form is legal whenever the corresponding long form would be legal and it has the same effect.\n\n\n\noperation\ncompound\nunfolded\n\n\n\n\naddition\nx += y\nx = x + y\n\n\nsubtraction\nx -= y\nx = x - y\n\n\nmultiplication\nx *= y\nx = x * y\n\n\ndivision\nx /= y\nx = x / y\n\n\nelementwise multiplication\nx .*= y\nx = x .* y\n\n\nelementwise division\nx ./= y\nx = x ./ y",
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-    "text": "Just as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are\nint, real, vector[K], row_vector[K], matrix[M, N].\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.",
+    "text": "Just as parentheses may be used to group expressions, curly brackets may be used to group a sequence of zero or more statements into a statement block. At the beginning of each block, local variables may be declared that are scoped over the rest of the statements in the block.\n\n\nBlocks are often used to group a sequence of statements together to be used in the body of a for loop. Because the body of a for loop can be any statement, for loops with bodies consisting of a single statement can be written as follows.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nTo put multiple statements inside the body of a for loop, a block is used, as in the following example.\nfor (n in 1:N) {\n  lambda[n] ~ gamma(alpha, beta);\n  y[n] ~ poisson(lambda[n]);\n}\nThe open curly bracket ({) is the first character of the block and the close curly bracket (}) is the last character.\nBecause whitespace is ignored in Stan, the following program will not compile.\nfor (n in 1:N)\n  y[n] ~ normal(mu, sigma);\n  z[n] ~ normal(mu, sigma); // ERROR!\nThe problem is that the body of the for loop is taken to be the statement directly following it, which is y[n] ~ normal(mu, sigma). This leaves the probability statement for z[n] hanging, as is clear from the following equivalent program.\nfor (n in 1:N) {\n  y[n] ~ normal(mu, sigma);\n}\nz[n] ~ normal(mu, sigma); // ERROR!\nNeither of these programs will compile. If the loop variable n was defined before the for loop, the for-loop declaration will raise an error. If the loop variable n was not defined before the for loop, then the use of the expression z[n] will raise an error.\n\n\n\nA for loop has a statement as a body. It is often convenient in writing programs to be able to define a local variable that will be used temporarily and then forgotten. For instance, the for loop example of repeated assignment should use a local variable for maximum clarity and efficiency, as in the following example.\nfor (n in 1:N) {\n  real theta;\n  theta = inv_logit(alpha + x[n] * beta);\n  y[n] ~ bernoulli(theta);\n}\nThe local variable theta is declared here inside the for loop. The scope of a local variable is just the block in which it is defined. Thus theta is available for use inside the for loop, but not outside of it. As in other situations, Stan does not allow variable hiding. So it is illegal to declare a local variable theta if the variable theta is already defined in the scope of the for loop. For instance, the following is not legal.\nfor (m in 1:M) {\n  real theta;\n  for (n in 1:N) {\n    real theta; // ERROR!\n    theta = inv_logit(alpha + x[m, n] * beta);\n    y[m, n] ~ bernoulli(theta);\n// ...\nThe compiler will flag the second declaration of theta with a message that it is already defined.\n\n\n\nLocal variables may not have constraints on their declaration. The only types that may be used are listed in the types table under “local”.\n\n\n\nA block is itself a statement, so anywhere a sequence of statements is allowed, one or more of the statements may be a block. For instance, in a for loop, it is legal to have the following\nfor (m in 1:M) {\n  {\n     int n = 2 * m;\n     sum += n;\n  }\n  for (n in 1:N) {\n    sum += x[m, n];\n  }\n}\nThe variable declaration int n; is the first element of an embedded block and so has scope within that block. The for loop defines its own local block implicitly over the statement following it in which the loop variable is defined. As far as Stan is concerned, these two uses of n are unrelated.",
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     "href": "reference-manual/statements.html#print-statements.section",
     "title": "Statements",
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-    "text": "Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).",
+    "text": "Stan provides print statements that can print literal strings and the values of expressions. Print statements accept any number of arguments. Consider the following for-each statement with a print statement in its body.\nfor (n in 1:N) { print(\"loop iteration: \", n); ... }\nThe print statement will execute every time the body of the loop does. Each time the loop body is executed, it will print the string “loop iteration:” (with the trailing space), followed by the value of the expression n, followed by a new line.\n\n\nThe text printed by a print statement varies based on its content. A literal (i.e., quoted) string in a print statement always prints exactly that string (without the quotes). Expressions in print statements result in the value of the expression being printed. But how the value of the expression is formatted will depend on its type.\nPrinting a simple real or int typed variable always prints the variable’s value.5\nFor array, vector, and matrix variables, the print format uses brackets. For example, a 3-vector will print as\n[1, 2, 3]\nand a \\(2 \\times 3\\)-matrix as\n[[1, 2, 3], [4, 5, 6]]\nComplex numbers print as pairs. For example, the pair of statements\ncomplex z = to_complex(1.2, -3.5);\nprint(z)\nwill print as (1.2,-3.5), with no space after the comma or within the parentheses.\nPrinting a more readable version of arrays or matrices can be done with loops. An example is the print statement in the following transformed data block.\ntransformed data {\n  matrix[2, 2] u;\n  u[1, 1] = 1.0;   u[1, 2] = 4.0;\n  u[2, 1] = 9.0;   u[2, 2] = 16.0;\n  for (n in 1:2) {\n    print(\"u[\", n, \"] = \", u[n]);\n  }\n}\nThis print statement executes twice, printing the following two lines of output.\nu[1] = [1, 4]\nu[2] = [9, 16]\n\n\n\nThe input type to a print function cannot be void. In particular, it can’t be the result of a user-defined void function. All other types are allowed as arguments to the print function.\n\n\n\nPrinting for a print statement happens every time it is executed. The transformed data block is executed once per chain, the transformed parameter and model blocks once per leapfrog step, and the generated quantities block once per iteration.\n\n\n\nString literals begin and end with a double quote character (\"). The characters between the double quote characters may be any byte sequence, with the exception of the double quote character.\nThe Stan interfaces preserve the byte sequences which they receive. The encoding of these byte sequences as characters and their rendering as glyphs will be handled by whatever display mechanism is being used to monitor Stan’s output (e.g., a terminal, a Jupyter notebook, RStudio, etc.). Stan does not enforce a character encoding for strings, and no attempt is made to validate the bytes as legal ASCII, UTF-8, etc.\n\n\n\nBecause Stan is an imperative language, print statements can be very useful for debugging. They can be used to display the values of variables or expressions at various points in the execution of a program. They are particularly useful for spotting problematic not-a-number of infinite values, both of which will be printed.\nIt is particularly useful to print the value of the target log density accumulator (through the target() function), as in the following example.\nvector[2] y;\ny[1] = 1;\nprint(\"log density before =\", target());\ny ~ normal(0,1);  // bug!  y[2] not defined\nprint(\"log density after =\", target());\nThe example has a bug in that y[2] is not defined before the vector y is used in the sampling statement. By printing the value of the log probability accumulator before and after each sampling statement, it’s possible to isolate where the log probability becomes ill-defined (i.e., becomes not-a-number).\nNote that print statements may not always be displayed immediately, but rather at the end of an operation (e.g., leapfrog step). As such, some issues such as infinite loops are difficult to debug effectively with this technique.",
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     "title": "User-Defined Functions",
     "section": "",
-    "text": "Stan allows users to define their own functions. The basic syntax is a simplified version of that used in C and C++. This chapter specifies how functions are declared, defined, and used in Stan.\n\n\nUser-defined functions appear in a special function-definition block before all of the other program blocks.\nfunctions {\n   // ... function declarations and definitions ...\n}\ndata {\n  // ...\nFunction definitions and declarations may appear in any order. Forward declarations are allowed but not required.\n\n\n\nThe rules for function naming and function-argument naming are the same as for other variables; see the section on variables for more information on valid identifiers. For example,\nreal foo(real mu, real sigma);\ndeclares a function named foo with two argument variables of types real and real. The arguments are named mu and sigma, but that is not part of the declaration.\n\n\nMultiple user-defined functions may have the same name if they have different sequences of argument types. This is known as function overloading.\nFor example, the following two functions are both defined with the name add_up\nreal add_up(real a, real b){\n  return a + b;\n}\n\nreal add_up(real a, real b, real c){\n  return a + b + c;\n}\nThe return types of overloaded functions do not need to be the same. One could define an additional add_up function as follows\nint add_up(int a, int b){\n  return a + b;\n}\nThat being said, functions may not use the same name if their signature only differs by the return type.\nFor example, the following is not permitted\n// illegal\nreal baz(int x);\nint baz(int x);\nFunction names used in the Stan standard library may be overloaded by user-defined functions. Exceptions to this are the reduce_sum family of functions and ODE integrators, which cannot be overloaded.\n\n\n\n\nAll function arguments are mandatory—there are no default values.\n\n\nFunctions with non-void return types are called just like any other built-in function in Stan—they are applied to appropriately typed arguments to produce an expression, which has a value when executed.\n\n\n\nFunctions with void return types may be applied to arguments and used as statements. These act like sampling statements or print statements. Such uses are only appropriate for functions that act through side effects, such as incrementing the log probability accumulator, printing, or raising exceptions.\n\n\n\nOverloaded functions alongside type promotion can result in situations where there are multiple valid interpretations of a function call. Stan requires that there be a unique signature which minimizes the number of promotions required.\nConsider the following two overloaded functions\nreal foo(int a, real b);\nreal foo(real a, int b);\nThese functions do not have a unique minimum when called with two integer arguments foo(1,2), and therefore cannot be called as such.\nPromotion of integers to complex numbers is considered as two separate promotions, one from int to real and a second from real to complex. Consider the following functions with real and complex signatures\nreal bar(real x);\nreal bar(complex z);\nA call bar(5) with an integer argument will be resolved to bar(real) because it only requires a single promotion, whereas the promotion to a complex number requires two promotions.\n\n\n\nThe rules for calling functions work the same way as assignment as far as promotion goes. This means that we can promote arguments to the type expected by function arguments. For example, the following will work.\nreal foo(real x) { return ... };\n...\nint a = 5;\nreal b = foo(a); // a promoted to type real\nIn addition to promoting int to real, Stan also promotes real to complex, and by transitivity, int to complex. This also works for containers, so an array of int may be assigned to an array of real of the same shape. And we can also promote vector to complex_vector and similarly for row vectors and matrices.\n\n\n\nFunctions whose name ends in _lpdf or _lpmf (density and mass functions) may be used as probability functions and may be used in place of parameterized distributions on the right-hand-side of sampling statements.\n\n\n\nFunctions of certain types are restricted on scope of usage. Functions whose names end in _lp assume access to the log probability accumulator and are only available in the transformed parameter and model blocks.\nFunctions whose names end in _rng assume access to the random number generator and may only be used within the generated quantities block, transformed data block, and within user-defined functions ending in _rng.\nFunctions whose names end in _lpdf and _lpmf can be used anywhere. However, _lupdf and _lupmf functions can only be used in the model block or user-defined probability functions.\nSee the section on function bodies for more information on these special types of function.\n\n\n\n\nStan’s functions all have declared types for both arguments and returned value. As with built-in functions, user-defined functions are only declared for base argument type and dimensionality. This requires a different syntax than for declaring other variables. The choice of language was made so that return types and argument types could use the same declaration syntax.\nThe type void may not be used as an argument type, only a return type for a function with side effects.\n\n\nThe base variable types are integer, real, complex, vector, row_vector, and matrix. No lower-bound or upper-bound constraints are allowed (e.g., real&lt;lower=0&gt; is illegal). Specialized constrained types are also not allowed (e.g., simplex is illegal).\nTuple types of the form tuple(T1, ..., TN) are also allowed, with all of the types T1 to TN being function argument types (i.e., no constraints and no sizes).\n\n\n\nArguments and return types may be arrays, and these are indicated with optional brackets and commas as would be used for indexing. For example, int denotes a single integer argument or return, whereas array[] real indicates a one-dimensional array of reals, array[,] real a two-dimensional array and array[,,] real a three-dimensional array; whitespace is optional, as usual.\nThe dimensions for vectors and matrices are not included, so that matrix is the type of a single matrix argument or return type. Thus if a variable is declared as matrix a, then a has two indexing dimensions, so that a[1] is a row vector and a[1, 1] a real value. Matrices implicitly have two indexing dimensions. The type declaration matrix[ , ] b specifies that b is a two-dimensional array of matrices, for a total of four indexing dimensions, with b[1, 1, 1, 1] picking out a real value.\n\n\n\nFunction argument and return types are not themselves checked for dimensionality. A matrix of any size may be passed in as a matrix argument. Nevertheless, a user-defined function might call a function (such as a multivariate normal density) that itself does dimensionality checks.\nDimensions of function return values will be checked if they’re assigned to a previously declared variable. They may also be checked if they are used as the argument to a function.\nAny errors raised by calls to functions inside user functions or return type mismatches are simply passed on; this typically results in a warning message and rejection of a proposal during sampling or optimization.\n\n\n\nSome of Stan’s built-in functions, like the differential equation solvers, have arguments that must be data. Such data-only arguments must be expressions involving only data, transformed data, and generated quantity variables.\nIn user-defined functions, the qualifier data may be placed before an argument type declaration to indicate that the argument must be data only. For example,\nreal foo(data real x) {\n  return x^2;\n}\nrequires the argument x to be data only.\nDeclaring an argument data only allows type inference to proceed in the body of the function so that, for example, the variable may be used as a data-only argument to a built-in function.\n\n\n\n\nThe body of a function is between an open curly brace ({) and close curly brace (}). The body may contain local variable declarations at the top of the function body’s block and these scope the same way as local variables used in any other statement block.\nAny user-defined function may be used in the function body regardless of the order in which the function definitions appear in the file. Self-recursive and mutually recursive functions are possible without any additional declarations.\nThe only restrictions on statements in function bodies are external, and determine whether the log probability accumulator or random number generators are available; see the rest of this section for details.\n\n\nFunctions that call random number generating functions in their bodies must have a name that ends in _rng; attempts to use random-number generators in other functions lead to a compile-time error.\nLike other random number generating functions, user-defined functions with names that end in _rng may be used only in the generated quantities block and transformed data block, or within the bodies of user-defined functions ending in _rng. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions that include sampling statements or log probability increment statements must have a name that ends in _lp. Attempts to use sampling statements or increment log probability statements in other functions lead to a compile-time error.\nLike the target log density increment statement and sampling statements, user-defined functions with names that end in _lp may only be used in blocks where the log probability accumulator is accessible, namely the transformed parameters and model blocks. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions whose names end in _lpdf and _lpmf (density and mass functions) can be used as probability functions in sampling statements. As with the built-in functions, the first argument will appear on the left of the sampling statement operator (~) in the sampling statement and the other arguments follow. For example, suppose a function returning the log of the density of y given parameter theta allows the use of the sampling statement is defined as follows.\nreal foo_lpdf(real y, vector theta) { ... }\nNote that for function definitions, the comma is used rather than the vertical bar.\nFor every custom _lpdf and _lpmf defined there is a corresponding _lupdf and _lupmf defined automatically. The _lupdf and _lupmf versions of the functions cannot be defined directly (to do so will produce an error). The difference in the _lpdf and _lpmf and the corresponding _lupdf and _lupmf functions is that if any other unnormalized density functions are used inside the user-defined function, the _lpdf and _lpmf forms of the user-defined function will change these densities to be normalized. The _lupdf and _lupmf forms of the user-defined functions will instead allow other unnormalized density functions to drop additive constants.\nThe sampling shorthand\nz ~ foo(phi);\nwill have the same effect as incrementing the target with the log of the unnormalized density:\ntarget += foo_lupdf(z | phi);\nOther _lupdf and _lupmf functions used in the definition of foo_lpdf will drop additive constants when foo_lupdf is called and will not drop additive constants when foo_lpdf is called.\nIf there are _lupdf and _lupmf functions used inside the following call to foo_lpdf, they will be forced to normalize (return the equivalent of their _lpdf and _lpmf forms):\ntarget += foo_lpdf(z | phi);\nIf there are no _lupdf or _lupmf functions used in the definition of foo_lpdf, then there will be no difference between a foo_lpdf or foo_lupdf call.\nThe unnormalized _lupdf and _lupmf functions can only be used in the model block or in user-defined probability functions (those ending in _lpdf or _lpmf).\nThe same syntax and shorthand that works for _lpdf also works for log probability mass functions with suffixes _lpmf.\nA function that is going to be accessed as distributions must return the log of the density or mass function it defines.\n\n\n\n\nWithin function definition bodies, the parameters may be used like any other variable. But the parameters are constant in the sense that they can’t be assigned to (i.e., can’t appear on the left side of an assignment (=) statement). In other words, their value remains constant throughout the function body. Attempting to assign a value to a function parameter value will raise a compile-time error.1\nLocal variables may be declared at the top of the function block and scope as usual.\n\n\n\nNon-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.\n\n\n\n\n\n\nA function can be declared without a return value by using void in place of a return type. Note that the type void may only be used as a return type—arguments may not be declared to be of type void.\n\n\n\nA void function may be used as a statement.\nBecause there is no return, such a usage is only for side effects, such as incrementing the log probability function, printing, or raising an error.\n\n\n\nIn a return statement within a void function’s definition, the return keyword is followed immediately by a semicolon (;) rather than by the expression whose value is returned.\n\n\n\n\nStan supports forward declarations, which look like function definitions without bodies. For example,\nreal unit_normal_lpdf(real y);\ndeclares a function named unit_normal_lpdf that consumes a single real-valued input and produces a real-valued output. Declaring a function without a definition is only really useful when using an extension which supplies the definition in C++ rather than in the Stan code itself. How exactly this can be accomplished will differ depending on your Stan interface.\nA function definition with a body simultaneously declares and defines the named function, as in\nreal unit_normal_lpdf(real y) {\n  return -0.5 * square(y);\n}\nA function can be declared and (perhaps separately) defined at most once. However, functions with different argument types are considered distinct even if they have the same name; see the section on function overloading.",
+    "text": "Stan allows users to define their own functions. The basic syntax is a simplified version of that used in C and C++. This chapter specifies how functions are declared, defined, and used in Stan.\n\n\nUser-defined functions appear in a special function-definition block before all of the other program blocks.\nfunctions {\n   // ... function declarations and definitions ...\n}\ndata {\n  // ...\nFunction definitions and declarations may appear in any order. Forward declarations are allowed but not required.\n\n\n\nThe rules for function naming and function-argument naming are the same as for other variables; see the section on variables for more information on valid identifiers. For example,\nreal foo(real mu, real sigma);\ndeclares a function named foo with two argument variables of types real and real. The arguments are named mu and sigma, but that is not part of the declaration.\n\n\nMultiple user-defined functions may have the same name if they have different sequences of argument types. This is known as function overloading.\nFor example, the following two functions are both defined with the name add_up\nreal add_up(real a, real b){\n  return a + b;\n}\n\nreal add_up(real a, real b, real c){\n  return a + b + c;\n}\nThe return types of overloaded functions do not need to be the same. One could define an additional add_up function as follows\nint add_up(int a, int b){\n  return a + b;\n}\nThat being said, functions may not use the same name if their signature only differs by the return type.\nFor example, the following is not permitted\n// illegal\nreal baz(int x);\nint baz(int x);\nFunction names used in the Stan standard library may be overloaded by user-defined functions. Exceptions to this are the reduce_sum family of functions and ODE integrators, which cannot be overloaded.\n\n\n\n\nAll function arguments are mandatory—there are no default values.\n\n\nFunctions with non-void return types are called just like any other built-in function in Stan—they are applied to appropriately typed arguments to produce an expression, which has a value when executed.\n\n\n\nFunctions with void return types may be applied to arguments and used as statements. These act like sampling statements or print statements. Such uses are only appropriate for functions that act through side effects, such as incrementing the log probability accumulator, printing, or raising exceptions.\n\n\n\nOverloaded functions alongside type promotion can result in situations where there are multiple valid interpretations of a function call. Stan requires that there be a unique signature which minimizes the number of promotions required.\nConsider the following two overloaded functions\nreal foo(int a, real b);\nreal foo(real a, int b);\nThese functions do not have a unique minimum when called with two integer arguments foo(1,2), and therefore cannot be called as such.\nPromotion of integers to complex numbers is considered as two separate promotions, one from int to real and a second from real to complex. Consider the following functions with real and complex signatures\nreal bar(real x);\nreal bar(complex z);\nA call bar(5) with an integer argument will be resolved to bar(real) because it only requires a single promotion, whereas the promotion to a complex number requires two promotions.\n\n\n\nThe rules for calling functions work the same way as assignment as far as promotion goes. This means that we can promote arguments to the type expected by function arguments. For example, the following will work.\nreal foo(real x) { return ... };\n...\nint a = 5;\nreal b = foo(a); // a promoted to type real\nIn addition to promoting int to real, Stan also promotes real to complex, and by transitivity, int to complex. This also works for containers, so an array of int may be assigned to an array of real of the same shape. And we can also promote vector to complex_vector and similarly for row vectors and matrices.\n\n\n\nFunctions whose name ends in _lpdf or _lpmf (density and mass functions) may be used as probability functions and may be used in place of parameterized distributions on the right-hand-side of sampling statements.\n\n\n\nFunctions of certain types are restricted on scope of usage. Functions whose names end in _lp assume access to the log probability accumulator and are only available in the transformed parameter and model blocks.\nFunctions whose names end in _rng assume access to the random number generator and may only be used within the generated quantities block, transformed data block, and within user-defined functions ending in _rng.\nFunctions whose names end in _lpdf and _lpmf can be used anywhere. However, _lupdf and _lupmf functions can only be used in the model block or user-defined probability functions.\nSee the section on function bodies for more information on these special types of function.\n\n\n\n\nStan’s functions all have declared types for both arguments and returned value. As with built-in functions, user-defined functions are only declared for base argument type and dimensionality. This requires a different syntax than for declaring other variables. The choice of language was made so that return types and argument types could use the same declaration syntax.\nThe type void may not be used as an argument type, only a return type for a function with side effects.\n\n\nThe base variable types are integer, real, complex, vector, row_vector, and matrix. No lower-bound or upper-bound constraints are allowed (e.g., real&lt;lower=0&gt; is illegal). Specialized constrained types are also not allowed (e.g., simplex is illegal).\nTuple types of the form tuple(T1, ..., TN) are also allowed, with all of the types T1 to TN being function argument types (i.e., no constraints and no sizes).\n\n\n\nArguments and return types may be arrays, and these are indicated with optional brackets and commas as would be used for indexing. For example, int denotes a single integer argument or return, whereas array[] real indicates a one-dimensional array of reals, array[,] real a two-dimensional array and array[,,] real a three-dimensional array; whitespace is optional, as usual.\nThe dimensions for vectors and matrices are not included, so that matrix is the type of a single matrix argument or return type. Thus if a variable is declared as matrix a, then a has two indexing dimensions, so that a[1] is a row vector and a[1, 1] a real value. Matrices implicitly have two indexing dimensions. The type declaration matrix[ , ] b specifies that b is a two-dimensional array of matrices, for a total of four indexing dimensions, with b[1, 1, 1, 1] picking out a real value.