lectures/Week-5-Thu.ipynb

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    "### ASTR-598, Winter 2018, Connolly & Ivezic, University of Washington\n",
    "https://github.com/dirac-institute/uw-astr598-w18/tree/master/lectures/Week-5-Thu.ipynb\n",
    "\n",
    "\n",
    "# Week 5, Thursday: Model Parameter Estimation and Model Selection in Bayesian Framework. II\n",
    "\n",
    "\n"
   ]
  },
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   "source": [
    "##### Resources for this notebook include:\n",
    "\n",
    "- [Textbook](http://press.princeton.edu/titles/10159.html) Chapter 5.  \n",
    "- [Gordon Richard's notebooks](https://github.com/gtrichards/PHYS_T480)\n",
    "- [Jake VanderPlas's workshop \"Bayesian Astronomy\"](https://github.com/jakevdp/BayesianAstronomy)\n",
    "- [Jake VanderPlas's blog \"Frequentism and Bayesianism: A Practical Introduction\"](http://jakevdp.github.io/blog/2014/03/11/frequentism-and-bayesianism-a-practical-intro/)\n",
    "\n",
    "\n",
    "##### Highly recommended supplemental background reading:\n",
    "\n",
    "- [Jake VanderPlas: ``Frequentism and Bayesianism: A Python-driven Primer\"](https://arxiv.org/abs/1411.5018)\n",
    "- [Hogg, Bovy and Lang: ``Data analysis recipes: Fitting a model to data\"](https://arxiv.org/abs/1008.4686)\n",
    "\n",
    "\n",
    "##### For those who want to dive deep:\n",
    "\n",
    "- [D. Sivia and J. Skilling: ``Data Analysis: A Bayesian Tutorial''](https://www.amazon.com/Data-Analysis-Bayesian-Devinderjit-Sivia/dp/0198568320)\n",
    "- [E.T. Jaynes: ``Probability Theory: The Logic of Science''](http://bayes.wustl.edu/etj/prob/book.pdf)\n",
    "- [E.T. Jaynes: ``Confidence Intervals vs. Bayesian intervals''](http://bayes.wustl.edu/etj/articles/confidence.pdf)"
   ]
  },
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   "source": [
    "##### Topics for this week: \n",
    "- Bayesian model parameter estimation \n",
    "- Bayesian model selection \n",
    "- optimal bin width for classical histograms\n",
    "- Bayesian Blocks algorithm for optimal histograms \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy import stats\n",
    "from scipy.stats import norm\n",
    "from astroML.plotting import hist\n",
    "np.random.seed(42)\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=8, usetex=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## Let's use Jake's excellent example from\n",
    "## http://jakevdp.github.io/blog/2015/08/07/frequentism-and-bayesianism-5-model-selection/\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns; sns.set() # set default plot styles\n",
    "from scipy import stats\n",
    "from scipy import optimize\n",
    "\n",
    "# generate (x,y, sigma_y) \"data\" \n",
    "mydata = np.array([[ 0.42,  0.72,  0.  ,  0.3 ,  0.15,\n",
    "                   0.09,  0.19,  0.35,  0.4 ,  0.54,\n",
    "                   0.42,  0.69,  0.2 ,  0.88,  0.03,\n",
    "                   0.67,  0.42,  0.56,  0.14,  0.2  ],\n",
    "                 [ 0.33,  0.41, -0.22,  0.01, -0.05,\n",
    "                  -0.05, -0.12,  0.26,  0.29,  0.39, \n",
    "                   0.31,  0.42, -0.01,  0.58, -0.2 ,\n",
    "                   0.52,  0.15,  0.32, -0.13, -0.09 ],\n",
    "                 [ 0.1 ,  0.1 ,  0.1 ,  0.1 ,  0.1 ,\n",
    "                   0.1 ,  0.1 ,  0.1 ,  0.1 ,  0.1 ,\n",
    "                   0.1 ,  0.1 ,  0.1 ,  0.1 ,  0.1 ,\n",
    "                   0.1 ,  0.1 ,  0.1 ,  0.1 ,  0.1  ]])\n",
    "\n",
    "\n",
    "# this function computes polynomial models given some data x\n",
    "# and parameters theta\n",
    "def polynomial_fit(theta, x):\n",
    "    \"\"\"Polynomial model of degree (len(theta) - 1)\"\"\"\n",
    "    return sum(t * x ** n for (n, t) in enumerate(theta))\n",
    "\n",
    "# compute the data log-likelihood given a model\n",
    "def logL(theta, model=polynomial_fit, data=mydata):\n",
    "    \"\"\"Gaussian log-likelihood of the model at theta\"\"\"\n",
    "    x, y, sigma_y = data\n",
    "    y_fit = model(theta, x)\n",
    "    return sum(stats.norm.logpdf(*args)\n",
    "               for args in zip(y, y_fit, sigma_y))\n",
    "\n",
    "# a direct optimization approach is used to get best model \n",
    "# parameters (which minimize -logL)\n",
    "def best_theta(degree, model=polynomial_fit, data=mydata):\n",
    "    theta_0 = (degree + 1) * [0]\n",
    "    neg_logL = lambda theta: -logL(theta, model, data)\n",
    "    return optimize.fmin_bfgs(neg_logL, theta_0, disp=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CHI2:\n",
      "   best linear model: 11.324128378349815 22.010867006612543\n",
      "best quadratic model: 9.462835218566857 22.94151358650403\n",
      "    best cubic model: 8.993608370385859 23.176127010594524\n",