\n\n\n\nFunction argument and return types are not themselves checked for dimensionality. A matrix of any size may be passed in as a matrix argument. Nevertheless, a user-defined function might call a function (such as a multivariate normal density) that itself does dimensionality checks.\nDimensions of function return values will be checked if they’re assigned to a previously declared variable. They may also be checked if they are used as the argument to a function.\nAny errors raised by calls to functions inside user functions or return type mismatches are simply passed on; this typically results in a warning message and rejection of a proposal during sampling or optimization.\n\n\n\nSome of Stan’s built-in functions, like the differential equation solvers, have arguments that must be data. Such data-only arguments must be expressions involving only data, transformed data, and generated quantity variables.\nIn user-defined functions, the qualifier data may be placed before an argument type declaration to indicate that the argument must be data only. For example,\nreal foo(data real x) {\n  return x^2;\n}\nrequires the argument x to be data only.\nDeclaring an argument data only allows type inference to proceed in the body of the function so that, for example, the variable may be used as a data-only argument to a built-in function.\n\n\n\n\nThe body of a function is between an open curly brace ({) and close curly brace (}). The body may contain local variable declarations at the top of the function body’s block and these scope the same way as local variables used in any other statement block.\nAny user-defined function may be used in the function body regardless of the order in which the function definitions appear in the file. Self-recursive and mutually recursive functions are possible without any additional declarations.\nThe only restrictions on statements in function bodies are external, and determine whether the log probability accumulator or random number generators are available; see the rest of this section for details.\n\n\nFunctions that call random number generating functions in their bodies must have a name that ends in _rng; attempts to use random-number generators in other functions lead to a compile-time error.\nLike other random number generating functions, user-defined functions with names that end in _rng may be used only in the generated quantities block and transformed data block, or within the bodies of user-defined functions ending in _rng. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions that include sampling statements or log probability increment statements must have a name that ends in _lp. Attempts to use sampling statements or increment log probability statements in other functions lead to a compile-time error.\nLike the target log density increment statement and sampling statements, user-defined functions with names that end in _lp may only be used in blocks where the log probability accumulator is accessible, namely the transformed parameters and model blocks. An attempt to use such a function elsewhere results in a compile-time error.\n\n\n\nFunctions whose names end in _lpdf and _lpmf (density and mass functions) can be used as probability functions in sampling statements. As with the built-in functions, the first argument will appear on the left of the sampling statement operator (~) in the sampling statement and the other arguments follow. For example, suppose a function returning the log of the density of y given parameter theta allows the use of the sampling statement is defined as follows.\nreal foo_lpdf(real y, vector theta) { ... }\nNote that for function definitions, the comma is used rather than the vertical bar.\nFor every custom _lpdf and _lpmf defined there is a corresponding _lupdf and _lupmf defined automatically. The _lupdf and _lupmf versions of the functions cannot be defined directly (to do so will produce an error). The difference in the _lpdf and _lpmf and the corresponding _lupdf and _lupmf functions is that if any other unnormalized density functions are used inside the user-defined function, the _lpdf and _lpmf forms of the user-defined function will change these densities to be normalized. The _lupdf and _lupmf forms of the user-defined functions will instead allow other unnormalized density functions to drop additive constants.\nThe sampling shorthand\nz ~ foo(phi);\nwill have the same effect as incrementing the target with the log of the unnormalized density:\ntarget += foo_lupdf(z | phi);\nOther _lupdf and _lupmf functions used in the definition of foo_lpdf will drop additive constants when foo_lupdf is called and will not drop additive constants when foo_lpdf is called.\nIf there are _lupdf and _lupmf functions used inside the following call to foo_lpdf, they will be forced to normalize (return the equivalent of their _lpdf and _lpmf forms):\ntarget += foo_lpdf(z | phi);\nIf there are no _lupdf or _lupmf functions used in the definition of foo_lpdf, then there will be no difference between a foo_lpdf or foo_lupdf call.\nThe unnormalized _lupdf and _lupmf functions can only be used in the model block or in user-defined probability functions (those ending in _lpdf or _lpmf).\nThe same syntax and shorthand that works for _lpdf also works for log probability mass functions with suffixes _lpmf.\nA function that is going to be accessed as distributions must return the log of the density or mass function it defines.\n\n\n\n\nWithin function definition bodies, the parameters may be used like any other variable. But the parameters are constant in the sense that they can’t be assigned to (i.e., can’t appear on the left side of an assignment (=) statement). In other words, their value remains constant throughout the function body. Attempting to assign a value to a function parameter value will raise a compile-time error.1\nLocal variables may be declared at the top of the function block and scope as usual.\n\n\n\nNon-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nAn exception is made for “obviously infinite” loops like while (1), which contain a return statement and no break statements. The only way to exit such a loop is to return, so they are considered as returning statements.\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.\n\n\n\n\n\n\nA function can be declared without a return value by using void in place of a return type. Note that the type void may only be used as a return type—arguments may not be declared to be of type void.\n\n\n\nA void function may be used as a statement.\nBecause there is no return, such a usage is only for side effects, such as incrementing the log probability function, printing, or raising an error.\n\n\n\nIn a return statement within a void function’s definition, the return keyword is followed immediately by a semicolon (;) rather than by the expression whose value is returned.\n\n\n\n\nStan supports forward declarations, which look like function definitions without bodies. For example,\nreal unit_normal_lpdf(real y);\ndeclares a function named unit_normal_lpdf that consumes a single real-valued input and produces a real-valued output. Declaring a function without a definition is only really useful when using an extension which supplies the definition in C++ rather than in the Stan code itself. How exactly this can be accomplished will differ depending on your Stan interface.\nA function definition with a body simultaneously declares and defines the named function, as in\nreal unit_normal_lpdf(real y) {\n  return -0.5 * square(y);\n}\nA function can be declared and (perhaps separately) defined at most once. However, functions with different argument types are considered distinct even if they have the same name; see the section on function overloading.",
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     "title": "User-Defined Functions",
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-    "text": "Non-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.",
+    "text": "Non-void functions must have a return statement that returns an appropriately typed expression. If the expression in a return statement does not have the same type as the return type declared for the function, a compile-time error is raised.\nVoid functions may use return only without an argument, but return statements are not mandatory.\n\n\nUnlike C++, Stan enforces a syntactic guarantee for non-void functions that ensures control will leave a non-void function through an appropriately typed return statement or because an exception is raised in the execution of the function. To enforce this condition, functions must have a return statement as the last statement in their body. This notion of last is defined recursively in terms of statements that qualify as bodies for functions. The base case is that\n\na return statement qualifies,\n\nand the recursive cases are that\n\na sequence of statements qualifies if its last statement qualifies,\na for loop or while loop qualifies if its body qualifies, and\na conditional statement qualifies if it has a default else clause and all of its body statements qualify.\n\nAn exception is made for “obviously infinite” loops like while (1), which contain a return statement and no break statements. The only way to exit such a loop is to return, so they are considered as returning statements.\nThese rules disqualify\nreal foo(real x) {\n  if (x &gt; 2) {\n    return 1.0;\n  } else if (x &lt;= 2) {\n    return -1.0;\n  }\n}\nbecause there is no default else clause, and disqualify\nreal foo(real x) {\n  real y;\n  y = x;\n  while (x &lt; 10) {\n    if (x &gt; 0) {\n      return x;\n    }\n    y = x / 2;\n  }\n}\nbecause the return statement is not the last statement in the while loop. A bogus dummy return could be placed after the while loop in this case. The rules for returns allow\nreal log_fancy(real x) {\n  if (x &lt; 1e-30) {\n    return x;\n  } else if (x &lt; 1e-14) {\n    return x * x;\n  } else {\n    return log(x);\n  }\n}\nbecause there’s a default else clause and each condition body has return as its final statement.",
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     "title": "Latent Discrete Parameters",
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-    "text": "Stan does not support sampling discrete parameters. So it is not possible to directly translate BUGS or JAGS models with discrete parameters (i.e., discrete stochastic nodes). Nevertheless, it is possible to code many models that involve bounded discrete parameters by marginalizing out the discrete parameters.1\nThis chapter shows how to code several widely-used models involving latent discrete parameters. The next chapter, the clustering chapter, on clustering models, considers further models involving latent discrete parameters.\n\n\nAlthough it requires some algebra on the joint probability function, a pleasant byproduct of the required calculations is the posterior expectation of the marginalized variable, which is often the quantity of interest for a model. This allows far greater exploration of the tails of the distribution as well as more efficient sampling on an iteration-by-iteration basis because the expectation at all possible values is being used rather than itself being estimated through sampling a discrete parameter.\nStandard optimization algorithms, including expectation maximization (EM), are often provided in applied statistics papers to describe maximum likelihood estimation algorithms. Such derivations provide exactly the marginalization needed for coding the model in Stan.\n\n\n\nThe first example is a model of coal mining disasters in the U.K. for the years 1851–1962.2\n\n\nFonnesbeck et al. (2013, sec. 3.1) provides a Poisson model of disaster \\(D_t\\) in year \\(t\\) with two rate parameters, an early rate (\\(e\\)) and late rate (\\(l\\)), that change at a given point in time \\(s\\). The full model expressed using a latent discrete parameter \\(s\\) is \\[\\begin{align*}\ne   &\\sim  \\textsf{exponential}(r_e) \\\\\nl   &\\sim  \\textsf{exponential}(r_l) \\\\\ns   &\\sim  \\textsf{uniform}(1, T) \\\\\nD_t &\\sim  \\textsf{Poisson}(t &lt; s \\; ? \\; e \\: : \\: l)\n\\end{align*}\\]\nThe last line uses the conditional operator (also known as the ternary operator), which is borrowed from C and related languages. The conditional operator has the same behavior as its counterpart in C++.3\nIt uses a compact notation involving separating its three arguments by a question mark (?) and a colon (:). The conditional operator is defined by \\[\nc \\; ? \\; x_1 \\: : \\: x_2\n=\n\\begin{cases}\n\\ x_1 & \\quad\\text{if } c \\text{ is true (i.e., non-zero), and} \\\\\n\\ x_2 & \\quad\\text{if } c \\text{ is false (i.e., zero).}\n\\end{cases}\n\\]\n\n\n\nTo code this model in Stan, the discrete parameter \\(s\\) must be marginalized out to produce a model defining the log of the probability function \\(p(e,l,D_t)\\). The full joint probability factors as \\[\\begin{align*}\np(e,l,s,D) &=  p(e) \\, p(l) \\, p(s) \\, p(D \\mid s, e, l) \\\\\n&= \\textsf{exponential}(e \\mid r_e) \\ \\textsf{exponential}(l \\mid r_l) \\,\n    \\textsf{uniform}(s \\mid 1, T) \\\\\n& \\qquad \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nTo marginalize, an alternative factorization into prior and likelihood is used, \\[\np(e,l,D) = p(e,l) \\, p(D \\mid e,l),\n\\]\nwhere the likelihood is defined by marginalizing \\(s\\) as \\[\\begin{align*}\np(D \\mid e,l) &= \\sum_{s=1}^T p(s, D \\mid e,l) \\\\\n&= \\sum_{s=1}^T p(s) \\, p(D \\mid s,e,l) \\\\\n&= \\sum_{s=1}^T \\textsf{uniform}(s \\mid 1,T) \\,\n    \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nStan operates on the log scale and thus requires the log likelihood, \\[\\begin{align*}\n\\log p(D \\mid e,l)\n&= \\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{s=1}^T\n    \\left( \\log \\textsf{uniform}(s \\mid 1, T) \\vphantom{\\sum_{t=1}^T}\\right. \\\\\n&\\qquad \\left.\n    + \\sum_{t=1}^T \\log \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l)\n\\right),\n\\end{align*}\\] where the log sum of exponents function is defined by \\[\n\\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{n=1}^N \\, \\alpha_n =\n\\log \\sum_{n=1}^N \\exp(\\alpha_n).\n\\]\nThe log sum of exponents function allows the model to be coded directly in Stan using the built-in function log_sum_exp, which provides both arithmetic stability and efficiency for mixture model calculations.\n\n\n\nThe Stan program for the change point model is shown in the figure below. The transformed parameter lp[s] stores the quantity \\(\\log p(s, D \\mid e, l)\\).\ndata {\n  real&lt;lower=0&gt; r_e;\n  real&lt;lower=0&gt; r_l;\n\n  int&lt;lower=1&gt; T;\n  array[T] int&lt;lower=0&gt; D;\n}\ntransformed data {\n  real log_unif;\n  log_unif = -log(T);\n}\nparameters {\n  real&lt;lower=0&gt; e;\n  real&lt;lower=0&gt; l;\n}\ntransformed parameters {\n  vector[T] lp;\n  lp = rep_vector(log_unif, T);\n  for (s in 1:T) {\n    for (t in 1:T) {\n      lp[s] = lp[s] + poisson_lpmf(D[t] | t &lt; s ? e : l);\n    }\n  }\n}\nmodel {\n  e ~ exponential(r_e);\n  l ~ exponential(r_l);\n  target += log_sum_exp(lp);\n}\nA change point model in which disaster rates D[t] have one rate, e, before the change point and a different rate, l, after the change point. The change point itself, s, is marginalized out as described in the text.\nAlthough the change-point model is coded directly, the doubly nested loop used for s and t is quadratic in T. Luke Wiklendt pointed out that a linear alternative can be achieved by the use of dynamic programming similar to the forward-backward algorithm for Hidden Markov models; he submitted a slight variant of the following code to replace the transformed parameters block of the above Stan program.\ntransformed parameters {\n    vector[T] lp;\n    {\n      vector[T + 1] lp_e;\n      vector[T + 1] lp_l;\n      lp_e[1] = 0;\n      lp_l[1] = 0;\n      for (t in 1:T) {\n        lp_e[t + 1] = lp_e[t] + poisson_lpmf(D[t] | e);\n        lp_l[t + 1] = lp_l[t] + poisson_lpmf(D[t] | l);\n      }\n      lp = rep_vector(log_unif + lp_l[T + 1], T)\n           + head(lp_e, T) - head(lp_l, T);\n    }\n  }\nAs should be obvious from looking at it, it has linear complexity in T rather than quadratic. The result for the mining-disaster data is about 20 times faster; the improvement will be greater for larger T.\nThe key to understanding Wiklendt’s dynamic programming version is to see that head(lp_e) holds the forward values, whereas lp_l[T + 1] - head(lp_l, T) holds the backward values; the clever use of subtraction allows lp_l to be accumulated naturally in the forward direction.\n\n\n\nThis model is easy to fit using MCMC with NUTS in its default configuration. Convergence is fast and sampling produces roughly one effective sample every two iterations. Because it is a relatively small model (the inner double loop over time is roughly 20,000 steps), it is fast.\nThe value of lp for each iteration for each change point is available because it is declared as a transformed parameter. If the value of lp were not of interest, it could be coded as a local variable in the model block and thus avoid the I/O overhead of saving values every iteration.\n\n\n\nThe value of lp[s] in a given iteration is given by \\(\\log\np(s,D \\mid e,l)\\) for the values of the early and late rates, \\(e\\) and \\(l\\), in the iteration. In each iteration after convergence, the early and late disaster rates, \\(e\\) and \\(l\\), are drawn from the posterior \\(p(e,l \\mid D)\\) by MCMC sampling and the associated lp calculated. The value of lp may be normalized to calculate \\(p(s \\mid e,l,D)\\) in each iteration, based on on the current values of \\(e\\) and \\(l\\). Averaging over iterations provides an unnormalized probability estimate of the change point being \\(s\\) (see below for the normalizing constant), \\[\\begin{align*}\np(s \\mid D) &\\propto q(s \\mid D) \\\\\n&= \\frac{1}{M} \\sum_{m=1}^{M} \\exp(\\texttt{lp}[m,s]).\n\\end{align*}\\] where \\(\\texttt{lp}[m,s]\\) represents the value of lp in posterior draw \\(m\\) for change point \\(s\\). By averaging over draws, \\(e\\) and \\(l\\) are themselves marginalized out, and the result has no dependence on a given iteration’s value for \\(e\\) and \\(l\\). A final normalization then produces the quantity of interest, the posterior probability of the change point being \\(s\\) conditioned on the data \\(D\\), \\[\np(s \\mid D) = \\frac{q(s \\mid D)}{\\sum_{s'=1}^T q(s' \\mid D)}.\n\\]\nA plot of the values of \\(\\log p(s \\mid D)\\) computed using Stan 2.4’s default MCMC implementation is shown in the posterior plot.\nLog probability of change point being in year, calculated analytically.\n\n\n\nAnalytical change-point posterior\n\n\nThe frequency of change points generated by sampling the discrete change points.\n\n\n\nSampled change-point posterior\n\n\nIn order their range of estimates be visible, the first plot is on the log scale and the second plot on the linear scale; note the narrower range of years in the second plot resulting from sampling. The posterior mean of \\(s\\) is roughly 1891.\n\n\n\nThe generated quantities block may be used to draw discrete parameter values using the built-in pseudo-random number generators. For example, with lp defined as above, the following program draws a random value for s at every iteration.\ngenerated quantities {\n  int&lt;lower=1, upper=T&gt; s;\n  s = categorical_logit_rng(lp);\n}\nA posterior histogram of draws for \\(s\\) is shown on the second change point posterior figure above.\nCompared to working in terms of expectations, discrete sampling is highly inefficient, especially for tails of distributions, so this approach should only be used if draws from a distribution are explicitly required. Otherwise, expectations should be computed in the generated quantities block based on the posterior distribution for s given by softmax(lp).\n\n\n\nThe discrete sample generated for \\(s\\) can be used to calculate covariance with other parameters. Although the sampling approach is straightforward, it is more statistically efficient (in the sense of requiring far fewer iterations for the same degree of accuracy) to calculate these covariances in expectation using lp.\n\n\n\nThere is no obstacle in principle to allowing multiple change points. The only issue is that computation increases from linear to quadratic in marginalizing out two change points, cubic for three change points, and so on. There are three parameters, e, m, and l, and two loops for the change point and then one over time, with log densities being stored in a matrix.\nmatrix[T, T] lp;\nlp = rep_matrix(log_unif, T);\nfor (s1 in 1:T) {\n  for (s2 in 1:T) {\n    for (t in 1:T) {\n      lp[s1,s2] = lp[s1,s2]\n        + poisson_lpmf(D[t] | t &lt; s1 ? e : (t &lt; s2 ? m : l));\n    }\n  }\n}\nThe matrix can then be converted back to a vector using to_vector before being passed to log_sum_exp.\n\n\n\n\nA widely applied field method in ecology is to capture (or sight) animals, mark them (e.g., by tagging), then release them. This process is then repeated one or more times, and is often done for populations on an ongoing basis. The resulting data may be used to estimate population size.\nThe first subsection describes a simple mark-recapture model that does not involve any latent discrete parameters. The following subsections describes the Cormack-Jolly-Seber model, which involves latent discrete parameters for animal death.\n\n\nIn the simplest case, a one-stage mark-recapture study produces the following data\n\n\\(M\\) : number of animals marked in first capture,\n\\(C\\) : number animals in second capture, and\n\\(R\\) : number of marked animals in second capture.\n\nThe estimand of interest is\n\n\\(N\\) : number of animals in the population.\n\nDespite the notation, the model will take \\(N\\) to be a continuous parameter; just because the population must be finite doesn’t mean the parameter representing it must be. The parameter will be used to produce a real-valued estimate of the population size.\nThe Lincoln-Petersen (Lincoln 1930; Petersen 1896) method for estimating population size is \\[\n\\hat{N} = \\frac{M C}{R}.\n\\]\nThis population estimate would arise from a probabilistic model in which the number of recaptured animals is distributed binomially, \\[\nR \\sim \\textsf{binomial}(C, M / N)\n\\] given the total number of animals captured in the second round (\\(C\\)) with a recapture probability of \\(M/N\\), the fraction of the total population \\(N\\) marked in the first round.\ndata {\n  int&lt;lower=0&gt; M;\n  int&lt;lower=0&gt; C;\n  int&lt;lower=0, upper=min(M, C)&gt; R;\n}\nparameters {\n  real&lt;lower=(C - R + M)&gt; N;\n}\nmodel {\n  R ~ binomial(C, M / N);\n}\nA probabilistic formulation of the Lincoln-Petersen estimator for population size based on data from a one-step mark-recapture study. The lower bound on \\(N\\) is necessary to efficiently eliminate impossible values.\nThe probabilistic variant of the Lincoln-Petersen estimator can be directly coded in Stan as shown in the Lincon-Petersen model figure. The Lincoln-Petersen estimate is the maximum likelihood estimate (MLE) for this model.\nTo ensure the MLE is the Lincoln-Petersen estimate, an improper uniform prior for \\(N\\) is used; this could (and should) be replaced with a more informative prior if possible, based on knowledge of the population under study.\nThe one tricky part of the model is the lower bound \\(C - R + M\\) placed on the population size \\(N\\). Values below this bound are impossible because it is otherwise not possible to draw \\(R\\) samples out of the \\(C\\) animals recaptured. Implementing this lower bound is necessary to ensure sampling and optimization can be carried out in an unconstrained manner with unbounded support for parameters on the transformed (unconstrained) space. The lower bound in the declaration for \\(C\\) implies a variable transform \\(f : (C-R+M,\\infty) \\rightarrow (-\\infty,+\\infty)\\) defined by \\(f(N) = \\log(N - (C - R + M))\\); the reference manual contains full details of all constrained parameter transforms.\n\n\n\nThe Cormack-Jolly-Seber (CJS) model (Cormack 1964; Jolly 1965; Seber 1965) is an open-population model in which the population may change over time due to death; the presentation here draws heavily on Schofield (2007).\nThe basic data are\n\n\\(I\\): number of individuals,\n\\(T\\): number of capture periods, and\n\\(y_{i,t}\\): Boolean indicating if individual \\(i\\) was captured at time \\(t\\).\n\nEach individual is assumed to have been captured at least once because an individual only contributes information conditionally after they have been captured the first time.\nThere are two Bernoulli parameters in the model,\n\n\\(\\phi_t\\) : probability that animal alive at time \\(t\\) survives until \\(t + 1\\) and\n\\(p_t\\) : probability that animal alive at time \\(t\\) is captured at time \\(t\\).\n\nThese parameters will both be given uniform priors, but information should be used to tighten these priors in practice.\nThe CJS model also employs a latent discrete parameter \\(z_{i,t}\\) indicating for each individual \\(i\\) whether it is alive at time \\(t\\), distributed as \\[\nz_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t-1} \\; ? \\; 0 \\: : \\: \\phi_{t-1}).\n\\]\nThe conditional prevents the model positing zombies; once an animal is dead, it stays dead. The data distribution is then simple to express conditional on \\(z\\) as \\[\ny_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t} \\; ? \\; 0 \\: : \\: p_t).\n\\]\nThe conditional enforces the constraint that dead animals cannot be captured.\n\n\n\nThis subsection presents an implementation of the model in terms of counts for different history profiles for individuals over three capture times. It assumes exchangeability of the animals in that each is assigned the same capture and survival probabilities.\nIn order to ease the marginalization of the latent discrete parameter \\(z_{i,t}\\), the Stan models rely on a derived quantity \\(\\chi_t\\) for the probability that an individual is never captured again if it is alive at time \\(t\\) (if it is dead, the recapture probability is zero). this quantity is defined recursively by \\[\n\\chi_t\n=\n\\begin{cases}\n1 & \\quad\\text{if } t = T \\\\\n(1 - \\phi_t) + \\phi_t (1 - p_{t+1}) \\chi_{t+1}\n  & \\quad\\text{if } t &lt; T\n\\end{cases}\n\\]\nThe base case arises because if an animal was captured in the last time period, the probability it is never captured again is 1 because there are no more capture periods. The recursive case defining \\(\\chi_{t}\\) in terms of \\(\\chi_{t+1}\\) involves two possibilities: (1) not surviving to the next time period, with probability \\((1 - \\phi_t)\\), or (2) surviving to the next time period with probability \\(\\phi_t\\), not being captured in the next time period with probability \\((1 - p_{t+1})\\), and not being captured again after being alive in period \\(t+1\\) with probability \\(\\chi_{t+1}\\).\nWith three capture times, there are eight captured/not-captured profiles an individual may have. These may be naturally coded as binary numbers as follows.\n\\[\n\\begin{array}{crclc}\n\\hline\n& \\qquad\\qquad & captures & \\qquad\\qquad &  \\\\\n\\mathrm{profile} & 1 & 2 & 3 & \\mathrm{probability} \\\\\n\\hline\n0 & - & - & - & n/a \\\\\n1 & - & - & + & n/a \\\\\n2 & - & + & - & \\chi_2 \\\\\n3 & - & + & + & \\phi_2 \\, p_3 \\\\\n4 & + & - & - & \\chi_1 \\\\\n5 & + & - & + & \\phi_1 \\, (1 - p_2) \\, \\phi_2 \\, p_3 \\\\\n6 & + & + & - & \\phi_1 \\, p_2 \\, \\chi_2 \\\\\n7 & + & + & + & \\phi_1 \\, p_2 \\, \\phi_2 \\, p_3 \\\\\n\\hline\n\\end{array}\n\\]\nHistory 0, for animals that are never captured, is unobservable because only animals that are captured are observed. History 1, for animals that are only captured in the last round, provides no information for the CJS model, because capture/non-capture status is only informative when conditioned on earlier captures. For the remaining cases, the contribution to the likelihood is provided in the final column.\nBy defining these probabilities in terms of \\(\\chi\\) directly, there is no need for a latent binary parameter indicating whether an animal is alive at time \\(t\\) or not. The definition of \\(\\chi\\) is typically used to define the likelihood (i.e., marginalize out the latent discrete parameter) for the CJS model (Schofield 2007).\nThe Stan model defines \\(\\chi\\) as a transformed parameter based on parameters \\(\\phi\\) and \\(p\\). In the model block, the log probability is incremented for each history based on its count. This second step is similar to collecting Bernoulli observations into a binomial or categorical observations into a multinomial, only it is coded directly in the Stan program using target += rather than being part of a built-in probability function.\nThe following is the Stan program for the Cormack-Jolly-Seber mark-recapture model that considers counts of individuals with observation histories of being observed or not in three capture periods\ndata {\n  array[7] int&lt;lower=0&gt; history;\n}\nparameters {\n  array[2] real&lt;lower=0, upper=1&gt; phi;\n  array[3] real&lt;lower=0, upper=1&gt; p;\n}\ntransformed parameters {\n  array[2] real&lt;lower=0, upper=1&gt; chi;\n  chi[2] = (1 - phi[2]) + phi[2] * (1 - p[3]);\n  chi[1] = (1 - phi[1]) + phi[1] * (1 - p[2]) * chi[2];\n}\nmodel {\n  target += history[2] * log(chi[2]);\n  target += history[3] * (log(phi[2]) + log(p[3]));\n  target += history[4] * (log(chi[1]));\n  target += history[5] * (log(phi[1]) + log1m(p[2])\n                            + log(phi[2]) + log(p[3]));\n  target += history[6] * (log(phi[1]) + log(p[2])\n                            + log(chi[2]));\n  target += history[7] * (log(phi[1]) + log(p[2])\n                            + log(phi[2]) + log(p[3]));\n}\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; beta3;\n  beta3 = phi[2] * p[3];\n}\n\n\nThe parameters \\(\\phi_2\\) and \\(p_3\\), the probability of death at time 2 and probability of capture at time 3 are not identifiable, because both may be used to account for lack of capture at time 3. Their product, \\(\\beta_3 = \\phi_2 \\, p_3\\), is identified. The Stan model defines beta3 as a generated quantity. Unidentified parameters pose a problem for Stan’s samplers’ adaptation. Although the problem posed for adaptation is mild here because the parameters are bounded and thus have proper uniform priors, it would be better to formulate an identified parameterization. One way to do this would be to formulate a hierarchical model for the \\(p\\) and \\(\\phi\\) parameters.\n\n\n\n\nThis section presents a version of the Cormack-Jolly-Seber (CJS) model cast at the individual level rather than collectively as in the previous subsection. It also extends the model to allow an arbitrary number of time periods. The data will consist of the number \\(T\\) of capture events, the number \\(I\\) of individuals, and a boolean flag \\(y_{i,t}\\) indicating if individual \\(i\\) was observed at time \\(t\\). In Stan,\ndata {\n  int&lt;lower=2&gt; T;\n  int&lt;lower=0&gt; I;\n  array[I, T] int&lt;lower=0, upper=1&gt; y;\n}\nThe advantages to the individual-level model is that it becomes possible to add individual “random effects” that affect survival or capture probability, as well as to avoid the combinatorics involved in unfolding \\(2^T\\) observation histories for \\(T\\) capture times.\n\n\nThe individual CJS model is written involves several function definitions. The first two are used in the transformed data block to compute the first and last time period in which an animal was captured.4\nfunctions {\n  int first_capture(array[] int y_i) {\n    for (k in 1:size(y_i)) {\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  int last_capture(array[] int y_i) {\n    for (k_rev in 0:(size(y_i) - 1)) {\n      int k;\n      k = size(y_i) - k_rev;\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  // ...\n}\nThese two functions are used to define the first and last capture time for each individual in the transformed data block.5\ntransformed data {\n  array[I] int&lt;lower=0, upper=T&gt; first;\n  array[I] int&lt;lower=0, upper=T&gt; last;\n  vector&lt;lower=0, upper=I&gt;[T] n_captured;\n  for (i in 1:I) {\n    first[i] = first_capture(y[i]);\n  }\n  for (i in 1:I) {\n    last[i] = last_capture(y[i]);\n  }\n  n_captured = rep_vector(0, T);\n  for (t in 1:T) {\n    for (i in 1:I) {\n      if (y[i, t]) {\n        n_captured[t] = n_captured[t] + 1;\n      }\n    }\n  }\n}\nThe transformed data block also defines n_captured[t], which is the total number of captures at time t. The variable n_captured is defined as a vector instead of an integer array so that it can be used in an elementwise vector operation in the generated quantities block to model the population estimates at each time point.\nThe parameters and transformed parameters are as before, but now there is a function definition for computing the entire vector chi, the probability that if an individual is alive at t that it will never be captured again.\nparameters {\n  vector&lt;lower=0, upper=1&gt;[T - 1] phi;\n  vector&lt;lower=0, upper=1&gt;[T] p;\n}\ntransformed parameters {\n  vector&lt;lower=0, upper=1&gt;[T] chi;\n  chi = prob_uncaptured(T, p, phi);\n}\nThe definition of prob_uncaptured, from the functions block, is\nfunctions {\n  // ...\n  vector prob_uncaptured(int T, vector p, vector phi) {\n    vector[T] chi;\n    chi[T] = 1.0;\n    for (t in 1:(T - 1)) {\n      int t_curr;\n      int t_next;\n      t_curr = T - t;\n      t_next = t_curr + 1;\n      chi[t_curr] = (1 - phi[t_curr])\n                     + phi[t_curr]\n                       * (1 - p[t_next])\n                       * chi[t_next];\n    }\n    return chi;\n  }\n}\nThe function definition directly follows the mathematical definition of \\(\\chi_t\\), unrolling the recursion into an iteration and defining the elements of chi from T down to 1.\n\n\n\nGiven the precomputed quantities, the model block directly encodes the CJS model’s log likelihood function. All parameters are left with their default uniform priors and the model simply encodes the log probability of the observations q given the parameters p and phi as well as the transformed parameter chi defined in terms of p and phi.\nmodel {\n  for (i in 1:I) {\n    if (first[i] &gt; 0) {\n      for (t in (first[i]+1):last[i]) {\n        1 ~ bernoulli(phi[t - 1]);\n        y[i, t] ~ bernoulli(p[t]);\n      }\n      1 ~ bernoulli(chi[last[i]]);\n    }\n  }\n}\nThe outer loop is over individuals, conditional skipping individuals i which are never captured. The never-captured check depends on the convention of the first-capture and last-capture functions returning 0 for first if an individual is never captured.\nThe inner loop for individual i first increments the log probability based on the survival of the individual with probability phi[t - 1]. The outcome of 1 is fixed because the individual must survive between the first and last capture (i.e., no zombies). The loop starts after the first capture, because all information in the CJS model is conditional on the first capture.\nIn the inner loop, the observed capture status y[i, t] for individual i at time t has a Bernoulli distribution based on the capture probability p[t] at time t.\nAfter the inner loop, the probability of an animal never being seen again after being observed at time last[i] is included, because last[i] was defined to be the last time period in which animal i was observed.\n\n\n\nAs with the collective model described in the previous subsection, this model does not identify phi[T - 1] and p[T], but does identify their product, beta. Thus beta is defined as a generated quantity to monitor convergence and report.\ngenerated quantities {\n  real beta;\n  // ...\n\n  beta = phi[T - 1] * p[T];\n  // ...\n}\nThe parameter p[1] is also not modeled and will just be uniform between 0 and 1. A more finely articulated model might have a hierarchical or time-series component, in which case p[1] would be an unknown initial condition and both phi[T - 1] and p[T] could be identified.\n\n\n\nThe generated quantities also calculates an estimate of the population mean at each time t in the same way as in the simple mark-recapture model as the number of individuals captured at time t divided by the probability of capture at time t. This is done with the elementwise division operation for vectors (./) in the generated quantities block.\ngenerated quantities {\n  // ...\n  vector&lt;lower=0&gt;[T] pop;\n  // ...\n  pop = n_captured ./ p;\n  pop[1] = -1;\n}\n\n\n\nAll individuals are modeled as having the same capture probability, but this model could be easily generalized to use a logistic regression here based on individual-level inputs to be used as predictors.\n\n\n\n\n\nAlthough seemingly disparate tasks, the rating/coding/annotation of items with categories and diagnostic testing for disease or other conditions, share several characteristics which allow their statistical properties to be modeled similarly.\n\n\nSuppose you have diagnostic tests for a condition of varying sensitivity and specificity. Sensitivity is the probability a test returns positive when the patient has the condition and specificity is the probability that a test returns negative when the patient does not have the condition. For example, mammograms and puncture biopsy tests both test for the presence of breast cancer. Mammograms have high sensitivity and low specificity, meaning lots of false positives, whereas puncture biopsies are the opposite, with low sensitivity and high specificity, meaning lots of false negatives.\nThere are several estimands of interest in such studies. An epidemiological study may be interested in the prevalence of a kind of infection, such as malaria, in a population. A test development study might be interested in the diagnostic accuracy of a new test. A health care worker performing tests might be interested in the disease status of a particular patient.\n\n\n\nHumans are often given the task of coding (equivalently rating or annotating) data. For example, journal or grant reviewers rate submissions, a political study may code campaign commercials as to whether they are attack ads or not, a natural language processing study might annotate Tweets as to whether they are positive or negative in overall sentiment, or a dentist looking at an X-ray classifies a patient as having a cavity or not. In all of these cases, the data coders play the role of the diagnostic tests and all of the same estimands are in play — data coder accuracy and bias, true categories of items being coded, or the prevalence of various categories of items in the data.\n\n\n\nIn this section, only categorical ratings are considered, and the challenge in the modeling for Stan is to marginalize out the discrete parameters.\nDawid and Skene (1979) introduce a noisy-measurement model for coding and apply it in the epidemiological setting of coding what doctors say about patient histories; the same model can be used for diagnostic procedures.\n\n\nThe data for the model consists of \\(J\\) raters (diagnostic tests), \\(I\\) items (patients), and \\(K\\) categories (condition statuses) to annotate, with \\(y_{i, j} \\in \\{1, \\dotsc, K\\}\\) being the rating provided by rater \\(j\\) for item \\(i\\). In a diagnostic test setting for a particular condition, the raters are diagnostic procedures and often \\(K=2\\), with values signaling the presence or absence of the condition.6\nIt is relatively straightforward to extend Dawid and Skene’s model to deal with the situation where not every rater rates each item exactly once.\n\n\n\n\nThe model is based on three parameters, the first of which is discrete:\n\n\\(z_i\\) : a value in \\(\\{1, \\dotsc, K\\}\\) indicating the true category of item \\(i\\),\n\\(\\pi\\) : a \\(K\\)-simplex for the prevalence of the \\(K\\) categories in the population, and\n\\(\\theta_{j,k}\\) : a \\(K\\)-simplex for the response of annotator \\(j\\) to an item of true category \\(k\\).\n\n\n\n\nThe true category of an item is assumed to be generated by a simple categorical distribution based on item prevalence, \\[\nz_i \\sim \\textsf{categorical}(\\pi).\n\\]\nThe rating \\(y_{i, j}\\) provided for item \\(i\\) by rater \\(j\\) is modeled as a categorical response of rater \\(i\\) to an item of category \\(z_i\\),7 \\[\ny_{i, j} \\sim \\textsf{categorical}(\\theta_{j,\\pi_{z[i]}}).\n\\]\n\n\nDawid and Skene provided maximum likelihood estimates for \\(\\theta\\) and \\(\\pi\\), which allows them to generate probability estimates for each \\(z_i\\).\nTo mimic Dawid and Skene’s maximum likelihood model, the parameters \\(\\theta_{j,k}\\) and \\(\\pi\\) can be given uniform priors over \\(K\\)-simplexes. It is straightforward to generalize to Dirichlet priors, \\[\n\\pi \\sim \\textsf{Dirichlet}(\\alpha)\n\\] and \\[\n\\theta_{j,k} \\sim \\textsf{Dirichlet}(\\beta_k)\n\\] with fixed hyperparameters \\(\\alpha\\) (a vector) and \\(\\beta\\) (a matrix or array of vectors). The prior for \\(\\theta_{j,k}\\) must be allowed to vary in \\(k\\), so that, for instance, \\(\\beta_{k,k}\\) is large enough to allow the prior to favor better-than-chance annotators over random or adversarial ones.\nBecause there are \\(J\\) coders, it would be natural to extend the model to include a hierarchical prior for \\(\\beta\\) and to partially pool the estimates of coder accuracy and bias.\n\n\n\nBecause the true category parameter \\(z\\) is discrete, it must be marginalized out of the joint posterior in order to carry out sampling or maximum likelihood estimation in Stan. The joint posterior factors as \\[\np(y, \\theta, \\pi) = p(y \\mid \\theta,\\pi) \\, p(\\pi) \\, p(\\theta),\n\\] where \\(p(y \\mid \\theta,\\pi)\\) is derived by marginalizing \\(z\\) out of \\[\np(z, y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\left( \\textsf{categorical}(z_i \\mid \\pi)\n                     \\prod_{j=1}^J\n                     \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, z[i]})\n              \\right).\n\\]\nThis can be done item by item, with \\[\np(y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\sum_{k=1}^K\n  \\left( \\textsf{categorical}(k \\mid \\pi)\n         \\prod_{j=1}^J\n         \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n  \\right).\n\\]\nIn the missing data model, only the observed labels would be used in the inner product.\nDawid and Skene (1979) derive exactly the same equation in their Equation (2.7), required for the E-step in their expectation maximization (EM) algorithm. Stan requires the marginalized probability function on the log scale, \\[\\begin{align*}\n\\log p(y \\mid \\theta, \\pi)\n&= \\sum_{i=1}^I \\log \\left( \\sum_{k=1}^K \\exp\n    \\left(\\log \\textsf{categorical}(k \\mid \\pi) \\vphantom{\\sum_{j=1}^J}\\right.\\right.\n    \\left.\\left. + \\ \\sum_{j=1}^J\n           \\log \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n    \\right) \\right),\n\\end{align*}\\] which can be directly coded using Stan’s built-in log_sum_exp function.\n\n\n\n\nThe Stan program for the Dawid and Skene model is provided below (Dawid and Skene 1979).\ndata {\n  int&lt;lower=2&gt; K;\n  int&lt;lower=1&gt; I;\n  int&lt;lower=1&gt; J;\n\n  array[I, J] int&lt;lower=1, upper=K&gt; y;\n\n  vector&lt;lower=0&gt;[K] alpha;\n  vector&lt;lower=0&gt;[K] beta[K];\n}\nparameters {\n  simplex[K] pi;\n  array[J, K] simplex[K] theta;\n}\ntransformed parameters {\n  array[I] vector[K] log_q_z;\n  for (i in 1:I) {\n    log_q_z[i] = log(pi);\n    for (j in 1:J) {\n      for (k in 1:K) {\n        log_q_z[i, k] = log_q_z[i, k]\n                         + log(theta[j, k, y[i, j]]);\n      }\n    }\n  }\n}\nmodel {\n  pi ~ dirichlet(alpha);\n  for (j in 1:J) {\n    for (k in 1:K) {\n      theta[j, k] ~ dirichlet(beta[k]);\n    }\n  }\n\n  for (i in 1:I) {\n    target += log_sum_exp(log_q_z[i]);\n  }\n}\nThe model marginalizes out the discrete parameter \\(z\\), storing the unnormalized conditional probability \\(\\log q(z_i=k|\\theta,\\pi)\\) in log_q_z[i, k].\nThe Stan model converges quickly and mixes well using NUTS starting at diffuse initial points, unlike the equivalent model implemented with Gibbs sampling over the discrete parameter. Reasonable weakly informative priors are \\(\\alpha_k = 3\\) and \\(\\beta_{k,k} = 2.5 K\\) and \\(\\beta_{k,k'} = 1\\) if \\(k \\neq k'\\). Taking \\(\\alpha\\) and \\(\\beta_k\\) to be unit vectors and applying optimization will produce the same answer as the expectation maximization (EM) algorithm of Dawid and Skene (1979).\n\n\nThe quantity log_q_z[i] is defined as a transformed parameter. It encodes the (unnormalized) log of \\(p(z_i \\mid \\theta,\n\\pi)\\). Each iteration provides a value conditioned on that iteration’s values for \\(\\theta\\) and \\(\\pi\\). Applying the softmax function to log_q_z[i] provides a simplex corresponding to the probability mass function of \\(z_i\\) in the posterior. These may be averaged across the iterations to provide the posterior probability distribution over each \\(z_i\\).\n\n\n\n\n\n\n\nThis section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z] \\mid \\Theta)] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
+    "text": "Stan does not support sampling discrete parameters. So it is not possible to directly translate BUGS or JAGS models with discrete parameters (i.e., discrete stochastic nodes). Nevertheless, it is possible to code many models that involve bounded discrete parameters by marginalizing out the discrete parameters.1\nThis chapter shows how to code several widely-used models involving latent discrete parameters. The next chapter, the clustering chapter, on clustering models, considers further models involving latent discrete parameters.\n\n\nAlthough it requires some algebra on the joint probability function, a pleasant byproduct of the required calculations is the posterior expectation of the marginalized variable, which is often the quantity of interest for a model. This allows far greater exploration of the tails of the distribution as well as more efficient sampling on an iteration-by-iteration basis because the expectation at all possible values is being used rather than itself being estimated through sampling a discrete parameter.\nStandard optimization algorithms, including expectation maximization (EM), are often provided in applied statistics papers to describe maximum likelihood estimation algorithms. Such derivations provide exactly the marginalization needed for coding the model in Stan.\n\n\n\nThe first example is a model of coal mining disasters in the U.K. for the years 1851–1962.2\n\n\nFonnesbeck et al. (2013, sec. 3.1) provides a Poisson model of disaster \\(D_t\\) in year \\(t\\) with two rate parameters, an early rate (\\(e\\)) and late rate (\\(l\\)), that change at a given point in time \\(s\\). The full model expressed using a latent discrete parameter \\(s\\) is \\[\\begin{align*}\ne   &\\sim  \\textsf{exponential}(r_e) \\\\\nl   &\\sim  \\textsf{exponential}(r_l) \\\\\ns   &\\sim  \\textsf{uniform}(1, T) \\\\\nD_t &\\sim  \\textsf{Poisson}(t &lt; s \\; ? \\; e \\: : \\: l)\n\\end{align*}\\]\nThe last line uses the conditional operator (also known as the ternary operator), which is borrowed from C and related languages. The conditional operator has the same behavior as its counterpart in C++.3\nIt uses a compact notation involving separating its three arguments by a question mark (?) and a colon (:). The conditional operator is defined by \\[\nc \\; ? \\; x_1 \\: : \\: x_2\n=\n\\begin{cases}\n\\ x_1 & \\quad\\text{if } c \\text{ is true (i.e., non-zero), and} \\\\\n\\ x_2 & \\quad\\text{if } c \\text{ is false (i.e., zero).}\n\\end{cases}\n\\]\n\n\n\nTo code this model in Stan, the discrete parameter \\(s\\) must be marginalized out to produce a model defining the log of the probability function \\(p(e,l,D_t)\\). The full joint probability factors as \\[\\begin{align*}\np(e,l,s,D) &=  p(e) \\, p(l) \\, p(s) \\, p(D \\mid s, e, l) \\\\\n&= \\textsf{exponential}(e \\mid r_e) \\ \\textsf{exponential}(l \\mid r_l) \\,\n    \\textsf{uniform}(s \\mid 1, T) \\\\\n& \\qquad \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nTo marginalize, an alternative factorization into prior and likelihood is used, \\[\np(e,l,D) = p(e,l) \\, p(D \\mid e,l),\n\\]\nwhere the likelihood is defined by marginalizing \\(s\\) as \\[\\begin{align*}\np(D \\mid e,l) &= \\sum_{s=1}^T p(s, D \\mid e,l) \\\\\n&= \\sum_{s=1}^T p(s) \\, p(D \\mid s,e,l) \\\\\n&= \\sum_{s=1}^T \\textsf{uniform}(s \\mid 1,T) \\,\n    \\prod_{t=1}^T \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l).\n\\end{align*}\\]\nStan operates on the log scale and thus requires the log likelihood, \\[\\begin{align*}\n\\log p(D \\mid e,l)\n&= \\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{s=1}^T\n    \\left( \\log \\textsf{uniform}(s \\mid 1, T) \\vphantom{\\sum_{t=1}^T}\\right. \\\\\n&\\qquad \\left.\n    + \\sum_{t=1}^T \\log \\textsf{Poisson}(D_t \\mid t &lt; s \\; ? \\; e \\: : \\: l)\n\\right),\n\\end{align*}\\] where the log sum of exponents function is defined by \\[\n\\texttt{log}\\mathtt{\\_}\\texttt{sum}\\mathtt{\\_}\\texttt{exp}_{n=1}^N \\, \\alpha_n =\n\\log \\sum_{n=1}^N \\exp(\\alpha_n).\n\\]\nThe log sum of exponents function allows the model to be coded directly in Stan using the built-in function log_sum_exp, which provides both arithmetic stability and efficiency for mixture model calculations.\n\n\n\nThe Stan program for the change point model is shown in the figure below. The transformed parameter lp[s] stores the quantity \\(\\log p(s, D \\mid e, l)\\).\ndata {\n  real&lt;lower=0&gt; r_e;\n  real&lt;lower=0&gt; r_l;\n\n  int&lt;lower=1&gt; T;\n  array[T] int&lt;lower=0&gt; D;\n}\ntransformed data {\n  real log_unif;\n  log_unif = -log(T);\n}\nparameters {\n  real&lt;lower=0&gt; e;\n  real&lt;lower=0&gt; l;\n}\ntransformed parameters {\n  vector[T] lp;\n  lp = rep_vector(log_unif, T);\n  for (s in 1:T) {\n    for (t in 1:T) {\n      lp[s] = lp[s] + poisson_lpmf(D[t] | t &lt; s ? e : l);\n    }\n  }\n}\nmodel {\n  e ~ exponential(r_e);\n  l ~ exponential(r_l);\n  target += log_sum_exp(lp);\n}\nA change point model in which disaster rates D[t] have one rate, e, before the change point and a different rate, l, after the change point. The change point itself, s, is marginalized out as described in the text.\nAlthough the change-point model is coded directly, the doubly nested loop used for s and t is quadratic in T. Luke Wiklendt pointed out that a linear alternative can be achieved by the use of dynamic programming similar to the forward-backward algorithm for Hidden Markov models; he submitted a slight variant of the following code to replace the transformed parameters block of the above Stan program.\ntransformed parameters {\n    vector[T] lp;\n    {\n      vector[T + 1] lp_e;\n      vector[T + 1] lp_l;\n      lp_e[1] = 0;\n      lp_l[1] = 0;\n      for (t in 1:T) {\n        lp_e[t + 1] = lp_e[t] + poisson_lpmf(D[t] | e);\n        lp_l[t + 1] = lp_l[t] + poisson_lpmf(D[t] | l);\n      }\n      lp = rep_vector(log_unif + lp_l[T + 1], T)\n           + head(lp_e, T) - head(lp_l, T);\n    }\n  }\nAs should be obvious from looking at it, it has linear complexity in T rather than quadratic. The result for the mining-disaster data is about 20 times faster; the improvement will be greater for larger T.\nThe key to understanding Wiklendt’s dynamic programming version is to see that head(lp_e) holds the forward values, whereas lp_l[T + 1] - head(lp_l, T) holds the backward values; the clever use of subtraction allows lp_l to be accumulated naturally in the forward direction.\n\n\n\nThis model is easy to fit using MCMC with NUTS in its default configuration. Convergence is fast and sampling produces roughly one effective sample every two iterations. Because it is a relatively small model (the inner double loop over time is roughly 20,000 steps), it is fast.\nThe value of lp for each iteration for each change point is available because it is declared as a transformed parameter. If the value of lp were not of interest, it could be coded as a local variable in the model block and thus avoid the I/O overhead of saving values every iteration.\n\n\n\nThe value of lp[s] in a given iteration is given by \\(\\log\np(s,D \\mid e,l)\\) for the values of the early and late rates, \\(e\\) and \\(l\\), in the iteration. In each iteration after convergence, the early and late disaster rates, \\(e\\) and \\(l\\), are drawn from the posterior \\(p(e,l \\mid D)\\) by MCMC sampling and the associated lp calculated. The value of lp may be normalized to calculate \\(p(s \\mid e,l,D)\\) in each iteration, based on on the current values of \\(e\\) and \\(l\\). Averaging over iterations provides an unnormalized probability estimate of the change point being \\(s\\) (see below for the normalizing constant), \\[\\begin{align*}\np(s \\mid D) &\\propto q(s \\mid D) \\\\\n&= \\frac{1}{M} \\sum_{m=1}^{M} \\exp(\\texttt{lp}[m,s]).\n\\end{align*}\\] where \\(\\texttt{lp}[m,s]\\) represents the value of lp in posterior draw \\(m\\) for change point \\(s\\). By averaging over draws, \\(e\\) and \\(l\\) are themselves marginalized out, and the result has no dependence on a given iteration’s value for \\(e\\) and \\(l\\). A final normalization then produces the quantity of interest, the posterior probability of the change point being \\(s\\) conditioned on the data \\(D\\), \\[\np(s \\mid D) = \\frac{q(s \\mid D)}{\\sum_{s'=1}^T q(s' \\mid D)}.\n\\]\nA plot of the values of \\(\\log p(s \\mid D)\\) computed using Stan 2.4’s default MCMC implementation is shown in the posterior plot.\nLog probability of change point being in year, calculated analytically.\n\n\n\nAnalytical change-point posterior\n\n\nThe frequency of change points generated by sampling the discrete change points.\n\n\n\nSampled change-point posterior\n\n\nIn order their range of estimates be visible, the first plot is on the log scale and the second plot on the linear scale; note the narrower range of years in the second plot resulting from sampling. The posterior mean of \\(s\\) is roughly 1891.\n\n\n\nThe generated quantities block may be used to draw discrete parameter values using the built-in pseudo-random number generators. For example, with lp defined as above, the following program draws a random value for s at every iteration.\ngenerated quantities {\n  int&lt;lower=1, upper=T&gt; s;\n  s = categorical_logit_rng(lp);\n}\nA posterior histogram of draws for \\(s\\) is shown on the second change point posterior figure above.\nCompared to working in terms of expectations, discrete sampling is highly inefficient, especially for tails of distributions, so this approach should only be used if draws from a distribution are explicitly required. Otherwise, expectations should be computed in the generated quantities block based on the posterior distribution for s given by softmax(lp).\n\n\n\nThe discrete sample generated for \\(s\\) can be used to calculate covariance with other parameters. Although the sampling approach is straightforward, it is more statistically efficient (in the sense of requiring far fewer iterations for the same degree of accuracy) to calculate these covariances in expectation using lp.\n\n\n\nThere is no obstacle in principle to allowing multiple change points. The only issue is that computation increases from linear to quadratic in marginalizing out two change points, cubic for three change points, and so on. There are three parameters, e, m, and l, and two loops for the change point and then one over time, with log densities being stored in a matrix.\nmatrix[T, T] lp;\nlp = rep_matrix(log_unif, T);\nfor (s1 in 1:T) {\n  for (s2 in 1:T) {\n    for (t in 1:T) {\n      lp[s1,s2] = lp[s1,s2]\n        + poisson_lpmf(D[t] | t &lt; s1 ? e : (t &lt; s2 ? m : l));\n    }\n  }\n}\nThe matrix can then be converted back to a vector using to_vector before being passed to log_sum_exp.\n\n\n\n\nA widely applied field method in ecology is to capture (or sight) animals, mark them (e.g., by tagging), then release them. This process is then repeated one or more times, and is often done for populations on an ongoing basis. The resulting data may be used to estimate population size.\nThe first subsection describes a simple mark-recapture model that does not involve any latent discrete parameters. The following subsections describes the Cormack-Jolly-Seber model, which involves latent discrete parameters for animal death.\n\n\nIn the simplest case, a one-stage mark-recapture study produces the following data\n\n\\(M\\) : number of animals marked in first capture,\n\\(C\\) : number animals in second capture, and\n\\(R\\) : number of marked animals in second capture.\n\nThe estimand of interest is\n\n\\(N\\) : number of animals in the population.\n\nDespite the notation, the model will take \\(N\\) to be a continuous parameter; just because the population must be finite doesn’t mean the parameter representing it must be. The parameter will be used to produce a real-valued estimate of the population size.\nThe Lincoln-Petersen (Lincoln 1930; Petersen 1896) method for estimating population size is \\[\n\\hat{N} = \\frac{M C}{R}.\n\\]\nThis population estimate would arise from a probabilistic model in which the number of recaptured animals is distributed binomially, \\[\nR \\sim \\textsf{binomial}(C, M / N)\n\\] given the total number of animals captured in the second round (\\(C\\)) with a recapture probability of \\(M/N\\), the fraction of the total population \\(N\\) marked in the first round.\ndata {\n  int&lt;lower=0&gt; M;\n  int&lt;lower=0&gt; C;\n  int&lt;lower=0, upper=min(M, C)&gt; R;\n}\nparameters {\n  real&lt;lower=(C - R + M)&gt; N;\n}\nmodel {\n  R ~ binomial(C, M / N);\n}\nA probabilistic formulation of the Lincoln-Petersen estimator for population size based on data from a one-step mark-recapture study. The lower bound on \\(N\\) is necessary to efficiently eliminate impossible values.\nThe probabilistic variant of the Lincoln-Petersen estimator can be directly coded in Stan as shown in the Lincon-Petersen model figure. The Lincoln-Petersen estimate is the maximum likelihood estimate (MLE) for this model.\nTo ensure the MLE is the Lincoln-Petersen estimate, an improper uniform prior for \\(N\\) is used; this could (and should) be replaced with a more informative prior if possible, based on knowledge of the population under study.\nThe one tricky part of the model is the lower bound \\(C - R + M\\) placed on the population size \\(N\\). Values below this bound are impossible because it is otherwise not possible to draw \\(R\\) samples out of the \\(C\\) animals recaptured. Implementing this lower bound is necessary to ensure sampling and optimization can be carried out in an unconstrained manner with unbounded support for parameters on the transformed (unconstrained) space. The lower bound in the declaration for \\(C\\) implies a variable transform \\(f : (C-R+M,\\infty) \\rightarrow (-\\infty,+\\infty)\\) defined by \\(f(N) = \\log(N - (C - R + M))\\); the reference manual contains full details of all constrained parameter transforms.\n\n\n\nThe Cormack-Jolly-Seber (CJS) model (Cormack 1964; Jolly 1965; Seber 1965) is an open-population model in which the population may change over time due to death; the presentation here draws heavily on Schofield (2007).\nThe basic data are\n\n\\(I\\): number of individuals,\n\\(T\\): number of capture periods, and\n\\(y_{i,t}\\): Boolean indicating if individual \\(i\\) was captured at time \\(t\\).\n\nEach individual is assumed to have been captured at least once because an individual only contributes information conditionally after they have been captured the first time.\nThere are two Bernoulli parameters in the model,\n\n\\(\\phi_t\\) : probability that animal alive at time \\(t\\) survives until \\(t + 1\\) and\n\\(p_t\\) : probability that animal alive at time \\(t\\) is captured at time \\(t\\).\n\nThese parameters will both be given uniform priors, but information should be used to tighten these priors in practice.\nThe CJS model also employs a latent discrete parameter \\(z_{i,t}\\) indicating for each individual \\(i\\) whether it is alive at time \\(t\\), distributed as \\[\nz_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t-1} \\; ? \\; 0 \\: : \\: \\phi_{t-1}).\n\\]\nThe conditional prevents the model positing zombies; once an animal is dead, it stays dead. The data distribution is then simple to express conditional on \\(z\\) as \\[\ny_{i,t} \\sim \\mathsf{Bernoulli}(z_{i,t} \\; ? \\; 0 \\: : \\: p_t).\n\\]\nThe conditional enforces the constraint that dead animals cannot be captured.\n\n\n\nThis subsection presents an implementation of the model in terms of counts for different history profiles for individuals over three capture times. It assumes exchangeability of the animals in that each is assigned the same capture and survival probabilities.\nIn order to ease the marginalization of the latent discrete parameter \\(z_{i,t}\\), the Stan models rely on a derived quantity \\(\\chi_t\\) for the probability that an individual is never captured again if it is alive at time \\(t\\) (if it is dead, the recapture probability is zero). this quantity is defined recursively by \\[\n\\chi_t\n=\n\\begin{cases}\n1 & \\quad\\text{if } t = T \\\\\n(1 - \\phi_t) + \\phi_t (1 - p_{t+1}) \\chi_{t+1}\n  & \\quad\\text{if } t &lt; T\n\\end{cases}\n\\]\nThe base case arises because if an animal was captured in the last time period, the probability it is never captured again is 1 because there are no more capture periods. The recursive case defining \\(\\chi_{t}\\) in terms of \\(\\chi_{t+1}\\) involves two possibilities: (1) not surviving to the next time period, with probability \\((1 - \\phi_t)\\), or (2) surviving to the next time period with probability \\(\\phi_t\\), not being captured in the next time period with probability \\((1 - p_{t+1})\\), and not being captured again after being alive in period \\(t+1\\) with probability \\(\\chi_{t+1}\\).\nWith three capture times, there are eight captured/not-captured profiles an individual may have. These may be naturally coded as binary numbers as follows.\n\\[\n\\begin{array}{crclc}\n\\hline\n& \\qquad\\qquad & captures & \\qquad\\qquad &  \\\\\n\\mathrm{profile} & 1 & 2 & 3 & \\mathrm{probability} \\\\\n\\hline\n0 & - & - & - & n/a \\\\\n1 & - & - & + & n/a \\\\\n2 & - & + & - & \\chi_2 \\\\\n3 & - & + & + & \\phi_2 \\, p_3 \\\\\n4 & + & - & - & \\chi_1 \\\\\n5 & + & - & + & \\phi_1 \\, (1 - p_2) \\, \\phi_2 \\, p_3 \\\\\n6 & + & + & - & \\phi_1 \\, p_2 \\, \\chi_2 \\\\\n7 & + & + & + & \\phi_1 \\, p_2 \\, \\phi_2 \\, p_3 \\\\\n\\hline\n\\end{array}\n\\]\nHistory 0, for animals that are never captured, is unobservable because only animals that are captured are observed. History 1, for animals that are only captured in the last round, provides no information for the CJS model, because capture/non-capture status is only informative when conditioned on earlier captures. For the remaining cases, the contribution to the likelihood is provided in the final column.\nBy defining these probabilities in terms of \\(\\chi\\) directly, there is no need for a latent binary parameter indicating whether an animal is alive at time \\(t\\) or not. The definition of \\(\\chi\\) is typically used to define the likelihood (i.e., marginalize out the latent discrete parameter) for the CJS model (Schofield 2007).\nThe Stan model defines \\(\\chi\\) as a transformed parameter based on parameters \\(\\phi\\) and \\(p\\). In the model block, the log probability is incremented for each history based on its count. This second step is similar to collecting Bernoulli observations into a binomial or categorical observations into a multinomial, only it is coded directly in the Stan program using target += rather than being part of a built-in probability function.\nThe following is the Stan program for the Cormack-Jolly-Seber mark-recapture model that considers counts of individuals with observation histories of being observed or not in three capture periods\ndata {\n  array[7] int&lt;lower=0&gt; history;\n}\nparameters {\n  array[2] real&lt;lower=0, upper=1&gt; phi;\n  array[3] real&lt;lower=0, upper=1&gt; p;\n}\ntransformed parameters {\n  array[2] real&lt;lower=0, upper=1&gt; chi;\n  chi[2] = (1 - phi[2]) + phi[2] * (1 - p[3]);\n  chi[1] = (1 - phi[1]) + phi[1] * (1 - p[2]) * chi[2];\n}\nmodel {\n  target += history[2] * log(chi[2]);\n  target += history[3] * (log(phi[2]) + log(p[3]));\n  target += history[4] * (log(chi[1]));\n  target += history[5] * (log(phi[1]) + log1m(p[2])\n                            + log(phi[2]) + log(p[3]));\n  target += history[6] * (log(phi[1]) + log(p[2])\n                            + log(chi[2]));\n  target += history[7] * (log(phi[1]) + log(p[2])\n                            + log(phi[2]) + log(p[3]));\n}\ngenerated quantities {\n  real&lt;lower=0, upper=1&gt; beta3;\n  beta3 = phi[2] * p[3];\n}\n\n\nThe parameters \\(\\phi_2\\) and \\(p_3\\), the probability of death at time 2 and probability of capture at time 3 are not identifiable, because both may be used to account for lack of capture at time 3. Their product, \\(\\beta_3 = \\phi_2 \\, p_3\\), is identified. The Stan model defines beta3 as a generated quantity. Unidentified parameters pose a problem for Stan’s samplers’ adaptation. Although the problem posed for adaptation is mild here because the parameters are bounded and thus have proper uniform priors, it would be better to formulate an identified parameterization. One way to do this would be to formulate a hierarchical model for the \\(p\\) and \\(\\phi\\) parameters.\n\n\n\n\nThis section presents a version of the Cormack-Jolly-Seber (CJS) model cast at the individual level rather than collectively as in the previous subsection. It also extends the model to allow an arbitrary number of time periods. The data will consist of the number \\(T\\) of capture events, the number \\(I\\) of individuals, and a boolean flag \\(y_{i,t}\\) indicating if individual \\(i\\) was observed at time \\(t\\). In Stan,\ndata {\n  int&lt;lower=2&gt; T;\n  int&lt;lower=0&gt; I;\n  array[I, T] int&lt;lower=0, upper=1&gt; y;\n}\nThe advantages to the individual-level model is that it becomes possible to add individual “random effects” that affect survival or capture probability, as well as to avoid the combinatorics involved in unfolding \\(2^T\\) observation histories for \\(T\\) capture times.\n\n\nThe individual CJS model is written involves several function definitions. The first two are used in the transformed data block to compute the first and last time period in which an animal was captured.4\nfunctions {\n  int first_capture(array[] int y_i) {\n    for (k in 1:size(y_i)) {\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  int last_capture(array[] int y_i) {\n    for (k_rev in 0:(size(y_i) - 1)) {\n      int k;\n      k = size(y_i) - k_rev;\n      if (y_i[k]) {\n        return k;\n      }\n    }\n    return 0;\n  }\n  // ...\n}\nThese two functions are used to define the first and last capture time for each individual in the transformed data block.5\ntransformed data {\n  array[I] int&lt;lower=0, upper=T&gt; first;\n  array[I] int&lt;lower=0, upper=T&gt; last;\n  vector&lt;lower=0, upper=I&gt;[T] n_captured;\n  for (i in 1:I) {\n    first[i] = first_capture(y[i]);\n  }\n  for (i in 1:I) {\n    last[i] = last_capture(y[i]);\n  }\n  n_captured = rep_vector(0, T);\n  for (t in 1:T) {\n    for (i in 1:I) {\n      if (y[i, t]) {\n        n_captured[t] = n_captured[t] + 1;\n      }\n    }\n  }\n}\nThe transformed data block also defines n_captured[t], which is the total number of captures at time t. The variable n_captured is defined as a vector instead of an integer array so that it can be used in an elementwise vector operation in the generated quantities block to model the population estimates at each time point.\nThe parameters and transformed parameters are as before, but now there is a function definition for computing the entire vector chi, the probability that if an individual is alive at t that it will never be captured again.\nparameters {\n  vector&lt;lower=0, upper=1&gt;[T - 1] phi;\n  vector&lt;lower=0, upper=1&gt;[T] p;\n}\ntransformed parameters {\n  vector&lt;lower=0, upper=1&gt;[T] chi;\n  chi = prob_uncaptured(T, p, phi);\n}\nThe definition of prob_uncaptured, from the functions block, is\nfunctions {\n  // ...\n  vector prob_uncaptured(int T, vector p, vector phi) {\n    vector[T] chi;\n    chi[T] = 1.0;\n    for (t in 1:(T - 1)) {\n      int t_curr;\n      int t_next;\n      t_curr = T - t;\n      t_next = t_curr + 1;\n      chi[t_curr] = (1 - phi[t_curr])\n                     + phi[t_curr]\n                       * (1 - p[t_next])\n                       * chi[t_next];\n    }\n    return chi;\n  }\n}\nThe function definition directly follows the mathematical definition of \\(\\chi_t\\), unrolling the recursion into an iteration and defining the elements of chi from T down to 1.\n\n\n\nGiven the precomputed quantities, the model block directly encodes the CJS model’s log likelihood function. All parameters are left with their default uniform priors and the model simply encodes the log probability of the observations q given the parameters p and phi as well as the transformed parameter chi defined in terms of p and phi.\nmodel {\n  for (i in 1:I) {\n    if (first[i] &gt; 0) {\n      for (t in (first[i]+1):last[i]) {\n        1 ~ bernoulli(phi[t - 1]);\n        y[i, t] ~ bernoulli(p[t]);\n      }\n      1 ~ bernoulli(chi[last[i]]);\n    }\n  }\n}\nThe outer loop is over individuals, conditional skipping individuals i which are never captured. The never-captured check depends on the convention of the first-capture and last-capture functions returning 0 for first if an individual is never captured.\nThe inner loop for individual i first increments the log probability based on the survival of the individual with probability phi[t - 1]. The outcome of 1 is fixed because the individual must survive between the first and last capture (i.e., no zombies). The loop starts after the first capture, because all information in the CJS model is conditional on the first capture.\nIn the inner loop, the observed capture status y[i, t] for individual i at time t has a Bernoulli distribution based on the capture probability p[t] at time t.\nAfter the inner loop, the probability of an animal never being seen again after being observed at time last[i] is included, because last[i] was defined to be the last time period in which animal i was observed.\n\n\n\nAs with the collective model described in the previous subsection, this model does not identify phi[T - 1] and p[T], but does identify their product, beta. Thus beta is defined as a generated quantity to monitor convergence and report.\ngenerated quantities {\n  real beta;\n  // ...\n\n  beta = phi[T - 1] * p[T];\n  // ...\n}\nThe parameter p[1] is also not modeled and will just be uniform between 0 and 1. A more finely articulated model might have a hierarchical or time-series component, in which case p[1] would be an unknown initial condition and both phi[T - 1] and p[T] could be identified.\n\n\n\nThe generated quantities also calculates an estimate of the population mean at each time t in the same way as in the simple mark-recapture model as the number of individuals captured at time t divided by the probability of capture at time t. This is done with the elementwise division operation for vectors (./) in the generated quantities block.\ngenerated quantities {\n  // ...\n  vector&lt;lower=0&gt;[T] pop;\n  // ...\n  pop = n_captured ./ p;\n  pop[1] = -1;\n}\n\n\n\nAll individuals are modeled as having the same capture probability, but this model could be easily generalized to use a logistic regression here based on individual-level inputs to be used as predictors.\n\n\n\n\n\nAlthough seemingly disparate tasks, the rating/coding/annotation of items with categories and diagnostic testing for disease or other conditions, share several characteristics which allow their statistical properties to be modeled similarly.\n\n\nSuppose you have diagnostic tests for a condition of varying sensitivity and specificity. Sensitivity is the probability a test returns positive when the patient has the condition and specificity is the probability that a test returns negative when the patient does not have the condition. For example, mammograms and puncture biopsy tests both test for the presence of breast cancer. Mammograms have high sensitivity and low specificity, meaning lots of false positives, whereas puncture biopsies are the opposite, with low sensitivity and high specificity, meaning lots of false negatives.\nThere are several estimands of interest in such studies. An epidemiological study may be interested in the prevalence of a kind of infection, such as malaria, in a population. A test development study might be interested in the diagnostic accuracy of a new test. A health care worker performing tests might be interested in the disease status of a particular patient.\n\n\n\nHumans are often given the task of coding (equivalently rating or annotating) data. For example, journal or grant reviewers rate submissions, a political study may code campaign commercials as to whether they are attack ads or not, a natural language processing study might annotate Tweets as to whether they are positive or negative in overall sentiment, or a dentist looking at an X-ray classifies a patient as having a cavity or not. In all of these cases, the data coders play the role of the diagnostic tests and all of the same estimands are in play — data coder accuracy and bias, true categories of items being coded, or the prevalence of various categories of items in the data.\n\n\n\nIn this section, only categorical ratings are considered, and the challenge in the modeling for Stan is to marginalize out the discrete parameters.\nDawid and Skene (1979) introduce a noisy-measurement model for coding and apply it in the epidemiological setting of coding what doctors say about patient histories; the same model can be used for diagnostic procedures.\n\n\nThe data for the model consists of \\(J\\) raters (diagnostic tests), \\(I\\) items (patients), and \\(K\\) categories (condition statuses) to annotate, with \\(y_{i, j} \\in \\{1, \\dotsc, K\\}\\) being the rating provided by rater \\(j\\) for item \\(i\\). In a diagnostic test setting for a particular condition, the raters are diagnostic procedures and often \\(K=2\\), with values signaling the presence or absence of the condition.6\nIt is relatively straightforward to extend Dawid and Skene’s model to deal with the situation where not every rater rates each item exactly once.\n\n\n\n\nThe model is based on three parameters, the first of which is discrete:\n\n\\(z_i\\) : a value in \\(\\{1, \\dotsc, K\\}\\) indicating the true category of item \\(i\\),\n\\(\\pi\\) : a \\(K\\)-simplex for the prevalence of the \\(K\\) categories in the population, and\n\\(\\theta_{j,k}\\) : a \\(K\\)-simplex for the response of annotator \\(j\\) to an item of true category \\(k\\).\n\n\n\n\nThe true category of an item is assumed to be generated by a simple categorical distribution based on item prevalence, \\[\nz_i \\sim \\textsf{categorical}(\\pi).\n\\]\nThe rating \\(y_{i, j}\\) provided for item \\(i\\) by rater \\(j\\) is modeled as a categorical response of rater \\(i\\) to an item of category \\(z_i\\),7 \\[\ny_{i, j} \\sim \\textsf{categorical}(\\theta_{j,\\pi_{z[i]}}).\n\\]\n\n\nDawid and Skene provided maximum likelihood estimates for \\(\\theta\\) and \\(\\pi\\), which allows them to generate probability estimates for each \\(z_i\\).\nTo mimic Dawid and Skene’s maximum likelihood model, the parameters \\(\\theta_{j,k}\\) and \\(\\pi\\) can be given uniform priors over \\(K\\)-simplexes. It is straightforward to generalize to Dirichlet priors, \\[\n\\pi \\sim \\textsf{Dirichlet}(\\alpha)\n\\] and \\[\n\\theta_{j,k} \\sim \\textsf{Dirichlet}(\\beta_k)\n\\] with fixed hyperparameters \\(\\alpha\\) (a vector) and \\(\\beta\\) (a matrix or array of vectors). The prior for \\(\\theta_{j,k}\\) must be allowed to vary in \\(k\\), so that, for instance, \\(\\beta_{k,k}\\) is large enough to allow the prior to favor better-than-chance annotators over random or adversarial ones.\nBecause there are \\(J\\) coders, it would be natural to extend the model to include a hierarchical prior for \\(\\beta\\) and to partially pool the estimates of coder accuracy and bias.\n\n\n\nBecause the true category parameter \\(z\\) is discrete, it must be marginalized out of the joint posterior in order to carry out sampling or maximum likelihood estimation in Stan. The joint posterior factors as \\[\np(y, \\theta, \\pi) = p(y \\mid \\theta,\\pi) \\, p(\\pi) \\, p(\\theta),\n\\] where \\(p(y \\mid \\theta,\\pi)\\) is derived by marginalizing \\(z\\) out of \\[\np(z, y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\left( \\textsf{categorical}(z_i \\mid \\pi)\n                     \\prod_{j=1}^J\n                     \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, z[i]})\n              \\right).\n\\]\nThis can be done item by item, with \\[\np(y \\mid \\theta, \\pi) =\n\\prod_{i=1}^I \\sum_{k=1}^K\n  \\left( \\textsf{categorical}(k \\mid \\pi)\n         \\prod_{j=1}^J\n         \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n  \\right).\n\\]\nIn the missing data model, only the observed labels would be used in the inner product.\nDawid and Skene (1979) derive exactly the same equation in their Equation (2.7), required for the E-step in their expectation maximization (EM) algorithm. Stan requires the marginalized probability function on the log scale, \\[\\begin{align*}\n\\log p(y \\mid \\theta, \\pi)\n&= \\sum_{i=1}^I \\log \\left( \\sum_{k=1}^K \\exp\n    \\left(\\log \\textsf{categorical}(k \\mid \\pi) \\vphantom{\\sum_{j=1}^J}\\right.\\right.\n    \\left.\\left. + \\ \\sum_{j=1}^J\n           \\log \\textsf{categorical}(y_{i, j} \\mid \\theta_{j, k})\n    \\right) \\right),\n\\end{align*}\\] which can be directly coded using Stan’s built-in log_sum_exp function.\n\n\n\n\nThe Stan program for the Dawid and Skene model is provided below (Dawid and Skene 1979).\ndata {\n  int&lt;lower=2&gt; K;\n  int&lt;lower=1&gt; I;\n  int&lt;lower=1&gt; J;\n\n  array[I, J] int&lt;lower=1, upper=K&gt; y;\n\n  vector&lt;lower=0&gt;[K] alpha;\n  vector&lt;lower=0&gt;[K] beta[K];\n}\nparameters {\n  simplex[K] pi;\n  array[J, K] simplex[K] theta;\n}\ntransformed parameters {\n  array[I] vector[K] log_q_z;\n  for (i in 1:I) {\n    log_q_z[i] = log(pi);\n    for (j in 1:J) {\n      for (k in 1:K) {\n        log_q_z[i, k] = log_q_z[i, k]\n                         + log(theta[j, k, y[i, j]]);\n      }\n    }\n  }\n}\nmodel {\n  pi ~ dirichlet(alpha);\n  for (j in 1:J) {\n    for (k in 1:K) {\n      theta[j, k] ~ dirichlet(beta[k]);\n    }\n  }\n\n  for (i in 1:I) {\n    target += log_sum_exp(log_q_z[i]);\n  }\n}\nThe model marginalizes out the discrete parameter \\(z\\), storing the unnormalized conditional probability \\(\\log q(z_i=k|\\theta,\\pi)\\) in log_q_z[i, k].\nThe Stan model converges quickly and mixes well using NUTS starting at diffuse initial points, unlike the equivalent model implemented with Gibbs sampling over the discrete parameter. Reasonable weakly informative priors are \\(\\alpha_k = 3\\) and \\(\\beta_{k,k} = 2.5 K\\) and \\(\\beta_{k,k'} = 1\\) if \\(k \\neq k'\\). Taking \\(\\alpha\\) and \\(\\beta_k\\) to be unit vectors and applying optimization will produce the same answer as the expectation maximization (EM) algorithm of Dawid and Skene (1979).\n\n\nThe quantity log_q_z[i] is defined as a transformed parameter. It encodes the (unnormalized) log of \\(p(z_i \\mid \\theta,\n\\pi)\\). Each iteration provides a value conditioned on that iteration’s values for \\(\\theta\\) and \\(\\pi\\). Applying the softmax function to log_q_z[i] provides a simplex corresponding to the probability mass function of \\(z_i\\) in the posterior. These may be averaged across the iterations to provide the posterior probability distribution over each \\(z_i\\).\n\n\n\n\n\n\n\nThis section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z)] \\mid \\Theta] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
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-    "text": "This section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z] \\mid \\Theta)] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
+    "text": "This section describes in more detail the mathematics of statistical inference using the output of marginalized Stan models, such as those presented in the last three sections. It provides a mathematical explanation of why and how certain manipulations of Stan’s output produce valid summaries of the posterior distribution when discrete parameters have been marginalized out of a statistical model. Ultimately, however, fully understanding the mathematics in this section is not necessary to fit models with discrete parameters using Stan.\nThroughout, the model under consideration consists of both continuous parameters, \\(\\Theta\\), and discrete parameters, \\(Z\\). It is also assumed that \\(Z\\) can only take finitely many values, as is the case for all the models described in this chapter of the User’s Guide. To simplify notation, any conditioning on data is suppressed in this section, except where specified. As with all Bayesian analyses, however, all inferences using models with marginalized parameters are made conditional on the observed data.\n\n\n\nWhen performing Bayesian inference, interest often centers on estimating some (constant) low-dimensional summary statistics of the posterior distribution. Mathematically, we are interested in estimating \\(\\mu\\), say, where \\(\\mu = \\mathbb{E}[g(\\Theta, Z)]\\) and \\(g(\\cdot)\\) is an arbitrary function. An example of such a quantity is \\(\\mathbb{E}[\\Theta]\\), the posterior mean of the continuous parameters, where we would take \\(g(\\theta, z) = \\theta\\). To estimate \\(\\mu\\) the most common approach is to sample a series of values, at least approximately, from the posterior distribution of the parameters of interest. The numerical values of these draws can then be used to calculate the quantities of interest. Often, this process of calculation is trivial, but more care is required when working with marginalized posteriors as we describe in this section.\nIf both \\(\\Theta\\) and \\(Z\\) were continuous, Stan could be used to sample \\(M\\) draws from the joint posterior \\(p_{\\Theta, Z}(\\theta, z)\\) and then estimate \\(\\mu\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M {g(\\theta^{(i)}, z^{(i)})}.\n\\] Given \\(Z\\) is discrete, however, Stan cannot be used to sample from the joint posterior (or even to do optimization). Instead, as outlined in the previous sections describing specific models, the user can first marginalize out \\(Z\\) from the joint posterior to give the marginalized posterior \\(p_\\Theta(\\theta)\\). This marginalized posterior can then be implemented in Stan as usual, and Stan will give draws \\(\\{\\theta^{(i)}\\}_{i = 1}^M\\) from the marginalized posterior.\nUsing only these draws, how can we estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\)? We can use a conditional estimator. We explain in more detail below, but at a high level the idea is that, for each function \\(g\\) of interest, we compute \\[\nh(\\Theta) = \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n\\] and then estimate \\(\\mathbb{E}[g(\\Theta, Z)]\\) with \\[\n\\hat{\\mu} = \\frac{1}{M} \\sum_{i = 1}^M h(\\theta^{(i)}).\n\\] This estimator is justified by the law of iterated expectation, the fact that \\[\n\\mathbb{E}[h(\\Theta)] = \\mathbb{E}[\\mathbb{E}[g(\\Theta, Z)] \\mid \\Theta] = \\mathbb{E}[g(\\Theta, Z)] = \\mu.\n\\] Using this marginalized estimator provides a way to estimate the expectation of any function \\(g(\\cdot)\\) for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\).\n\n\n\nFortunately, the discrete nature of \\(Z\\) makes evaluating \\(\\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\\) easy. The function \\(h(\\Theta)\\) can be written as: \\[\nh(\\Theta)\n= \\mathbb{E}[g(\\Theta, Z) \\mid \\Theta]\n= \\sum_{k} g(\\Theta, k) \\Pr[Z = k \\mid \\Theta],\n\\] where we sum over the possible values of the latent discrete parameters. An essential part of this formula is the probability of the discrete parameters conditional on the continuous parameters, \\(\\Pr[Z = k \\mid \\Theta]\\). More detail on how this quantity can be calculated is included below. Note that if \\(Z\\) takes infinitely many values then computing the infinite sums will involve, potentially computationally expensive, approximation.\nWhen \\(g(\\theta, z)\\) is a function of either \\(\\theta\\) or \\(z\\) only, the above formula simplifies further.\nIn the first case, where \\(g(\\theta, z) = g(\\theta)\\), we have: \\[\\begin{align*}\nh(\\Theta)\n&= \\sum_{k} g(\\Theta) \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta) \\sum_{k} \\Pr[Z = k \\mid \\Theta] \\\\\n&= g(\\Theta).\n\\end{align*}\\] This means that we can estimate \\(\\mathbb{E}[g(\\Theta)]\\) with the standard, seemingly unconditional, estimator: \\[\n\\frac{1}{M} \\sum_{i = 1}^M g(\\theta^{(i)}).\n\\] Even after marginalization, computing expectations of functions of the continuous parameters can be performed as if no marginalization had taken place.\nIn the second case, where \\(g(\\theta, z) = g(z)\\), the conditional expectation instead simplifies as follows: \\[\nh(\\Theta) = \\sum_{k} g(k) \\Pr[Z = k \\mid \\Theta].\n\\] An important special case of this result is when \\(g(\\theta, z) = \\textrm{I}(z = k)\\), where \\(\\textrm{I}\\) is the indicator function. This choice allows us to recover the probability mass function of the discrete random variable \\(Z\\), since \\(\\mathbb{E}[\\textrm{I}(Z = k)] = \\Pr[Z = k]\\). In this case, \\[\nh(\\Theta)\n= \\sum_{k} \\textrm{I}(z = k) \\Pr[Z = k \\mid \\Theta]\n= \\Pr[Z = k \\mid \\Theta].\n\\] The quantity \\(\\Pr[Z = k]\\) can therefore be estimated with: \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}].\n\\] When calculating this conditional probability it is important to remember that we are also conditioning on the observed data, \\(Y\\). That is, we are really estimating \\(\\Pr[Z = k \\mid Y]\\) with \\[\n\\frac{1}{M} \\sum_{i = 1}^M \\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y].\n\\] This point is important as it suggests one of the main ways of calculating the required conditional probability. Using Bayes’s theorem gives us \\[\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}, Y]\n= \\frac{\\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}\n{\\sum_{k = 1}^K \\Pr[Y \\mid Z = k, \\Theta = \\theta^{(i)}]\n\\Pr[Z = k \\mid \\Theta = \\theta^{(i)}]}.\n\\] Here, \\(\\Pr[Y \\mid \\Theta = \\theta^{(i)}, Z = k]\\) is the likelihood conditional on a particular value of the latent variables. Crucially, all elements of the expression can be calculated using the draws from the posterior of the continuous parameters and knowledge of the model structure.\nOther than the use of Bayes’s theorem, \\(\\Pr[Z = k \\mid \\theta = \\theta^{(i)}, Y]\\) can also be extracted by coding the Stan model to include the conditional probability explicitly (as is done for the Dawid–Skene model).\nFor a longer introduction to the mathematics of marginalization in Stan, which also covers the connections between Rao–Blackwellization and marginalization, see Pullin, Gurrin, and Vukcevic (2021).",
     "crumbs": [
       "Stan Users Guide",
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+@import"https://fonts.googleapis.com/css2?family=Source+Sans+Pro:wght@300;400;700&display=swap";:root{--stan-bg: #242424;--stan-highlight: #053A42;--stan-secondary: #AED1E4}/*!
+ * Bootstrap  v5.3.1 (https://getbootstrap.com/)
+ * Copyright 2011-2023 The Bootstrap Authors
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
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none}.btn-outline-primary{--bs-btn-color: #2780e3;--bs-btn-border-color: #2780e3;--bs-btn-hover-color: #fff;--bs-btn-hover-bg: #2780e3;--bs-btn-hover-border-color: #2780e3;--bs-btn-focus-shadow-rgb: 39, 128, 227;--bs-btn-active-color: #fff;--bs-btn-active-bg: #2780e3;--bs-btn-active-border-color: #2780e3;--bs-btn-active-shadow: inset 0 3px 5px rgba(0, 0, 0, 0.125);--bs-btn-disabled-color: #2780e3;--bs-btn-disabled-bg: transparent;--bs-btn-disabled-border-color: #2780e3;--bs-btn-bg: transparent;--bs-gradient: none}.btn-outline-secondary{--bs-btn-color: #343a40;--bs-btn-border-color: #343a40;--bs-btn-hover-color: #fff;--bs-btn-hover-bg: #343a40;--bs-btn-hover-border-color: #343a40;--bs-btn-focus-shadow-rgb: 52, 58, 64;--bs-btn-active-color: #fff;--bs-btn-active-bg: #343a40;--bs-btn-active-border-color: #343a40;--bs-btn-active-shadow: inset 0 3px 5px rgba(0, 0, 0, 0.125);--bs-btn-disabled-color: #343a40;--bs-btn-disabled-bg: transparent;--bs-btn-disabled-border-color: #343a40;--bs-btn-bg: 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var(--bs-dropdown-item-padding-x);clear:both;font-weight:400;color:var(--bs-dropdown-link-color);text-align:inherit;text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;white-space:nowrap;background-color:rgba(0,0,0,0);border:0}.dropdown-item:hover,.dropdown-item:focus{color:var(--bs-dropdown-link-hover-color);background-color:var(--bs-dropdown-link-hover-bg)}.dropdown-item.active,.dropdown-item:active{color:var(--bs-dropdown-link-active-color);text-decoration:none;background-color:var(--bs-dropdown-link-active-bg)}.dropdown-item.disabled,.dropdown-item:disabled{color:var(--bs-dropdown-link-disabled-color);pointer-events:none;background-color:rgba(0,0,0,0)}.dropdown-menu.show{display:block}.dropdown-header{display:block;padding:var(--bs-dropdown-header-padding-y) 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auto}.btn-group>.btn-check:checked+.btn,.btn-group>.btn-check:focus+.btn,.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active,.btn-group-vertical>.btn-check:checked+.btn,.btn-group-vertical>.btn-check:focus+.btn,.btn-group-vertical>.btn:hover,.btn-group-vertical>.btn:focus,.btn-group-vertical>.btn:active,.btn-group-vertical>.btn.active{z-index:1}.btn-toolbar{display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;justify-content:flex-start;-webkit-justify-content:flex-start}.btn-toolbar .input-group{width:auto}.btn-group>:not(.btn-check:first-child)+.btn,.btn-group>.btn-group:not(:first-child){margin-left:calc(1px*-1)}.dropdown-toggle-split{padding-right:.5625rem;padding-left:.5625rem}.dropdown-toggle-split::after,.dropup .dropdown-toggle-split::after,.dropend .dropdown-toggle-split::after{margin-left:0}.dropstart .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:.375rem;padding-left:.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:.75rem;padding-left:.75rem}.btn-group-vertical{flex-direction:column;-webkit-flex-direction:column;align-items:flex-start;-webkit-align-items:flex-start;justify-content:center;-webkit-justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:calc(1px*-1)}.nav{--bs-nav-link-padding-x: 1rem;--bs-nav-link-padding-y: 0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color: #AED1E4;--bs-nav-link-hover-color: #8ba7b6;--bs-nav-link-disabled-color: rgba(250, 241, 228, 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#2780e3}.nav-pills .nav-link.active,.nav-pills .show>.nav-link{color:var(--bs-nav-pills-link-active-color);background-color:var(--bs-nav-pills-link-active-bg)}.nav-underline{--bs-nav-underline-gap: 1rem;--bs-nav-underline-border-width: 0.125rem;--bs-nav-underline-link-active-color: #000;gap:var(--bs-nav-underline-gap)}.nav-underline .nav-link{padding-right:0;padding-left:0;border-bottom:var(--bs-nav-underline-border-width) solid rgba(0,0,0,0)}.nav-underline .nav-link:hover,.nav-underline .nav-link:focus{border-bottom-color:currentcolor}.nav-underline .nav-link.active,.nav-underline .show>.nav-link{font-weight:700;color:var(--bs-nav-underline-link-active-color);border-bottom-color:currentcolor}.nav-fill>.nav-link,.nav-fill .nav-item{flex:1 1 auto;-webkit-flex:1 1 auto;text-align:center}.nav-justified>.nav-link,.nav-justified .nav-item{flex-basis:0;-webkit-flex-basis:0;flex-grow:1;-webkit-flex-grow:1;text-align:center}.nav-fill .nav-item .nav-link,.nav-justified .nav-item 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var(--bs-navbar-hover-color);--bs-nav-link-disabled-color: var(--bs-navbar-disabled-color);display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;list-style:none}.navbar-nav .nav-link.active,.navbar-nav .nav-link.show{color:var(--bs-navbar-active-color)}.navbar-nav .dropdown-menu{position:static}.navbar-text{padding-top:.5rem;padding-bottom:.5rem;color:var(--bs-navbar-color)}.navbar-text a,.navbar-text a:hover,.navbar-text a:focus{color:var(--bs-navbar-active-color)}.navbar-collapse{flex-basis:100%;-webkit-flex-basis:100%;flex-grow:1;-webkit-flex-grow:1;align-items:center;-webkit-align-items:center}.navbar-toggler{padding:var(--bs-navbar-toggler-padding-y) var(--bs-navbar-toggler-padding-x);font-size:var(--bs-navbar-toggler-font-size);line-height:1;color:var(--bs-navbar-color);background-color:rgba(0,0,0,0);border:var(--bs-border-width) solid 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992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #bdbdbd;--bs-navbar-hover-color: rgba(174, 209, 228, 0.8);--bs-navbar-disabled-color: rgba(189, 189, 189, 0.75);--bs-navbar-active-color: #aed1e4;--bs-navbar-brand-color: #bdbdbd;--bs-navbar-brand-hover-color: #aed1e4;--bs-navbar-toggler-border-color: rgba(189, 189, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 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0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #FAF1E4;--bs-accordion-bg: #181818;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #FAF1E4;--bs-accordion-btn-bg: #181818;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23FAF1E4'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(250, 241, 228, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(250, 241, 228, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #AED1E4;--bs-pagination-bg: #181818;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #8ba7b6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #8ba7b6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(250, 241, 228, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #FAF1E4;--bs-list-group-bg: #181818;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(250, 241, 228, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #FAF1E4;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(250, 241, 228, 0.75);--bs-list-group-disabled-bg: #181818;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(24, 24, 24, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(250, 241, 228, 0.75);--bs-toast-header-bg: rgba(24, 24, 24, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #181818;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 0.5;position:fixed;top:0;left:0;z-index:var(--bs-backdrop-zindex);width:100vw;height:100vh;background-color:var(--bs-backdrop-bg)}.modal-backdrop.fade{opacity:0}.modal-backdrop.show{opacity:var(--bs-backdrop-opacity)}.modal-header{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;align-items:center;-webkit-align-items:center;justify-content:space-between;-webkit-justify-content:space-between;padding:var(--bs-modal-header-padding);border-bottom:var(--bs-modal-header-border-width) solid var(--bs-modal-header-border-color)}.modal-header .btn-close{padding:calc(var(--bs-modal-header-padding-y)*.5) calc(var(--bs-modal-header-padding-x)*.5);margin:calc(-0.5*var(--bs-modal-header-padding-y)) calc(-0.5*var(--bs-modal-header-padding-x)) calc(-0.5*var(--bs-modal-header-padding-y)) auto}.modal-title{margin-bottom:0;line-height:var(--bs-modal-title-line-height)}.modal-body{position:relative;flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-modal-padding)}.modal-footer{display:flex;display:-webkit-flex;flex-shrink:0;-webkit-flex-shrink:0;flex-wrap:wrap;-webkit-flex-wrap:wrap;align-items:center;-webkit-align-items:center;justify-content:flex-end;-webkit-justify-content:flex-end;padding:calc(var(--bs-modal-padding) - var(--bs-modal-footer-gap)*.5);background-color:var(--bs-modal-footer-bg);border-top:var(--bs-modal-footer-border-width) solid var(--bs-modal-footer-border-color)}.modal-footer>*{margin:calc(var(--bs-modal-footer-gap)*.5)}@media(min-width: 576px){.modal{--bs-modal-margin: 1.75rem;--bs-modal-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15)}.modal-dialog{max-width:var(--bs-modal-width);margin-right:auto;margin-left:auto}.modal-sm{--bs-modal-width: 300px}}@media(min-width: 992px){.modal-lg,.modal-xl{--bs-modal-width: 800px}}@media(min-width: 1200px){.modal-xl{--bs-modal-width: 1140px}}.modal-fullscreen{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen .modal-content{height:100%;border:0}.modal-fullscreen .modal-body{overflow-y:auto}@media(max-width: 575.98px){.modal-fullscreen-sm-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-sm-down .modal-content{height:100%;border:0}.modal-fullscreen-sm-down .modal-body{overflow-y:auto}}@media(max-width: 767.98px){.modal-fullscreen-md-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-md-down .modal-content{height:100%;border:0}.modal-fullscreen-md-down .modal-body{overflow-y:auto}}@media(max-width: 991.98px){.modal-fullscreen-lg-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-lg-down .modal-content{height:100%;border:0}.modal-fullscreen-lg-down .modal-body{overflow-y:auto}}@media(max-width: 1199.98px){.modal-fullscreen-xl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xl-down .modal-content{height:100%;border:0}.modal-fullscreen-xl-down .modal-body{overflow-y:auto}}@media(max-width: 1399.98px){.modal-fullscreen-xxl-down{width:100vw;max-width:none;height:100%;margin:0}.modal-fullscreen-xxl-down .modal-content{height:100%;border:0}.modal-fullscreen-xxl-down .modal-body{overflow-y:auto}}.tooltip{--bs-tooltip-zindex: 1080;--bs-tooltip-max-width: 200px;--bs-tooltip-padding-x: 0.5rem;--bs-tooltip-padding-y: 0.25rem;--bs-tooltip-margin: ;--bs-tooltip-font-size:0.875rem;--bs-tooltip-color: #181818;--bs-tooltip-bg: #000;--bs-tooltip-border-radius: 0.25rem;--bs-tooltip-opacity: 0.9;--bs-tooltip-arrow-width: 0.8rem;--bs-tooltip-arrow-height: 0.4rem;z-index:var(--bs-tooltip-zindex);display:block;margin:var(--bs-tooltip-margin);font-family:"Source Sans Pro",-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,"Helvetica Neue",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI 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var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout 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.nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-sm .navbar-nav-scroll{overflow:visible}.navbar-expand-sm .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-sm .navbar-toggler{display:none}.navbar-expand-sm .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-sm .offcanvas .offcanvas-header{display:none}.navbar-expand-sm .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 768px){.navbar-expand-md{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-md .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-md .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-md .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-md .navbar-nav-scroll{overflow:visible}.navbar-expand-md .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-md .navbar-toggler{display:none}.navbar-expand-md .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-md .offcanvas .offcanvas-header{display:none}.navbar-expand-md .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1200px){.navbar-expand-xl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xl .navbar-nav-scroll{overflow:visible}.navbar-expand-xl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xl .navbar-toggler{display:none}.navbar-expand-xl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xl .offcanvas .offcanvas-header{display:none}.navbar-expand-xl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1400px){.navbar-expand-xxl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xxl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xxl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xxl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xxl .navbar-nav-scroll{overflow:visible}.navbar-expand-xxl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xxl .navbar-toggler{display:none}.navbar-expand-xxl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #F2E5BD;--bs-navbar-hover-color: rgba(147, 176, 241, 0.8);--bs-navbar-disabled-color: rgba(242, 229, 189, 0.75);--bs-navbar-active-color: #93b0f1;--bs-navbar-brand-color: #F2E5BD;--bs-navbar-brand-hover-color: #93b0f1;--bs-navbar-toggler-border-color: rgba(242, 229, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #fff;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #fff;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #343a40;--bs-accordion-btn-bg: #fff;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23343a40'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #fff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #fff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #fff;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #fff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 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140deg), #20c997 var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #3c7fbb;color:#fff}.bg-gradient-teal-purple{--bslib-color-fg: #fff;--bslib-color-bg: #3a918c;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #3a918c;color:#fff}.bg-gradient-teal-pink{--bslib-color-fg: #fff;--bslib-color-bg: #709193;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #709193;color:#fff}.bg-gradient-teal-red{--bslib-color-fg: #fff;--bslib-color-bg: #797971;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #797971;color:#fff}.bg-gradient-teal-orange{--bslib-color-fg: #000;--bslib-color-bg: #73be7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #73be7a;color:#000}.bg-gradient-teal-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #79a764;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #79a764;color:#fff}.bg-gradient-teal-green{--bslib-color-fg: #000;--bslib-color-bg: #2cc164;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #2cc164;color:#000}.bg-gradient-teal-cyan{--bslib-color-fg: #fff;--bslib-color-bg: #509aa5;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #9954bb var(--bg-gradient-end, 180%)) #509aa5;color:#fff}.bg-gradient-cyan-blue{--bslib-color-fg: #fff;--bslib-color-bg: #6b66cb;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #2780e3 var(--bg-gradient-end, 180%)) #6b66cb;color:#fff}.bg-gradient-cyan-indigo{--bslib-color-fg: #fff;--bslib-color-bg: #8539d1;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #8539d1;color:#fff}.bg-gradient-cyan-purple{--bslib-color-fg: #fff;--bslib-color-bg: #834ba2;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #834ba2;color:#fff}.bg-gradient-cyan-pink{--bslib-color-fg: #fff;--bslib-color-bg: #b94ba8;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #b94ba8;color:#fff}.bg-gradient-cyan-red{--bslib-color-fg: #fff;--bslib-color-bg: #c23287;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #c23287;color:#fff}.bg-gradient-cyan-orange{--bslib-color-fg: #fff;--bslib-color-bg: #bc788f;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #bc788f;color:#fff}.bg-gradient-cyan-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #c2617a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #c2617a;color:#fff}.bg-gradient-cyan-green{--bslib-color-fg: #fff;--bslib-color-bg: #757b7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #757b7a;color:#fff}.bg-gradient-cyan-teal{--bslib-color-fg: #fff;--bslib-color-bg: #6983ad;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #20c997 var(--bg-gradient-end, 180%)) #6983ad;color:#fff}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--bslib-sidebar-fg: var(--bs-emphasis-color, black);--bslib-sidebar-main-fg: var(--bs-card-color, var(--bs-body-color));--bslib-sidebar-main-bg: var(--bs-card-bg, var(--bs-body-bg));--bslib-sidebar-toggle-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.1);--bslib-sidebar-padding: calc(var(--bslib-spacer) * 1.5);--bslib-sidebar-icon-size: var(--bslib-spacer, 1rem);--bslib-sidebar-icon-button-size: calc(var(--bslib-sidebar-icon-size, 1rem) * 2);--bslib-sidebar-padding-icon: calc(var(--bslib-sidebar-icon-button-size, 2rem) * 1.5);--bslib-collapse-toggle-border-radius: var(--bs-border-radius, 0.25rem);--bslib-collapse-toggle-transform: 0deg;--bslib-sidebar-toggle-transition-easing: cubic-bezier(1, 0, 0, 1);--bslib-collapse-toggle-right-transform: 180deg;--bslib-sidebar-column-main: minmax(0, 1fr);display:grid !important;grid-template-columns:min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px)) var(--bslib-sidebar-column-main);position:relative;transition:grid-template-columns ease-in-out var(--bslib-sidebar-transition-duration);border:var(--bslib-sidebar-border);border-radius:var(--bslib-sidebar-border-radius)}@media(prefers-reduced-motion: reduce){.bslib-sidebar-layout{transition:none}}.bslib-sidebar-layout[data-bslib-sidebar-border=false]{border:none}.bslib-sidebar-layout[data-bslib-sidebar-border-radius=false]{border-radius:initial}.bslib-sidebar-layout>.main,.bslib-sidebar-layout>.sidebar{grid-row:1/2;border-radius:inherit;overflow:auto}.bslib-sidebar-layout>.main{grid-column:2/3;border-top-left-radius:0;border-bottom-left-radius:0;padding:var(--bslib-sidebar-padding);transition:padding var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout .sidebar-title{font-size:1.25rem;line-height:1.25;margin-top:0;margin-bottom:1rem;padding-bottom:1rem;border-bottom:var(--bslib-sidebar-border)}.bslib-sidebar-layout.sidebar-right{grid-template-columns:var(--bslib-sidebar-column-main) min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px))}.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/2;border-top-right-radius:0;border-bottom-right-radius:0;border-top-left-radius:inherit;border-bottom-left-radius:inherit}.bslib-sidebar-layout.sidebar-right>.sidebar{grid-column:2/3;border-right:none;border-left:var(--bslib-sidebar-vert-border);border-top-left-radius:0;border-bottom-left-radius:0}.bslib-sidebar-layout.sidebar-right>.collapse-toggle{grid-column:2/3;left:var(--bslib-sidebar-icon-size);right:unset;border:var(--bslib-collapse-toggle-border)}.bslib-sidebar-layout.sidebar-right>.collapse-toggle>.collapse-icon{transform:rotateY(var(--bslib-collapse-toggle-right-transform))}.bslib-sidebar-layout.sidebar-collapsed{--bslib-collapse-toggle-transform: 180deg;--bslib-collapse-toggle-right-transform: 0deg;--bslib-sidebar-vert-border: none;grid-template-columns:0 minmax(0, 1fr)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right{grid-template-columns:minmax(0, 1fr) 0}.bslib-sidebar-layout.sidebar-collapsed:not(.transitioning)>.sidebar>*{display:none}.bslib-sidebar-layout.sidebar-collapsed>.main{border-radius:inherit}.bslib-sidebar-layout.sidebar-collapsed:not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed>.collapse-toggle{color:var(--bslib-sidebar-main-fg);top:calc(var(--bslib-sidebar-overlap-counter, 0)*(var(--bslib-sidebar-icon-size) + var(--bslib-sidebar-padding)) + var(--bslib-sidebar-icon-size, 1rem)/2);right:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px))}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.collapse-toggle{left:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px));right:unset}@media(min-width: 576px){.bslib-sidebar-layout.transitioning>.sidebar>.sidebar-content{display:none}}@media(max-width: 575.98px){.bslib-sidebar-layout[data-bslib-sidebar-open=desktop]{--bslib-sidebar-js-init-collapsed: true}.bslib-sidebar-layout>.sidebar,.bslib-sidebar-layout.sidebar-right>.sidebar{border:none}.bslib-sidebar-layout>.main,.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/3}.bslib-sidebar-layout[data-bslib-sidebar-open=always]{display:block !important}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar{max-height:var(--bslib-sidebar-max-height-mobile);overflow-y:auto;border-top:var(--bslib-sidebar-vert-border)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]){grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.sidebar{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.collapse-toggle{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed.sidebar-right{grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always])>.main{opacity:0;transition:opacity var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed>.main{opacity:1}}.accordion .accordion-header{font-size:calc(1.29rem + 0.48vw);margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2;color:var(--bs-heading-color);margin-bottom:0}@media(min-width: 1200px){.accordion .accordion-header{font-size:1.65rem}}.accordion .accordion-icon:not(:empty){margin-right:.75rem;display:flex}.accordion .accordion-button:not(.collapsed){box-shadow:none}.accordion .accordion-button:not(.collapsed):focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow)}:root{--bslib-page-sidebar-title-bg: #052744;--bslib-page-sidebar-title-color: #fff}.bslib-page-title{background-color:var(--bslib-page-sidebar-title-bg);color:var(--bslib-page-sidebar-title-color);font-size:1.25rem;font-weight:300;padding:var(--bslib-spacer, 1rem);padding-left:1.5rem;margin-bottom:0;border-bottom:1px solid #dee2e6}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.form-inline{margin-top:auto;margin-bottom:auto}.nav:not(.nav-hidden).nav-stacked{flex-direction:column;-webkit-flex-direction:column;height:100%}.nav:not(.nav-hidden).nav-stacked>.bslib-nav-spacer{margin-top:auto !important}}.bslib-card{overflow:auto}.bslib-card .card-body+.card-body{padding-top:0}.bslib-card .card-body{overflow:auto}.bslib-card .card-body p{margin-top:0}.bslib-card .card-body p:last-child{margin-bottom:0}.bslib-card .card-body{max-height:var(--bslib-card-body-max-height, none)}.bslib-card[data-full-screen=true]>.card-body{max-height:var(--bslib-card-body-max-height-full-screen, none)}.bslib-card .card-header .form-group{margin-bottom:0}.bslib-card .card-header .selectize-control{margin-bottom:0}.bslib-card .card-header .selectize-control .item{margin-right:1.15rem}.bslib-card .card-footer{margin-top:auto}.bslib-card .bslib-navs-card-title{display:flex;flex-wrap:wrap;justify-content:space-between;align-items:center}.bslib-card .bslib-navs-card-title .nav{margin-left:auto}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border=true]){border:none}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border-radius=true]){border-top-left-radius:0;border-top-right-radius:0}[data-full-screen=true]{position:fixed;inset:3.5rem 1rem 1rem;height:auto !important;max-height:none !important;width:auto !important;z-index:1070}.bslib-full-screen-enter{display:none;position:absolute;bottom:var(--bslib-full-screen-enter-bottom, 0.2rem);right:var(--bslib-full-screen-enter-right, 0);top:var(--bslib-full-screen-enter-top);left:var(--bslib-full-screen-enter-left);color:var(--bslib-color-fg, var(--bs-card-color));background-color:var(--bslib-color-bg, var(--bs-card-bg, var(--bs-body-bg)));border:var(--bs-card-border-width) solid var(--bslib-color-fg, var(--bs-card-border-color));box-shadow:0 2px 4px rgba(0,0,0,.15);margin:.2rem .4rem;padding:.55rem !important;font-size:.8rem;cursor:pointer;opacity:.7;z-index:1070}.bslib-full-screen-enter:hover{opacity:1}.card[data-full-screen=false]:hover>*>.bslib-full-screen-enter{display:block}.bslib-has-full-screen .card:hover>*>.bslib-full-screen-enter{display:none}@media(max-width: 575.98px){.bslib-full-screen-enter{display:none !important}}.bslib-full-screen-exit{position:relative;top:1.35rem;font-size:.9rem;cursor:pointer;text-decoration:none;display:flex;float:right;margin-right:2.15rem;align-items:center;color:rgba(var(--bs-body-bg-rgb), 0.8)}.bslib-full-screen-exit:hover{color:rgba(var(--bs-body-bg-rgb), 1)}.bslib-full-screen-exit svg{margin-left:.5rem;font-size:1.5rem}#bslib-full-screen-overlay{position:fixed;inset:0;background-color:rgba(var(--bs-body-color-rgb), 0.6);backdrop-filter:blur(2px);-webkit-backdrop-filter:blur(2px);z-index:1069;animation:bslib-full-screen-overlay-enter 400ms cubic-bezier(0.6, 0.02, 0.65, 1) 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.nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-sm .navbar-nav-scroll{overflow:visible}.navbar-expand-sm .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-sm .navbar-toggler{display:none}.navbar-expand-sm .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-sm .offcanvas .offcanvas-header{display:none}.navbar-expand-sm .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 768px){.navbar-expand-md{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-md .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-md .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-md .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-md .navbar-nav-scroll{overflow:visible}.navbar-expand-md .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-md .navbar-toggler{display:none}.navbar-expand-md .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-md .offcanvas .offcanvas-header{display:none}.navbar-expand-md .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 992px){.navbar-expand-lg{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-lg .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-lg .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-lg .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-lg .navbar-nav-scroll{overflow:visible}.navbar-expand-lg .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-lg .navbar-toggler{display:none}.navbar-expand-lg .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-lg .offcanvas .offcanvas-header{display:none}.navbar-expand-lg .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1200px){.navbar-expand-xl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xl .navbar-nav-scroll{overflow:visible}.navbar-expand-xl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xl .navbar-toggler{display:none}.navbar-expand-xl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xl .offcanvas .offcanvas-header{display:none}.navbar-expand-xl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}@media(min-width: 1400px){.navbar-expand-xxl{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand-xxl .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand-xxl .navbar-nav .dropdown-menu{position:absolute}.navbar-expand-xxl .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand-xxl .navbar-nav-scroll{overflow:visible}.navbar-expand-xxl .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand-xxl .navbar-toggler{display:none}.navbar-expand-xxl .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #F2E5BD;--bs-navbar-hover-color: rgba(147, 176, 241, 0.8);--bs-navbar-disabled-color: rgba(242, 229, 189, 0.75);--bs-navbar-active-color: #93b0f1;--bs-navbar-brand-color: #F2E5BD;--bs-navbar-brand-hover-color: #93b0f1;--bs-navbar-toggler-border-color: rgba(242, 229, 189, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23F2E5BD' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #FEFBF2;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0}.card>.list-group:last-child{border-bottom-width:0}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}}.accordion{--bs-accordion-color: #343a40;--bs-accordion-bg: #FEFBF2;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #343a40;--bs-accordion-btn-bg: #FEFBF2;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23343a40'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%2310335b'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #93c0f1;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #10335b;--bs-accordion-active-bg: #d4e6f9}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:not(:first-of-type){border-top:0}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%237db3ee'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(52, 58, 64, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(52, 58, 64, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #2761e3;--bs-pagination-bg: #FEFBF2;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #1f4eb6;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #1f4eb6;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #2780e3;--bs-pagination-active-border-color: #2780e3;--bs-pagination-disabled-color: rgba(52, 58, 64, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 0 solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:.5rem}}.progress,.progress-stacked{--bs-progress-height: 0.5rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #2780e3;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #343a40;--bs-list-group-bg: #FEFBF2;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(52, 58, 64, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #343a40;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(52, 58, 64, 0.75);--bs-list-group-disabled-bg: #FEFBF2;--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #2780e3;--bs-list-group-active-border-color: #2780e3;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(39, 128, 227, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(254, 251, 242, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(52, 58, 64, 0.75);--bs-toast-header-bg: rgba(254, 251, 242, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color)}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #FEFBF2;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 1px;position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;min-height:calc(100% - var(--bs-modal-margin)*2)}.modal-content{position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;width:100%;color:var(--bs-modal-color);pointer-events:auto;background-color:var(--bs-modal-bg);background-clip:padding-box;border:var(--bs-modal-border-width) solid var(--bs-modal-border-color);outline:0}.modal-backdrop{--bs-backdrop-zindex: 1050;--bs-backdrop-bg: #000;--bs-backdrop-opacity: 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140deg), #20c997 var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #3c7fbb;color:#fff}.bg-gradient-teal-purple{--bslib-color-fg: #fff;--bslib-color-bg: #3a918c;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #3a918c;color:#fff}.bg-gradient-teal-pink{--bslib-color-fg: #fff;--bslib-color-bg: #709193;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #709193;color:#fff}.bg-gradient-teal-red{--bslib-color-fg: #fff;--bslib-color-bg: #797971;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #797971;color:#fff}.bg-gradient-teal-orange{--bslib-color-fg: #000;--bslib-color-bg: #73be7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #73be7a;color:#000}.bg-gradient-teal-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #79a764;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #79a764;color:#fff}.bg-gradient-teal-green{--bslib-color-fg: #000;--bslib-color-bg: #2cc164;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #2cc164;color:#000}.bg-gradient-teal-cyan{--bslib-color-fg: #fff;--bslib-color-bg: #509aa5;background:linear-gradient(var(--bg-gradient-deg, 140deg), #20c997 var(--bg-gradient-start, 36%), #9954bb var(--bg-gradient-end, 180%)) #509aa5;color:#fff}.bg-gradient-cyan-blue{--bslib-color-fg: #fff;--bslib-color-bg: #6b66cb;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #2780e3 var(--bg-gradient-end, 180%)) #6b66cb;color:#fff}.bg-gradient-cyan-indigo{--bslib-color-fg: #fff;--bslib-color-bg: #8539d1;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #6610f2 var(--bg-gradient-end, 180%)) #8539d1;color:#fff}.bg-gradient-cyan-purple{--bslib-color-fg: #fff;--bslib-color-bg: #834ba2;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #613d7c var(--bg-gradient-end, 180%)) #834ba2;color:#fff}.bg-gradient-cyan-pink{--bslib-color-fg: #fff;--bslib-color-bg: #b94ba8;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #e83e8c var(--bg-gradient-end, 180%)) #b94ba8;color:#fff}.bg-gradient-cyan-red{--bslib-color-fg: #fff;--bslib-color-bg: #c23287;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff0039 var(--bg-gradient-end, 180%)) #c23287;color:#fff}.bg-gradient-cyan-orange{--bslib-color-fg: #fff;--bslib-color-bg: #bc788f;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #f0ad4e var(--bg-gradient-end, 180%)) #bc788f;color:#fff}.bg-gradient-cyan-yellow{--bslib-color-fg: #fff;--bslib-color-bg: #c2617a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #ff7518 var(--bg-gradient-end, 180%)) #c2617a;color:#fff}.bg-gradient-cyan-green{--bslib-color-fg: #fff;--bslib-color-bg: #757b7a;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #3fb618 var(--bg-gradient-end, 180%)) #757b7a;color:#fff}.bg-gradient-cyan-teal{--bslib-color-fg: #fff;--bslib-color-bg: #6983ad;background:linear-gradient(var(--bg-gradient-deg, 140deg), #9954bb var(--bg-gradient-start, 36%), #20c997 var(--bg-gradient-end, 180%)) #6983ad;color:#fff}.bslib-sidebar-layout{--bslib-sidebar-transition-duration: 500ms;--bslib-sidebar-transition-easing-x: cubic-bezier(0.8, 0.78, 0.22, 1.07);--bslib-sidebar-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-border-radius: var(--bs-border-radius);--bslib-sidebar-vert-border: var(--bs-card-border-width, 1px) solid var(--bs-card-border-color, rgba(0, 0, 0, 0.175));--bslib-sidebar-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.05);--bslib-sidebar-fg: var(--bs-emphasis-color, black);--bslib-sidebar-main-fg: var(--bs-card-color, var(--bs-body-color));--bslib-sidebar-main-bg: var(--bs-card-bg, var(--bs-body-bg));--bslib-sidebar-toggle-bg: rgba(var(--bs-emphasis-color-rgb, 0, 0, 0), 0.1);--bslib-sidebar-padding: calc(var(--bslib-spacer) * 1.5);--bslib-sidebar-icon-size: var(--bslib-spacer, 1rem);--bslib-sidebar-icon-button-size: calc(var(--bslib-sidebar-icon-size, 1rem) * 2);--bslib-sidebar-padding-icon: calc(var(--bslib-sidebar-icon-button-size, 2rem) * 1.5);--bslib-collapse-toggle-border-radius: var(--bs-border-radius, 0.25rem);--bslib-collapse-toggle-transform: 0deg;--bslib-sidebar-toggle-transition-easing: cubic-bezier(1, 0, 0, 1);--bslib-collapse-toggle-right-transform: 180deg;--bslib-sidebar-column-main: minmax(0, 1fr);display:grid !important;grid-template-columns:min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px)) var(--bslib-sidebar-column-main);position:relative;transition:grid-template-columns ease-in-out var(--bslib-sidebar-transition-duration);border:var(--bslib-sidebar-border);border-radius:var(--bslib-sidebar-border-radius)}@media(prefers-reduced-motion: reduce){.bslib-sidebar-layout{transition:none}}.bslib-sidebar-layout[data-bslib-sidebar-border=false]{border:none}.bslib-sidebar-layout[data-bslib-sidebar-border-radius=false]{border-radius:initial}.bslib-sidebar-layout>.main,.bslib-sidebar-layout>.sidebar{grid-row:1/2;border-radius:inherit;overflow:auto}.bslib-sidebar-layout>.main{grid-column:2/3;border-top-left-radius:0;border-bottom-left-radius:0;padding:var(--bslib-sidebar-padding);transition:padding var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration);color:var(--bslib-sidebar-main-fg);background-color:var(--bslib-sidebar-main-bg)}.bslib-sidebar-layout>.sidebar{grid-column:1/2;width:100%;height:100%;border-right:var(--bslib-sidebar-vert-border);border-top-right-radius:0;border-bottom-right-radius:0;color:var(--bslib-sidebar-fg);background-color:var(--bslib-sidebar-bg);backdrop-filter:blur(5px)}.bslib-sidebar-layout>.sidebar>.sidebar-content{display:flex;flex-direction:column;gap:var(--bslib-spacer, 1rem);padding:var(--bslib-sidebar-padding);padding-top:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout>.sidebar>.sidebar-content>:last-child:not(.sidebar-title){margin-bottom:0}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion{margin-left:calc(-1*var(--bslib-sidebar-padding));margin-right:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:last-child{margin-bottom:calc(-1*var(--bslib-sidebar-padding))}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child){margin-bottom:1rem}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion .accordion-body{display:flex;flex-direction:column}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:first-child) .accordion-item:first-child{border-top:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content>.accordion:not(:last-child) .accordion-item:last-child{border-bottom:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.bslib-sidebar-layout>.sidebar>.sidebar-content.has-accordion>.sidebar-title{border-bottom:none;padding-bottom:0}.bslib-sidebar-layout>.sidebar .shiny-input-container{width:100%}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar>.sidebar-content{padding-top:var(--bslib-sidebar-padding)}.bslib-sidebar-layout>.collapse-toggle{grid-row:1/2;grid-column:1/2;display:inline-flex;align-items:center;position:absolute;right:calc(var(--bslib-sidebar-icon-size));top:calc(var(--bslib-sidebar-icon-size, 1rem)/2);border:none;border-radius:var(--bslib-collapse-toggle-border-radius);height:var(--bslib-sidebar-icon-button-size, 2rem);width:var(--bslib-sidebar-icon-button-size, 2rem);display:flex;align-items:center;justify-content:center;padding:0;color:var(--bslib-sidebar-fg);background-color:unset;transition:color var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),top var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),right var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration),left var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover{background-color:var(--bslib-sidebar-toggle-bg)}.bslib-sidebar-layout>.collapse-toggle>.collapse-icon{opacity:.8;width:var(--bslib-sidebar-icon-size);height:var(--bslib-sidebar-icon-size);transform:rotateY(var(--bslib-collapse-toggle-transform));transition:transform var(--bslib-sidebar-toggle-transition-easing) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout>.collapse-toggle:hover>.collapse-icon{opacity:1}.bslib-sidebar-layout .sidebar-title{font-size:1.25rem;line-height:1.25;margin-top:0;margin-bottom:1rem;padding-bottom:1rem;border-bottom:var(--bslib-sidebar-border)}.bslib-sidebar-layout.sidebar-right{grid-template-columns:var(--bslib-sidebar-column-main) min(100% - var(--bslib-sidebar-icon-size),var(--bslib-sidebar-width, 250px))}.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/2;border-top-right-radius:0;border-bottom-right-radius:0;border-top-left-radius:inherit;border-bottom-left-radius:inherit}.bslib-sidebar-layout.sidebar-right>.sidebar{grid-column:2/3;border-right:none;border-left:var(--bslib-sidebar-vert-border);border-top-left-radius:0;border-bottom-left-radius:0}.bslib-sidebar-layout.sidebar-right>.collapse-toggle{grid-column:2/3;left:var(--bslib-sidebar-icon-size);right:unset;border:var(--bslib-collapse-toggle-border)}.bslib-sidebar-layout.sidebar-right>.collapse-toggle>.collapse-icon{transform:rotateY(var(--bslib-collapse-toggle-right-transform))}.bslib-sidebar-layout.sidebar-collapsed{--bslib-collapse-toggle-transform: 180deg;--bslib-collapse-toggle-right-transform: 0deg;--bslib-sidebar-vert-border: none;grid-template-columns:0 minmax(0, 1fr)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right{grid-template-columns:minmax(0, 1fr) 0}.bslib-sidebar-layout.sidebar-collapsed:not(.transitioning)>.sidebar>*{display:none}.bslib-sidebar-layout.sidebar-collapsed>.main{border-radius:inherit}.bslib-sidebar-layout.sidebar-collapsed:not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout.sidebar-collapsed>.collapse-toggle{color:var(--bslib-sidebar-main-fg);top:calc(var(--bslib-sidebar-overlap-counter, 0)*(var(--bslib-sidebar-icon-size) + var(--bslib-sidebar-padding)) + var(--bslib-sidebar-icon-size, 1rem)/2);right:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px))}.bslib-sidebar-layout.sidebar-collapsed.sidebar-right>.collapse-toggle{left:calc(-2.5*var(--bslib-sidebar-icon-size) - var(--bs-card-border-width, 1px));right:unset}@media(min-width: 576px){.bslib-sidebar-layout.transitioning>.sidebar>.sidebar-content{display:none}}@media(max-width: 575.98px){.bslib-sidebar-layout[data-bslib-sidebar-open=desktop]{--bslib-sidebar-js-init-collapsed: true}.bslib-sidebar-layout>.sidebar,.bslib-sidebar-layout.sidebar-right>.sidebar{border:none}.bslib-sidebar-layout>.main,.bslib-sidebar-layout.sidebar-right>.main{grid-column:1/3}.bslib-sidebar-layout[data-bslib-sidebar-open=always]{display:block !important}.bslib-sidebar-layout[data-bslib-sidebar-open=always]>.sidebar{max-height:var(--bslib-sidebar-max-height-mobile);overflow-y:auto;border-top:var(--bslib-sidebar-vert-border)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]){grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.sidebar{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-collapsed)>.collapse-toggle{z-index:1}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed{grid-template-columns:0 100%}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed.sidebar-right{grid-template-columns:100% 0}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]):not(.sidebar-right)>.main{padding-left:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-right>.main{padding-right:var(--bslib-sidebar-padding-icon)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always])>.main{opacity:0;transition:opacity var(--bslib-sidebar-transition-easing-x) var(--bslib-sidebar-transition-duration)}.bslib-sidebar-layout:not([data-bslib-sidebar-open=always]).sidebar-collapsed>.main{opacity:1}}.accordion .accordion-header{font-size:calc(1.29rem + 0.48vw);margin-top:0;margin-bottom:.5rem;font-weight:400;line-height:1.2;color:var(--bs-heading-color);margin-bottom:0}@media(min-width: 1200px){.accordion .accordion-header{font-size:1.65rem}}.accordion .accordion-icon:not(:empty){margin-right:.75rem;display:flex}.accordion .accordion-button:not(.collapsed){box-shadow:none}.accordion .accordion-button:not(.collapsed):focus{box-shadow:var(--bs-accordion-btn-focus-box-shadow)}:root{--bslib-page-sidebar-title-bg: #052744;--bslib-page-sidebar-title-color: #fff}.bslib-page-title{background-color:var(--bslib-page-sidebar-title-bg);color:var(--bslib-page-sidebar-title-color);font-size:1.25rem;font-weight:300;padding:var(--bslib-spacer, 1rem);padding-left:1.5rem;margin-bottom:0;border-bottom:1px solid #dee2e6}@media(min-width: 576px){.nav:not(.nav-hidden){display:flex !important;display:-webkit-flex !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column){float:none !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.bslib-nav-spacer{margin-left:auto !important}.nav:not(.nav-hidden):not(.nav-stacked):not(.flex-column)>.form-inline{margin-top:auto;margin-bottom:auto}.nav:not(.nav-hidden).nav-stacked{flex-direction:column;-webkit-flex-direction:column;height:100%}.nav:not(.nav-hidden).nav-stacked>.bslib-nav-spacer{margin-top:auto !important}}.bslib-card{overflow:auto}.bslib-card .card-body+.card-body{padding-top:0}.bslib-card .card-body{overflow:auto}.bslib-card .card-body p{margin-top:0}.bslib-card .card-body p:last-child{margin-bottom:0}.bslib-card .card-body{max-height:var(--bslib-card-body-max-height, none)}.bslib-card[data-full-screen=true]>.card-body{max-height:var(--bslib-card-body-max-height-full-screen, none)}.bslib-card .card-header .form-group{margin-bottom:0}.bslib-card .card-header .selectize-control{margin-bottom:0}.bslib-card .card-header .selectize-control .item{margin-right:1.15rem}.bslib-card .card-footer{margin-top:auto}.bslib-card .bslib-navs-card-title{display:flex;flex-wrap:wrap;justify-content:space-between;align-items:center}.bslib-card .bslib-navs-card-title .nav{margin-left:auto}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border=true]){border:none}.bslib-card .bslib-sidebar-layout:not([data-bslib-sidebar-border-radius=true]){border-top-left-radius:0;border-top-right-radius:0}[data-full-screen=true]{position:fixed;inset:3.5rem 1rem 1rem;height:auto !important;max-height:none !important;width:auto !important;z-index:1070}.bslib-full-screen-enter{display:none;position:absolute;bottom:var(--bslib-full-screen-enter-bottom, 0.2rem);right:var(--bslib-full-screen-enter-right, 0);top:var(--bslib-full-screen-enter-top);left:var(--bslib-full-screen-enter-left);color:var(--bslib-color-fg, var(--bs-card-color));background-color:var(--bslib-color-bg, var(--bs-card-bg, var(--bs-body-bg)));border:var(--bs-card-border-width) solid var(--bslib-color-fg, var(--bs-card-border-color));box-shadow:0 2px 4px rgba(0,0,0,.15);margin:.2rem .4rem;padding:.55rem !important;font-size:.8rem;cursor:pointer;opacity:.7;z-index:1070}.bslib-full-screen-enter:hover{opacity:1}.card[data-full-screen=false]:hover>*>.bslib-full-screen-enter{display:block}.bslib-has-full-screen .card:hover>*>.bslib-full-screen-enter{display:none}@media(max-width: 575.98px){.bslib-full-screen-enter{display:none !important}}.bslib-full-screen-exit{position:relative;top:1.35rem;font-size:.9rem;cursor:pointer;text-decoration:none;display:flex;float:right;margin-right:2.15rem;align-items:center;color:rgba(var(--bs-body-bg-rgb), 0.8)}.bslib-full-screen-exit:hover{color:rgba(var(--bs-body-bg-rgb), 1)}.bslib-full-screen-exit svg{margin-left:.5rem;font-size:1.5rem}#bslib-full-screen-overlay{position:fixed;inset:0;background-color:rgba(var(--bs-body-color-rgb), 0.6);backdrop-filter:blur(2px);-webkit-backdrop-filter:blur(2px);z-index:1069;animation:bslib-full-screen-overlay-enter 400ms cubic-bezier(0.6, 0.02, 0.65, 1) 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diff --git a/docs/stan-users-guide/algebraic-equations.html b/docs/stan-users-guide/algebraic-equations.html
index 66b4b9d4c..4685967df 100644
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -704,25 +707,6 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
 Powell, Michael J. D. 1970. <span>“A Hybrid Method for Nonlinear Equations.”</span> In <em>Numerical Methods for Nonlinear Algebraic Equations</em>, edited by P. Rabinowitz. Gordon; Breach.
 </div>
 </div></section></div></main> <!-- /main -->
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@@ -743,6 +727,150 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
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+      stylesheet.rel = 'stylesheet';
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+  const manageTransitions = (selector, allowTransitions) => {
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+  const toggleColorMode = (alternate) => {
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
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+      window.scrollTo(0, 1);
+      setTimeout(() => {
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+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
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+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
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+  const setStyleSentinel = (alternate) => {
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+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
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+      localAlternateSentinel = value;
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+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/stan-users-guide/bootstrap.html b/docs/stan-users-guide/bootstrap.html
index 3ba7e8f61..ef510d5a9 100644
--- a/docs/stan-users-guide/bootstrap.html
+++ b/docs/stan-users-guide/bootstrap.html
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -729,25 +732,6 @@ <h2 class="anchored" data-anchor-id="bayesian-bootstrap-and-bagging">Bayesian bo
 Huggins, Jonathan H, and Jeffrey W Miller. 2019. <span>“Using Bagged Posteriors for Robust Inference and Model Criticism.”</span> <em>arXiv</em>, no. 1912.07104.
 </div>
 </div></section></div></main> <!-- /main -->
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+  const enableStylesheet = (stylesheets) => {
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+      stylesheet.rel = 'stylesheet';
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+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
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+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
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+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/clustering.html b/docs/stan-users-guide/clustering.html
index 7cd709a81..9285c7f0c 100644
--- a/docs/stan-users-guide/clustering.html
+++ b/docs/stan-users-guide/clustering.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1021,25 +1024,6 @@ <h4 class="unnumbered anchored" data-anchor-id="full-bayes-correlated-topic-mode
 <li id="fn1"><p>For clustering, the non-identifiability problems for all mixture models present a problem, whereas there is no such problem for classification. Despite the difficulties with full Bayesian inference for clustering, researchers continue to use it, often in an exploratory data analysis setting rather than for predictive modeling.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1060,6 +1044,150 @@ <h4 class="unnumbered anchored" data-anchor-id="full-bayes-correlated-topic-mode
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/complex-numbers.html b/docs/stan-users-guide/complex-numbers.html
index 618cb9722..f45729806 100644
--- a/docs/stan-users-guide/complex-numbers.html
+++ b/docs/stan-users-guide/complex-numbers.html
@@ -72,10 +72,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -716,25 +719,6 @@ <h3 class="unnumbered anchored" data-anchor-id="dependent-complex-error">Depende
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -755,6 +739,150 @@ <h3 class="unnumbered anchored" data-anchor-id="dependent-complex-error">Depende
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/cross-validation.html b/docs/stan-users-guide/cross-validation.html
index 3a716d95c..62837d868 100644
--- a/docs/stan-users-guide/cross-validation.html
+++ b/docs/stan-users-guide/cross-validation.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
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 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -886,25 +889,6 @@ <h3 class="unnumbered anchored" data-anchor-id="approximate-cross-validation">Ap
 <li id="fn1"><p>The traditional approach is to walk through a vector and replace each item with a random element from the remaining elements, which is guaranteed to only move each item once. This was not done here as it’d require new categorical <code>theta</code> because Stan does not have a uniform discrete RNG built in.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -925,6 +909,150 @@ <h3 class="unnumbered anchored" data-anchor-id="approximate-cross-validation">Ap
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+  const disableStylesheet = (stylesheets) => {
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    a.classList.add('top-right');
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+    const i = window.document.createElement("i");
+    i.classList.add('bi');
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+    window.document.body.appendChild(a);
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diff --git a/docs/stan-users-guide/custom-probability.html b/docs/stan-users-guide/custom-probability.html
index 588ecfba8..b0c533ca7 100644
--- a/docs/stan-users-guide/custom-probability.html
+++ b/docs/stan-users-guide/custom-probability.html
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+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
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 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -133,7 +135,7 @@
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -665,25 +668,6 @@ <h3 class="unnumbered anchored" data-anchor-id="bivariate-normal-cumulative-dist
 <li id="fn1"><p>The problem is the (extremely!) light tails of the triangle distribution. The standard HMC and NUTS samplers can’t get into the corners of the triangle properly. Because the Stan code declares <code>y</code> to be of type <code>real&lt;lower=-1, upper=1&gt;</code>, the inverse logit transform is applied to the unconstrained variable and its log absolute derivative added to the log probability. The resulting distribution on the logit-transformed <code>y</code> is well behaved.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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@@ -704,6 +688,150 @@ <h3 class="unnumbered anchored" data-anchor-id="bivariate-normal-cumulative-dist
     }
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+  const disableStylesheet = (stylesheets) => {
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+  const toggleColorMode = (alternate) => {
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
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diff --git a/docs/stan-users-guide/dae.html b/docs/stan-users-guide/dae.html
index a700283b5..337d1e808 100644
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -725,25 +728,6 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
 Serban, Radu, Cosmin Petra, Alan C. Hindmarsh, Cody J. Balos, David J. Gardner, Daniel R. Reynolds, and Carol S. Woodward. 2021. <span>“User <span>Documentation</span> for <span>IDAS</span> V5.0.0.”</span> Lawrence Livermore National Laboratory.
 </div>
 </div></section></div></main> <!-- /main -->
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@@ -764,6 +748,150 @@ <h3 class="unnumbered anchored" data-anchor-id="maximum-number-of-steps">Maximum
     }
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+        el.classList.remove('notransition');
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+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    } else {
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+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
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+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
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@@ -738,25 +741,6 @@ <h2 class="anchored" data-anchor-id="continuous-choices">Continuous choices</h2>
 Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. <em>Bayesian Data Analysis</em>. Third Edition. London: Chapman &amp; Hall / CRC Press.
 </div>
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diff --git a/docs/stan-users-guide/efficiency-tuning.html b/docs/stan-users-guide/efficiency-tuning.html
index f3867eea3..6c00116ac 100644
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-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
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@@ -268,6 +270,7 @@
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     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
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@@ -1446,25 +1449,6 @@ <h2 class="anchored" data-anchor-id="using-map-reduce">Using map-reduce</h2>
 <li id="fn4"><p>Future versions of Stan may remove this inefficiency by more fully exploiting expression templates inside the Eigen C++ matrix library. This will require enhancing Eigen to deal with mixed-type arguments, such as the type <code>double</code> used for constants and the algorithmic differentiation type <code>stan::math::var</code> used for variables.<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
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diff --git a/docs/stan-users-guide/finite-mixtures.html b/docs/stan-users-guide/finite-mixtures.html
index ae4993415..96e76b77e 100644
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-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1031,25 +1034,6 @@ <h3 class="unnumbered anchored" data-anchor-id="comparison-to-model-averaging">C
 <li id="fn1"><p>Imposing a constraint such as <span class="math inline">\(\theta &lt; 0.5\)</span> will resolve the symmetry, but fundamentally changes the model and its posterior inferences.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1070,6 +1054,150 @@ <h3 class="unnumbered anchored" data-anchor-id="comparison-to-model-averaging">C
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/floating-point.html b/docs/stan-users-guide/floating-point.html
index b494f90a7..b00745985 100644
--- a/docs/stan-users-guide/floating-point.html
+++ b/docs/stan-users-guide/floating-point.html
@@ -38,10 +38,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -99,7 +101,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -214,6 +216,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -753,25 +756,6 @@ <h2 class="anchored" data-anchor-id="comparing-floating-point-numbers">Comparing
 <li id="fn1"><p>The notable exception is Intel’s optimizing compilers under certain optimization settings.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -792,6 +776,150 @@ <h2 class="anchored" data-anchor-id="comparing-floating-point-numbers">Comparing
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/for-bugs-users.html b/docs/stan-users-guide/for-bugs-users.html
index 5813d2430..90d8a3d1b 100644
--- a/docs/stan-users-guide/for-bugs-users.html
+++ b/docs/stan-users-guide/for-bugs-users.html
@@ -91,10 +91,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -152,7 +154,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -267,6 +269,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -819,25 +822,6 @@ <h2 class="anchored" data-anchor-id="the-stan-community">The Stan community</h2>
 <li id="fn2"><p>Most distributions have been vectorized, but currently the truncated versions may not exist.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -858,6 +842,150 @@ <h2 class="anchored" data-anchor-id="the-stan-community">The Stan community</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
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+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
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+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
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+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
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+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/stan-users-guide/gaussian-processes.html b/docs/stan-users-guide/gaussian-processes.html
index 444823659..d01b83588 100644
--- a/docs/stan-users-guide/gaussian-processes.html
+++ b/docs/stan-users-guide/gaussian-processes.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1326,25 +1329,6 @@ <h3 class="unnumbered anchored" data-anchor-id="multiple-output-gaussian-process
 <li id="fn2"><p>A boundary-avoiding prior is just one where the limit of the density is zero at the boundary, the result of which is estimates that are pushed away from the boundary.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1365,6 +1349,150 @@ <h3 class="unnumbered anchored" data-anchor-id="multiple-output-gaussian-process
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
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+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
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+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
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+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/hyperspherical-models.html b/docs/stan-users-guide/hyperspherical-models.html
index 35f97f0c6..f0e233347 100644
--- a/docs/stan-users-guide/hyperspherical-models.html
+++ b/docs/stan-users-guide/hyperspherical-models.html
@@ -92,10 +92,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -675,25 +678,6 @@ <h2 class="anchored" data-anchor-id="circular-representations-of-days-and-years"
 Muller, Mervin E. 1959. <span>“A Note on a Method for Generating Points Uniformly on n-Dimensional Spheres.”</span> <em>Commun. ACM</em> 2 (4): 19–20. <a href="https://doi.org/10.1145/377939.377946">https://doi.org/10.1145/377939.377946</a>.
 </div>
 </div></section></div></main> <!-- /main -->
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-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -714,6 +698,150 @@ <h2 class="anchored" data-anchor-id="circular-representations-of-days-and-years"
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/index.html b/docs/stan-users-guide/index.html
index 61e6882d0..b45503c2e 100644
--- a/docs/stan-users-guide/index.html
+++ b/docs/stan-users-guide/index.html
@@ -37,10 +37,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -69,7 +71,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed slimcontent">
@@ -184,6 +186,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -516,7 +519,7 @@ <h1 class="title">Stan User’s Guide</h1>
 
 
 <div class="page-columns page-full"><p></p><div class="no-row-height column-margin column-container"><img src="img/logo_tm.png" title="Stan Logo" class="d-none d-md-block img-fluid" width="225"></div></div>
-<p>This is the official user’s guide for Stan. It provides example models and programming techniques for coding statistical models in Stan.</p>
+<p>This is the official user’s guide for <a href="https://mc-stan.org/">Stan</a>. It provides example models and programming techniques for coding statistical models in Stan.</p>
 <ul>
 <li><p>Part 1 gives Stan code and discussions for several important classes of models.</p></li>
 <li><p>Part 2 discusses various general Stan programming techniques that are not tied to any particular model.</p></li>
@@ -544,25 +547,6 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-
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 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -583,6 +567,150 @@ <h4 class="unnumbered anchored" data-anchor-id="licensing">Licensing</h4>
     }
   }
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+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
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+    const a = window.document.createElement('a');
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+    a.classList.add('quarto-color-scheme-toggle');
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+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
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diff --git a/docs/stan-users-guide/latent-discrete.html b/docs/stan-users-guide/latent-discrete.html
index 188bbc4d4..f64b04dab 100644
--- a/docs/stan-users-guide/latent-discrete.html
+++ b/docs/stan-users-guide/latent-discrete.html
@@ -92,10 +92,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
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@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1207,7 +1210,7 @@ <h3 class="unnumbered anchored" data-anchor-id="estimating-expectations">Estimat
 \]</span> and then estimate <span class="math inline">\(\mathbb{E}[g(\Theta, Z)]\)</span> with <span class="math display">\[
 \hat{\mu} = \frac{1}{M} \sum_{i = 1}^M h(\theta^{(i)}).
 \]</span> This estimator is justified by the law of iterated expectation, the fact that <span class="math display">\[
-\mathbb{E}[h(\Theta)] = \mathbb{E}[\mathbb{E}[g(\Theta, Z] \mid \Theta)] = \mathbb{E}[g(\Theta, Z)] = \mu.
+\mathbb{E}[h(\Theta)] = \mathbb{E}[\mathbb{E}[g(\Theta, Z)] \mid \Theta] = \mathbb{E}[g(\Theta, Z)] = \mu.
 \]</span> Using this marginalized estimator provides a way to estimate the expectation of any function <span class="math inline">\(g(\cdot)\)</span> for all combinations of discrete or continuous parameters in the model. However, it presents a possible new challenge: evaluating the conditional expectation <span class="math inline">\(\mathbb{E}[g(\Theta, Z) \mid \Theta]\)</span>.</p>
 </section>
 <section id="evaluating-the-conditional-expectation" class="level3 unnumbered">
@@ -1299,25 +1302,6 @@ <h3 class="unnumbered anchored" data-anchor-id="evaluating-the-conditional-expec
 <li id="fn7"><p>In the subscript, <span class="math inline">\(z_i\)</span> is written as <span class="math inline">\(z[i]\)</span> to improve legibility.<a href="#fnref7" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
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-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1338,6 +1322,150 @@ <h3 class="unnumbered anchored" data-anchor-id="evaluating-the-conditional-expec
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+      const sendMessage = (message) => {
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+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
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+    // properly recolor the scrollbar when toggling (#1455)
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diff --git a/docs/stan-users-guide/matrices-arrays.html b/docs/stan-users-guide/matrices-arrays.html
index 3f55070be..7967a6659 100644
--- a/docs/stan-users-guide/matrices-arrays.html
+++ b/docs/stan-users-guide/matrices-arrays.html
@@ -72,10 +72,12 @@
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 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
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@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -802,25 +805,6 @@ <h2 class="anchored" data-anchor-id="aliasing-in-stan-containers">Aliasing in St
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -841,6 +825,150 @@ <h2 class="anchored" data-anchor-id="aliasing-in-stan-containers">Aliasing in St
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/measurement-error.html b/docs/stan-users-guide/measurement-error.html
index 4cdc3c903..7c528cb0b 100644
--- a/docs/stan-users-guide/measurement-error.html
+++ b/docs/stan-users-guide/measurement-error.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -842,25 +845,6 @@ <h4 class="unnumbered anchored" data-anchor-id="extensions-and-alternatives">Ext
 <li id="fn2"><p>When dividing two integers, the result type is an integer and rounding will ensue if the result is not exact. See the discussion of primitive arithmetic types in the reference manual for more information.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -881,6 +865,150 @@ <h4 class="unnumbered anchored" data-anchor-id="extensions-and-alternatives">Ext
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/missing-data.html b/docs/stan-users-guide/missing-data.html
index efd0f8bf8..acb397fe4 100644
--- a/docs/stan-users-guide/missing-data.html
+++ b/docs/stan-users-guide/missing-data.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -781,25 +784,6 @@ <h2 class="anchored" data-anchor-id="missing-multivariate-data">Missing multivar
 <li id="fn2"><p>This is not the same as missing components of a multivariate predictor in a regression problem; in that case, you will need to represent the missing data as a parameter and impute missing values in order to feed them into the regression.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -820,6 +804,150 @@ <h2 class="anchored" data-anchor-id="missing-multivariate-data">Missing multivar
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/multi-indexing.html b/docs/stan-users-guide/multi-indexing.html
index 5e275a2a4..6fcbfdcf2 100644
--- a/docs/stan-users-guide/multi-indexing.html
+++ b/docs/stan-users-guide/multi-indexing.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -826,25 +829,6 @@ <h2 class="anchored" data-anchor-id="matrices-with-parameters-and-constants">Mat
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -865,6 +849,150 @@ <h2 class="anchored" data-anchor-id="matrices-with-parameters-and-constants">Mat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+          theme: newTheme
+        }
+      });
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
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+      for (const sheetNode of alternateStylesheets) {
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+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+      const toggle = toggles[i];
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+  const setStyleSentinel = (alternate) => {
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+    if (!isFileUrl()) {
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+    } else {
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
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+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/odes.html b/docs/stan-users-guide/odes.html
index 5647b9efc..d5d16fa43 100644
--- a/docs/stan-users-guide/odes.html
+++ b/docs/stan-users-guide/odes.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
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   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -909,25 +912,6 @@ <h2 class="anchored" data-anchor-id="solving-a-system-of-linear-odes-using-a-mat
 <li id="fn1"><p>This example is drawn from the documentation for the Boost Numeric Odeint library <span class="citation" data-cites="AhnertMulansky:2011">(<a href="#ref-AhnertMulansky:2011" role="doc-biblioref">Ahnert and Mulansky 2011</a>)</span>, which Stan uses to implement the <code>rk45</code> and <code>ckrk</code> solver.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -948,6 +932,150 @@ <h2 class="anchored" data-anchor-id="solving-a-system-of-linear-odes-using-a-mat
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+        }
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
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+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
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+  const hasAlternateSentinel = () => {  
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+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+      return false;
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+  const setStyleSentinel = (alternate) => {
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+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
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+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/stan-users-guide/one-dimensional-integrals.html b/docs/stan-users-guide/one-dimensional-integrals.html
index cb4b9f203..4863bce40 100644
--- a/docs/stan-users-guide/one-dimensional-integrals.html
+++ b/docs/stan-users-guide/one-dimensional-integrals.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -780,25 +783,6 @@ <h3 class="unnumbered anchored" data-anchor-id="integral-precision">Avoiding pre
 Agrawal, Nikhar, Anton Bikineev, Paul Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, et al. 2017. <span>“Double-Exponential Quadrature.”</span> https://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/double_exponential.html.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
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-
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -819,6 +803,150 @@ <h3 class="unnumbered anchored" data-anchor-id="integral-precision">Avoiding pre
     }
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   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
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diff --git a/docs/stan-users-guide/parallelization.html b/docs/stan-users-guide/parallelization.html
index f38b772e1..1271b2ae7 100644
--- a/docs/stan-users-guide/parallelization.html
+++ b/docs/stan-users-guide/parallelization.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1002,25 +1005,6 @@ <h2 class="anchored" data-anchor-id="opencl">OpenCL</h2>
 <li id="fn3"><p>This makes the strong assumption that each group has the same number of observations!<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1041,6 +1025,150 @@ <h2 class="anchored" data-anchor-id="opencl">OpenCL</h2>
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/posterior-prediction.html b/docs/stan-users-guide/posterior-prediction.html
index 9230403f4..ae4a032c5 100644
--- a/docs/stan-users-guide/posterior-prediction.html
+++ b/docs/stan-users-guide/posterior-prediction.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -842,25 +845,6 @@ <h2 class="anchored" data-anchor-id="stand-alone-generated-quantities-and-ongoin
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -881,6 +865,150 @@ <h2 class="anchored" data-anchor-id="stand-alone-generated-quantities-and-ongoin
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/posterior-predictive-checks.html b/docs/stan-users-guide/posterior-predictive-checks.html
index 80bc40cac..01b646378 100644
--- a/docs/stan-users-guide/posterior-predictive-checks.html
+++ b/docs/stan-users-guide/posterior-predictive-checks.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script src="../site_libs/quarto-contrib/glightbox/glightbox.min.js"></script>
 <link href="../site_libs/quarto-contrib/glightbox/glightbox.min.css" rel="stylesheet">
 <link href="../site_libs/quarto-contrib/glightbox/lightbox.css" rel="stylesheet">
@@ -156,7 +158,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
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@@ -271,6 +273,7 @@
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@@ -586,7 +589,7 @@ <h2 id="toc-title">On this page</h2>
   <ul class="collapse">
   <li><a href="#simulating-from-the-posterior-predictive-distribution" id="toc-simulating-from-the-posterior-predictive-distribution" class="nav-link" data-scroll-target="#simulating-from-the-posterior-predictive-distribution">Simulating from the posterior predictive distribution</a></li>
   <li><a href="#plotting-multiples" id="toc-plotting-multiples" class="nav-link" data-scroll-target="#plotting-multiples">Plotting multiples</a></li>
-  <li><a href="#posterior-p-values" id="toc-posterior-p-values" class="nav-link" data-scroll-target="#posterior-p-values">Posterior ``p-values’’</a>
+  <li><a href="#posterior-p-values" id="toc-posterior-p-values" class="nav-link" data-scroll-target="#posterior-p-values">Posterior ‘’p-values’’</a>
   <ul class="collapse">
   <li><a href="#which-statistics-to-test" id="toc-which-statistics-to-test" class="nav-link" data-scroll-target="#which-statistics-to-test">Which statistics to test?</a></li>
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@@ -679,7 +682,7 @@ <h2 class="anchored" data-anchor-id="plotting-multiples">Plotting multiples</h2>
 <p>Now consider generating data <span class="math inline">\(y \sim \textrm{Poisson}(5)\)</span>. The resulting small multiples plot shows the original data plotted in the upper left and eight different posterior replications plotted in the remaining boxes.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pois-pois.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-1" data-gallery="quarto-lightbox-gallery-1" title="Posterior predictive checks for Poisson data generating process and Poisson model."><img src="./img/ppc-pois-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for Poisson data generating process and Poisson model."></a></p>
+<p><a href="./img/ppc-pois-pois.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-1" data-glightbox="description: .lightbox-desc-1" title="Posterior predictive checks for Poisson data generating process and Poisson model."><img src="./img/ppc-pois-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for Poisson data generating process and Poisson model."></a></p>
 <figcaption>Posterior predictive checks for Poisson data generating process and Poisson model.</figcaption>
 </figure>
 </div>
@@ -687,14 +690,14 @@ <h2 class="anchored" data-anchor-id="plotting-multiples">Plotting multiples</h2>
 <p>Now consider generating over-dispersed data <span class="math inline">\(y \sim \textrm{negative-binomial2}(5, 1).\)</span> This has the same mean as <span class="math inline">\(\textrm{Poisson}(5)\)</span>, namely <span class="math inline">\(5\)</span>, but a standard deviation of <span class="math inline">\(\sqrt{5 + 5^2 /1} \approx 5.5.\)</span> There is no way to fit this data with the Poisson model, because a variable distributed as <span class="math inline">\(\textrm{Poisson}(\lambda)\)</span> has mean <span class="math inline">\(\lambda\)</span> and standard deviation <span class="math inline">\(\sqrt{\lambda},\)</span> which is <span class="math inline">\(\sqrt{5}\)</span> for <span class="math inline">\(\textrm{Poisson}(5).\)</span> Here’s the resulting small multiples plot, again with original data in the upper left.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-nb-pois.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-2" data-gallery="quarto-lightbox-gallery-2" title="Posterior predictive checks for negative binomial data generating process and Poisson model."><img src="./img/ppc-nb-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for negative binomial data generating process and Poisson model."></a></p>
+<p><a href="./img/ppc-nb-pois.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-2" data-glightbox="description: .lightbox-desc-2" title="Posterior predictive checks for negative binomial data generating process and Poisson model."><img src="./img/ppc-nb-pois.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Posterior predictive checks for negative binomial data generating process and Poisson model."></a></p>
 <figcaption>Posterior predictive checks for negative binomial data generating process and Poisson model.</figcaption>
 </figure>
 </div>
 <p>This time, the original data stands out in stark contrast to the replicated data sets, all of which are clearly more symmetric and lower variance than the original data. That is, the model’s not appropriately capturing the variance of the data.</p>
 </section>
 <section id="posterior-p-values" class="level2">
-<h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’’</h2>
+<h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ‘’p-values’’</h2>
 <p>If a model captures the data well, summary statistics such as sample mean and standard deviation, should have similar values in the original and replicated data sets. This can be tested by means of a p-value-like statistic, which here is just the probability the test statistic <span class="math inline">\(s(\cdot)\)</span> in a replicated data set exceeds that in the original data, <span class="math display">\[
 \Pr\!\left[ s(y^{\textrm{rep}}) \geq s(y) \mid y \right]
 =
@@ -702,7 +705,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 \textrm{I}\left( s(y^{\textrm{rep}}) \geq s(y) \mid y \right)
 \cdot p\left( y^{\textrm{rep}} \mid y \right)
 \, \textrm{d}{y^{\textrm{rep}}}.
-\]</span> It is important to note that``p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified <span class="citation" data-cites="BayarriBerger:2000">(<a href="#ref-BayarriBerger:2000" role="doc-biblioref">Bayarri and Berger 2000</a>)</span>.</p>
+\]</span> It is important to note that ‘’p-values’’ is in quotes because these statistics are not classically calibrated, and thus will not in general have a uniform distribution even when the model is well specified <span class="citation" data-cites="BayarriBerger:2000">(<a href="#ref-BayarriBerger:2000" role="doc-biblioref">Bayarri and Berger 2000</a>)</span>.</p>
 <p>Nevertheless, values of this statistic very close to zero or one are cause for concern that the model is not fitting the data well. Unlike a visual test, this p-value-like test is easily automated for bulk model fitting.</p>
 <p>To calculate event probabilities in Stan, it suffices to define indicator variables that take on value 1 if the event occurs and 0 if it does not. The posterior mean is then the event probability. For efficiency, indicator variables are defined in the generated quantities block.</p>
 <div class="sourceCode" id="cb4"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">generated quantities</span> {</span>
@@ -718,7 +721,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 <p>For the example in the previous section, where over-dispersed data generated by a negative binomial distribution was fit with a simple Poisson model, the following plot illustrates the posterior p-value calculation for the mean statistic.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pvalue-nb-pois-mean.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-3" data-gallery="quarto-lightbox-gallery-3" title="Histogram of means of replicated data sets; vertical red line at mean of original data."><img src="./img/ppc-pvalue-nb-pois-mean.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Histogram of means of replicated data sets; vertical red line at mean of original data."></a></p>
+<p><a href="./img/ppc-pvalue-nb-pois-mean.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-3" data-glightbox="description: .lightbox-desc-3" title="Histogram of means of replicated data sets; vertical red line at mean of original data."><img src="./img/ppc-pvalue-nb-pois-mean.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Histogram of means of replicated data sets; vertical red line at mean of original data."></a></p>
 <figcaption>Histogram of means of replicated data sets; vertical red line at mean of original data.</figcaption>
 </figure>
 </div>
@@ -726,7 +729,7 @@ <h2 class="anchored" data-anchor-id="posterior-p-values">Posterior ``p-values’
 <p>The standard deviation statistic tells a different story.</p>
 <div class="quarto-figure quarto-figure-center">
 <figure class="figure">
-<p><a href="./img/ppc-pvalue-nb-pois-sd.jpg" class="lightbox" data-glightbox="description: .lightbox-desc-4" data-gallery="quarto-lightbox-gallery-4" title="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."><img src="./img/ppc-pvalue-nb-pois-sd.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."></a></p>
+<p><a href="./img/ppc-pvalue-nb-pois-sd.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-4" data-glightbox="description: .lightbox-desc-4" title="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."><img src="./img/ppc-pvalue-nb-pois-sd.jpg" class="img-fluid figure-img" style="width:50.0%" alt="Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data."></a></p>
 <figcaption>Scatterplot of standard deviations of replicated data sets; the vertical red line is at standard deviation of original data.</figcaption>
 </figure>
 </div>
@@ -933,25 +936,6 @@ <h3 class="unnumbered anchored" data-anchor-id="mixed-replication-for-hierarchic
 Rubin, Donald B. 1984. <span>“Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician.”</span> <em>The Annals of Statistics</em>, 1151–72.
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diff --git a/docs/stan-users-guide/poststratification.html b/docs/stan-users-guide/poststratification.html
index 0355e7019..fa5fde33f 100644
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@@ -877,25 +880,6 @@ <h2 class="anchored" data-anchor-id="adding-group-level-predictors">Adding group
 Paustian, Keith, Elissa Levine, Wilfred M Post, and Irene M Ryzhova. 1997. <span>“The Use of Models to Integrate Information and Understanding of Soil <span>C</span> at the Regional Scale.”</span> <em>Geoderma</em> 79 (1-4): 227–60.
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+      toggleBodyColorPrimary();
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+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
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+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
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+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   anchorJS.options = {
diff --git a/docs/stan-users-guide/problematic-posteriors.html b/docs/stan-users-guide/problematic-posteriors.html
index 9760a01e8..4cc39bda9 100644
--- a/docs/stan-users-guide/problematic-posteriors.html
+++ b/docs/stan-users-guide/problematic-posteriors.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -985,25 +988,6 @@ <h3 class="unnumbered anchored" data-anchor-id="examples-fits-in-stan">Examples:
 <li id="fn4"><p>Tempering methods may be viewed as automated ways to carry out such a search for modes, though most MCMC tempering methods continue to search for modes on an ongoing basis; see <span class="citation" data-cites="SwendsenWang:1986 Neal:1996b">(<a href="#ref-SwendsenWang:1986" role="doc-biblioref">Swendsen and Wang 1986</a>; <a href="#ref-Neal:1996b" role="doc-biblioref">Neal 1996</a>)</span>.<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1024,6 +1008,150 @@ <h3 class="unnumbered anchored" data-anchor-id="examples-fits-in-stan">Examples:
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/proportionality-constants.html b/docs/stan-users-guide/proportionality-constants.html
index 712c364aa..4eb783670 100644
--- a/docs/stan-users-guide/proportionality-constants.html
+++ b/docs/stan-users-guide/proportionality-constants.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -665,25 +668,6 @@ <h2 class="anchored" data-anchor-id="limitations-on-using-_lupdf-and-_lupmf-func
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -704,6 +688,150 @@ <h2 class="anchored" data-anchor-id="limitations-on-using-_lupdf-and-_lupmf-func
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/references.html b/docs/stan-users-guide/references.html
index df2ece9e2..29e726062 100644
--- a/docs/stan-users-guide/references.html
+++ b/docs/stan-users-guide/references.html
@@ -36,10 +36,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -68,7 +70,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-fixed">
@@ -183,6 +185,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -218,25 +221,6 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
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-
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
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-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -257,6 +241,150 @@ <h1 class="unnumbered"><i style="font-size: 110%; color:var(--bs-body-color)">Re
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
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+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
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+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
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+    for (let i=0; i < toggles.length; i++) {
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+        } else {
+          toggle.classList.remove("alternate");
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+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
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+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/regression.html b/docs/stan-users-guide/regression.html
index a2f766840..66d39c4e8 100644
--- a/docs/stan-users-guide/regression.html
+++ b/docs/stan-users-guide/regression.html
@@ -92,10 +92,12 @@
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1723,25 +1726,6 @@ <h3 class="unnumbered anchored" data-anchor-id="posterior-predictive-checks">Pos
 <li id="fn6"><p>Thanks to Mike Lawrence for pointing this out in the GitHub issue for the manual.<a href="#fnref6" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1762,6 +1746,150 @@ <h3 class="unnumbered anchored" data-anchor-id="posterior-predictive-checks">Pos
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+        }
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
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+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
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+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
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+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/stan-users-guide/reparameterization.html b/docs/stan-users-guide/reparameterization.html
index 2759e6399..a5ce918ad 100644
--- a/docs/stan-users-guide/reparameterization.html
+++ b/docs/stan-users-guide/reparameterization.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -995,25 +998,6 @@ <h3 class="unnumbered anchored" data-anchor-id="varying-upper-and-lower-bounds">
 <li id="fn2"><p>This example is for illustrative purposes only; the recommended way to implement the lognormal distribution in Stan is with the built-in <code>lognormal</code> probability function; see the functions reference manual for details.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1034,6 +1018,150 @@ <h3 class="unnumbered anchored" data-anchor-id="varying-upper-and-lower-bounds">
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/simulation-based-calibration.html b/docs/stan-users-guide/simulation-based-calibration.html
index 1cf6fd15c..e8747779d 100644
--- a/docs/stan-users-guide/simulation-based-calibration.html
+++ b/docs/stan-users-guide/simulation-based-calibration.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
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-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -979,25 +982,6 @@ <h3 class="unnumbered anchored" data-anchor-id="when-stans-sampler-goes-wrong">W
 Talts, Sean, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. 2018. <span>“Validating <span>B</span>ayesian Inference Algorithms with Simulation-Based Calibration.”</span> <em>arXiv</em>, no. 1804.06788.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1018,6 +1002,150 @@ <h3 class="unnumbered anchored" data-anchor-id="when-stans-sampler-goes-wrong">W
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/sparse-ragged.html b/docs/stan-users-guide/sparse-ragged.html
index 0f7417b64..3b7860ff8 100644
--- a/docs/stan-users-guide/sparse-ragged.html
+++ b/docs/stan-users-guide/sparse-ragged.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -737,25 +740,6 @@ <h2 class="anchored" data-anchor-id="ragged-data-structs.section">Ragged data st
 Gelman, Andrew, and Jennifer Hill. 2007. <em>Data Analysis Using Regression and Multilevel-Hierarchical Models</em>. Cambridge, United Kingdom: Cambridge University Press.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -776,6 +760,150 @@ <h2 class="anchored" data-anchor-id="ragged-data-structs.section">Ragged data st
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/style-guide.html b/docs/stan-users-guide/style-guide.html
index a6f6bd5d5..a85683fc9 100644
--- a/docs/stan-users-guide/style-guide.html
+++ b/docs/stan-users-guide/style-guide.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -833,25 +836,6 @@ <h4 class="unnumbered anchored" data-anchor-id="platform-specificity-of-newlines
 <li id="fn2"><p>This is the usual convention in both typesetting and other programming languages. Neither R nor BUGS allows breaks before an operator because they allow newlines to signal the end of an expression or statement.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
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-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
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-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
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 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
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@@ -872,6 +856,150 @@ <h4 class="unnumbered anchored" data-anchor-id="platform-specificity-of-newlines
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
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+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
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+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
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+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
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+      sendMessage({
+        setConfig: {
+          theme: newTheme
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+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
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+    }
+  }
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+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
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+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
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diff --git a/docs/stan-users-guide/survival.html b/docs/stan-users-guide/survival.html
index 275201183..0c0f7f590 100644
--- a/docs/stan-users-guide/survival.html
+++ b/docs/stan-users-guide/survival.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1030,25 +1033,6 @@ <h3 class="unnumbered anchored" data-anchor-id="stan-model-for-tied-survival-tim
 Plackett, Robin L. 1975. <span>“The Analysis of Permutations.”</span> <em>Journal of the Royal Statistical Society Series C: Applied Statistics</em> 24 (2): 193–202.
 </div>
 </div></section></div></main> <!-- /main -->
-<script type="text/javascript">
-
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-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
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-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1069,6 +1053,150 @@ <h3 class="unnumbered anchored" data-anchor-id="stan-model-for-tied-survival-tim
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/time-series.html b/docs/stan-users-guide/time-series.html
index 5727d3037..4c1e1bf65 100644
--- a/docs/stan-users-guide/time-series.html
+++ b/docs/stan-users-guide/time-series.html
@@ -92,10 +92,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -153,7 +155,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -268,6 +270,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -1181,25 +1184,6 @@ <h3 class="unnumbered anchored" data-anchor-id="predictive-inference">Predictive
 <li id="fn3"><p>This subsection is a lightly edited comment of Jonathan Gilligan’s on GitHub; see <a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1220,6 +1204,150 @@ <h3 class="unnumbered anchored" data-anchor-id="predictive-inference">Predictive
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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diff --git a/docs/stan-users-guide/truncation-censoring.html b/docs/stan-users-guide/truncation-censoring.html
index e5ccafcf3..f692658e5 100644
--- a/docs/stan-users-guide/truncation-censoring.html
+++ b/docs/stan-users-guide/truncation-censoring.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -720,25 +723,6 @@ <h3 class="unnumbered anchored" data-anchor-id="integrating-out-censored-values"
 </section>
 
 <a onclick="window.scrollTo(0, 0); return false;" role="button" id="quarto-back-to-top"><i class="bi bi-arrow-up"></i> Back to top</a></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -759,6 +743,150 @@ <h3 class="unnumbered anchored" data-anchor-id="integrating-out-censored-values"
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
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+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
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   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/user-functions.html b/docs/stan-users-guide/user-functions.html
index c79e897c1..61d8b06bd 100644
--- a/docs/stan-users-guide/user-functions.html
+++ b/docs/stan-users-guide/user-functions.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -133,7 +135,7 @@
 };
 </script>
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -248,6 +250,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -981,25 +984,6 @@ <h4 class="unnumbered anchored" data-anchor-id="truncating-above-and-below">Trun
 <li id="fn5"><p>The original code and impetus for including this in the manual came from the <a href="http://discourse.mc-stan.org/t/rng-for-truncated-distributions/3122/7">Stan forums post</a>; by user <code>lcomm</code>, who also explained truncation above and below.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main> <!-- /main -->
-<script type="text/javascript">
-
-// replace cmd keyboard shortcut w/ control on non-Mac platforms
-const kPlatformMac = typeof navigator !== 'undefined' ? /Mac/.test(navigator.platform) : false;
-if (!kPlatformMac) {
-   var kbds = document.querySelectorAll("kbd")
-   kbds.forEach(function(kbd) {
-      kbd.innerHTML = kbd.innerHTML.replace(/⌘/g, '⌃');
-   });
-}
-
-// tweak headings in pymd
-document.querySelectorAll(".pymd span.co").forEach(el => {
-   if (!el.innerText.startsWith("#|")) {
-      el.style.fontWeight = 1000;
-   }
-});
-
-</script>
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
@@ -1020,6 +1004,150 @@ <h4 class="unnumbered anchored" data-anchor-id="truncating-above-and-below">Trun
     }
   }
   toggleBodyColorPrimary();  
+  const disableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'prefetch';
+    }
+  }
+  const enableStylesheet = (stylesheets) => {
+    for (let i=0; i < stylesheets.length; i++) {
+      const stylesheet = stylesheets[i];
+      stylesheet.rel = 'stylesheet';
+    }
+  }
+  const manageTransitions = (selector, allowTransitions) => {
+    const els = window.document.querySelectorAll(selector);
+    for (let i=0; i < els.length; i++) {
+      const el = els[i];
+      if (allowTransitions) {
+        el.classList.remove('notransition');
+      } else {
+        el.classList.add('notransition');
+      }
+    }
+  }
+  const toggleGiscusIfUsed = (isAlternate, darkModeDefault) => {
+    const baseTheme = document.querySelector('#giscus-base-theme')?.value ?? 'light';
+    const alternateTheme = document.querySelector('#giscus-alt-theme')?.value ?? 'dark';
+    let newTheme = '';
+    if(darkModeDefault) {
+      newTheme = isAlternate ? baseTheme : alternateTheme;
+    } else {
+      newTheme = isAlternate ? alternateTheme : baseTheme;
+    }
+    const changeGiscusTheme = () => {
+      // From: https://github.com/giscus/giscus/issues/336
+      const sendMessage = (message) => {
+        const iframe = document.querySelector('iframe.giscus-frame');
+        if (!iframe) return;
+        iframe.contentWindow.postMessage({ giscus: message }, 'https://giscus.app');
+      }
+      sendMessage({
+        setConfig: {
+          theme: newTheme
+        }
+      });
+    }
+    const isGiscussLoaded = window.document.querySelector('iframe.giscus-frame') !== null;
+    if (isGiscussLoaded) {
+      changeGiscusTheme();
+    }
+  }
+  const toggleColorMode = (alternate) => {
+    // Switch the stylesheets
+    const alternateStylesheets = window.document.querySelectorAll('link.quarto-color-scheme.quarto-color-alternate');
+    manageTransitions('#quarto-margin-sidebar .nav-link', false);
+    if (alternate) {
+      enableStylesheet(alternateStylesheets);
+      for (const sheetNode of alternateStylesheets) {
+        if (sheetNode.id === "quarto-bootstrap") {
+          toggleBodyColorMode(sheetNode);
+        }
+      }
+    } else {
+      disableStylesheet(alternateStylesheets);
+      toggleBodyColorPrimary();
+    }
+    manageTransitions('#quarto-margin-sidebar .nav-link', true);
+    // Switch the toggles
+    const toggles = window.document.querySelectorAll('.quarto-color-scheme-toggle');
+    for (let i=0; i < toggles.length; i++) {
+      const toggle = toggles[i];
+      if (toggle) {
+        if (alternate) {
+          toggle.classList.add("alternate");     
+        } else {
+          toggle.classList.remove("alternate");
+        }
+      }
+    }
+    // Hack to workaround the fact that safari doesn't
+    // properly recolor the scrollbar when toggling (#1455)
+    if (navigator.userAgent.indexOf('Safari') > 0 && navigator.userAgent.indexOf('Chrome') == -1) {
+      manageTransitions("body", false);
+      window.scrollTo(0, 1);
+      setTimeout(() => {
+        window.scrollTo(0, 0);
+        manageTransitions("body", true);
+      }, 40);  
+    }
+  }
+  const isFileUrl = () => { 
+    return window.location.protocol === 'file:';
+  }
+  const hasAlternateSentinel = () => {  
+    let styleSentinel = getColorSchemeSentinel();
+    if (styleSentinel !== null) {
+      return styleSentinel === "alternate";
+    } else {
+      return false;
+    }
+  }
+  const setStyleSentinel = (alternate) => {
+    const value = alternate ? "alternate" : "default";
+    if (!isFileUrl()) {
+      window.localStorage.setItem("quarto-color-scheme", value);
+    } else {
+      localAlternateSentinel = value;
+    }
+  }
+  const getColorSchemeSentinel = () => {
+    if (!isFileUrl()) {
+      const storageValue = window.localStorage.getItem("quarto-color-scheme");
+      return storageValue != null ? storageValue : localAlternateSentinel;
+    } else {
+      return localAlternateSentinel;
+    }
+  }
+  const darkModeDefault = false;
+  let localAlternateSentinel = darkModeDefault ? 'alternate' : 'default';
+  // Dark / light mode switch
+  window.quartoToggleColorScheme = () => {
+    // Read the current dark / light value 
+    let toAlternate = !hasAlternateSentinel();
+    toggleColorMode(toAlternate);
+    setStyleSentinel(toAlternate);
+    toggleGiscusIfUsed(toAlternate, darkModeDefault);
+  };
+  // Ensure there is a toggle, if there isn't float one in the top right
+  if (window.document.querySelector('.quarto-color-scheme-toggle') === null) {
+    const a = window.document.createElement('a');
+    a.classList.add('top-right');
+    a.classList.add('quarto-color-scheme-toggle');
+    a.href = "";
+    a.onclick = function() { try { window.quartoToggleColorScheme(); } catch {} return false; };
+    const i = window.document.createElement("i");
+    i.classList.add('bi');
+    a.appendChild(i);
+    window.document.body.appendChild(a);
+  }
+  // Switch to dark mode if need be
+  if (hasAlternateSentinel()) {
+    toggleColorMode(true);
+  } else {
+    toggleColorMode(false);
+  }
   const icon = "";
   const anchorJS = new window.AnchorJS();
   anchorJS.options = {
diff --git a/docs/stan-users-guide/using-stanc.html b/docs/stan-users-guide/using-stanc.html
index ca3393940..0bc241490 100644
--- a/docs/stan-users-guide/using-stanc.html
+++ b/docs/stan-users-guide/using-stanc.html
@@ -72,10 +72,12 @@
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
 <script src="../site_libs/quarto-html/anchor.min.js"></script>
 <link href="../site_libs/quarto-html/tippy.css" rel="stylesheet">
-<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-text-highlighting-styles">
+<link href="../site_libs/quarto-html/quarto-syntax-highlighting-dark.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-text-highlighting-styles">
 <script src="../site_libs/bootstrap/bootstrap.min.js"></script>
 <link href="../site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
-<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" class="quarto-color-scheme" id="quarto-bootstrap" data-mode="light">
+<link href="../site_libs/bootstrap/bootstrap-dark.min.css" rel="prefetch" class="quarto-color-scheme quarto-color-alternate" id="quarto-bootstrap" data-mode="dark">
 <script id="quarto-search-options" type="application/json">{
   "location": "navbar",
   "copy-button": false,
@@ -104,7 +106,7 @@
 }</script>
 
 
-<link rel="stylesheet" href="../quarto_styles.css">
+<link rel="stylesheet" href="../theming/quarto_styles.css">
 </head>
 
 <body class="nav-sidebar floating nav-fixed">
@@ -219,6 +221,7 @@
           <div class="quarto-navbar-tools tools-wide">
     <a href="https://github.com/stan-dev" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-github"></i></a>
     <a href="https://discourse.mc-stan.org" title="" class="quarto-navigation-tool px-1" aria-label=""><i class="bi bi-chat-text-fill"></i></a>
+  <a href="" class="quarto-color-scheme-toggle quarto-navigation-tool  px-1" onclick="window.quartoToggleColorScheme(); return false;" title="Toggle dark mode"><i class="bi"></i></a>
   <a href="" class="quarto-reader-toggle quarto-navigation-tool px-1" onclick="window.quartoToggleReader(); return false;" title="Toggle reader mode">
   <div class="quarto-reader-toggle-btn">
   <i class="bi"></i>
@@ -943,7 +946,7 @@ <h3 class="unnumbered anchored" data-anchor-id="strict-or-nonsensical-parameter-
 </section>
 <section id="nonlinear-transformations" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="nonlinear-transformations">Nonlinear transformations</h3>
-<p>When a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the [Changes of variables] section.</p>
+<p>When a parameter is transformed in a non-linear fashion, an adjustment must be applied to account for distortion caused by the transform. This is discussed in depth in the <a href="../stan-users-guide/reparameterization.html">Changes of variables</a> section.</p>
 <p>This portion of pedantic mode tries to detect instances where such an adjustment would be necessary and remind the user.</p>
 <p>For example, consider the following program.</p>
 <div class="sourceCode" id="cb29"><pre class="sourceCode stan code-with-copy"><code class="sourceCode stan"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="kw">parameters</span> {</span>
@@ -979,7 +982,7 @@ <h2 class="anchored" data-anchor-id="stanc-pretty-printing">Automatic updating a
 <p>These flags work for both <code>.stan</code> model files and <code>.stanfunctions</code> function files. They can be combined with <code>--o</code> to redirect the formatted output to a new file.</p>
 <section id="automatic-formatting" class="level3 unnumbered">
 <h3 class="unnumbered anchored" data-anchor-id="automatic-formatting">Automatic formatting</h3>
-<p>Invoking <code>stanc --auto-format &lt;model_file&gt;</code> will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the [Stan Program Style Guide]. This means spacing, indentation, and line length are all regularized. <em>Some</em> deprecated features, like the use of <code>#</code> for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.</p>
+<p>Invoking <code>stanc --auto-format &lt;model_file&gt;</code> will print a version of your model which has been re-formatted. The goal is to have this automatic formatting stay as close as possible to the <a href="../stan-users-guide/style-guide.html">Stan Program Style Guide</a>. This means spacing, indentation, and line length are all regularized. <em>Some</em> deprecated features, like the use of <code>#</code> for line comments, are replaced, but the goal is mainly to preserve the program while formatting it.</p>
 <p>By default, this will try to split lines at or before column 78. This number can be changed using <code>--max-line-length</code>.</p>
 </section>
 <section id="canonicalizing" class="level3 unnumbered">
@@ -1386,25 +1389,6 @@ <h4 class="unnumbered anchored" data-anchor-id="static-loop-unrolling">Static lo
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@@ -1425,6 +1409,150 @@ <h4 class="unnumbered anchored" data-anchor-id="static-loop-unrolling">Static lo
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diff --git a/docs/theming/quarto_styles.css b/docs/theming/quarto_styles.css
new file mode 100644
index 000000000..29078a7fc
--- /dev/null
+++ b/docs/theming/quarto_styles.css
@@ -0,0 +1,406 @@
+/* quarto website styling */
+
+@media (min-width: 1020px) {
+.navbar-brand-container {
+  margin-right: 1em;
+}
+}
+
+@media (max-width: 1060px) and (min-width: 991.98px) {
+
+#navbarCollapse ul:last-of-type a.nav-link {
+  padding-left: .25em;
+  padding-right: .25em;
+}
+
+.navbar #quarto-search {
+  margin-left: .1em;
+}
+}
+
+
+@media (min-width: 991.98px) {
+#quarto-header {
+  border-bottom: 1px solid
+}
+}
+
+.navbar-brand > img {
+  max-height: 36px;
+}
+
+.platform-table td {
+  vertical-align: middle;
+}
+
+.platform-table td > div.sourceCode {
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+  margin-bottom: 0.3rem;
+}
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+  padding: 6px;
+  font-weight: 500;
+  margin-bottom: 1rem;
+}
+
+.document-example div {
+  padding: 5px;
+}
+
+.document-example .citation {
+    color: var(--stan-secondary);
+}
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+  display: inline-block;
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+}
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+}
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+}
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+  font-size: .775em;
+  cursor: pointer;
+  padding-top: 4px;
+}
+
+.download-button {
+  display:flex;
+  padding-bottom: 10px;
+  padding-top: 10px;
+}
+
+.download-button .secondary {
+  font-size: .775em;
+  margin-bottom: 0;
+}
+
+.download-button .container {
+  display: flex;
+  padding-left: 10px;
+  padding-right: 40px;
+}
+
+.download-button .icon-container {
+  fill: white;
+  width: 30px;
+  margin-right: 15px;
+}
+
+iframe.reveal-demo {
+  width: 100%;
+  height: 350px;
+  outline: none;
+}
+
+.slide-deck {
+  border: 3px solid #dee2e6;
+  width: 100%;
+  height: 475px;
+}
+
+@media only screen and (max-width: 600px) {
+ .slide-deck {
+    height: 400px;
+  }
+}
+
+@media (max-width: 575px) {
+.link-cards .card {
+  margin-bottom: 20px;
+  margin-right: 35px;
+}
+
+}
+
+@media (min-width: 576px) {
+.link-cards {
+  display: flex;
+  flex-direction: row;
+  flex-wrap: wrap;
+}
+
+.link-cards .card {
+  width: 190px;
+  margin: 0 20px 12px 0;
+}
+
+
+}
+
+.link-cards .card {
+  border: none;
+  padding: 0;
+}
+
+.link-cards .card-title h4 {
+  margin-top: 0;
+}
+
+.link-cards .card-title p {
+  margin-bottom: 0;
+}
+
+.link-cards .card-subtitle {
+  margin-bottom: 0.7rem;
+}
+
+.link-cards .card-body {
+  padding: 0.5rem;
+  padding-left: 0.1rem;
+}
+
+.link-cards .card-body ul {
+  margin-bottom: 0;
+  padding-left: 0;
+  list-style-type: none;
+}
+
+.link-cards .card-body ul a {
+  text-decoration: none;
+}
+
+.link-cards .card-body ul li {
+  padding-bottom: 0.2rem;
+}
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+}
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+}
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+}
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+}
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+}
+
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+}
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+  color: var(--stan-bg);
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+}
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+}
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+}
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+
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+.panel-tabset[data-group="tools-tabset"] .nav-tabs .nav-link.active,
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+
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+
+
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+    font-style: italic;
+    color: var(--stan-secondary);
+}
+
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+  margin-top: 2em;
+}
+
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+    white-space: inherit;
+    color: var(--bs-dark-warm);
+    background-color: var(--stan-bg);
+}
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+    color: var(--bs-dark-warm);
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+
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+
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+    background-color: var(--stan-bg);
+}
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+    margin: 0;
+    padding: 0;
+    color: var(--bs-dark-warm);
+    background-color: var(--stan-bg);
+}
+
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+    white-space: inherit;
+    margin: 0;
+    padding: 0;
+    color: var(--bs-dark-warm);
+    background-color: var(--stan-bg);
+}
+
+#quarto-appendix {
+    color: var(--bs-dark-warm);
+    background-color: var(--stan-bg);
+}
+
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+    text-align: left;
+    margin-left: 2em; /* Adjust as needed */
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+    color: var(--stan-secondary);
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+}