      "CHI2 per degree of freedom:\n",
      "   best linear model: 0.6291182432416564\n",
      "best quadratic model: 0.5566373657980505\n",
      "    best cubic model: 0.5621005231491162\n"
     ]
    },
    {
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Uhf25ZXyyNQ9jdSMh/l7MmRZDvAsOX7tZ0sbuoi6HV3h4BHl5rY/dnj//JdLTd/Dyy79j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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1139b7510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# generate (x,y, sigma_y) \"data\" \n",
    "x, y, sigma_y = mydata\n",
    "Ndata = x.size\n",
    "\n",
    "# get best-fit parameters for linear, quadratic and cubic models\n",
    "theta1 = best_theta(1)\n",
    "theta2 = best_theta(2)\n",
    "theta3 = best_theta(3)\n",
    "# generate best fit lines on a fine grid \n",
    "xfit = np.linspace(-0.1, 1.1, 1000)\n",
    "yfit1 = polynomial_fit(theta1, xfit)\n",
    "yfit2 = polynomial_fit(theta2, xfit)\n",
    "yfit3 = polynomial_fit(theta3, xfit)\n",
    "# and compute chi2: sum{[(y-yfit)/sigma_y]^2} \n",
    "chi21 = np.sum(((y-polynomial_fit(theta1, x))/sigma_y)**2) \n",
    "chi22 = np.sum(((y-polynomial_fit(theta2, x))/sigma_y)**2) \n",
    "chi23 = np.sum(((y-polynomial_fit(theta3, x))/sigma_y)**2) \n",
    "# and chi2 per degree of freedom (the number of fitted parameters is 2, 3, 4)\n",
    "chi2dof1 = chi21/(Ndata - 2)\n",
    "chi2dof2 = chi22/(Ndata - 3)\n",
    "chi2dof3 = chi23/(Ndata - 4)\n",
    "\n",
    "print \"CHI2:\"\n",
    "print '   best linear model:', chi21, zi21\n",
    "print 'best quadratic model:', chi22, zi22\n",
    "print '    best cubic model:', chi23, zi23\n",
    "print \"CHI2 per degree of freedom:\"\n",
    "print '   best linear model:', chi2dof1\n",
    "print 'best quadratic model:', chi2dof2\n",
    "print '    best cubic model:', chi2dof3\n",
    "\n",
    "# plot \n",
    "fig, ax = plt.subplots()\n",
    "ax.errorbar(x, y, sigma_y, fmt='ok', ecolor='gray')\n",
    "ax.plot(xfit, polynomial_fit(theta1, xfit), label='best linear model')\n",
    "ax.plot(xfit, polynomial_fit(theta2, xfit), label='best quadratic model')\n",
    "ax.plot(xfit, polynomial_fit(theta3, xfit), label='best cubic model')\n",
    "ax.legend(loc='best', fontsize=14)\n",
    "ax.set(xlabel='x', ylabel='y', title='data');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "\n",
    "$\\chi^2$ per degree of freedom was computed as \n",
    "$$ \\chi^2_{dof}  = \\frac{1}{N-k} \\sum_i^N \\left( \\frac{y - y_{fit}}{\\sigma_y} \\right)^2, $$\n",
    "where $N$ is the number of data points and $k$ is the number of free model parameters (2, 3, and 4).\n",
    "\n",
    "For large values of $(N-k)$ (larger than about 10), the distribution of \n",
    "$\\chi^2$ per degre of freedom is approximately Gaussian centered on $\\mu=1$ and the width of\n",
    "$\\sigma=\\sqrt{2/(N-k)}$  (here, $\\sigma = 0.33, 0.34, 0.35$). \n",
    "\n",
    "The values of $\\chi^2$ per degre of freedom for the three models are: \n",
    "\n",
    "   best linear model: 0.629 \n",
    "\n",
    "best quadratic model: 0.557\n",
    "\n",
    "best cubic model: 0.562 \n",
    "\n",
    " \n",
    "**Which model should we adopt?**\n",
    "\n",
    "Obviously, the cubic model has the lowest $\\chi2$ (\"fits the best\"), but \n",
    "it has 4 free parameters while the linear model has only 2 free parameters.\n",
    "\n",
    "**How do we trade improvement in $\\chi^2$ with increasing model\n",
    "complexity? **\n",
    "\n",
    "**Occam’s razor:** select the simplest model that is in fair agreement with the data.\n",
    "\n",
    "This principle was already known to Ptolemy who said: “We consider it a good principle \n",
    "to explain the phenomena by the simplest hypothesis possible.”\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Model Comparison\n",
    "\n",
    "Up to now we have concerned ourselves with determining the optimal parameters of a given model fit.  But what if *another* model would be a better fit (regardless of how you choose the parameters of the first model).  \n",
    "\n",
    "That leads us to a discussion of model comparison (see textbook $\\S$5.4 and $\\S$5.7.1-3).\n",
    "We start with the Bayes theorem,\n",
    "$$p(M,\\theta \\,|\\,D,I) = \\frac{p(D\\,|\\,M,\\theta,I)\\,p(M,\\theta\\,|\\,I)}{p(D\\,|\\,I)},$$\n",
    "\n",
    "and marginalize over model parameter space $\\theta$\n",
    "to obtain **the probability of model $M$** given the data $D$ and prior information $I$:\n",
    "\n",
    "$$p(M\\,|\\,D,I) \\equiv \\int p(M,\\theta \\,|\\,D,I) \\, d\\theta = \n",
    "      \\int \\frac{p(D\\,|\\,M,\\theta,I)\\,p(M,\\theta\\,|\\,I)}{p(D\\,|\\,I)} \\, d\\theta =\n",
    "      \\frac{p(M\\,|\\,I)}{p(D\\,|\\,I)} \\int p(D\\,|\\,M,\\theta,I)\\,p(\\theta\\,|\\,M,I) \\, d\\theta  $$\n",
    " \n",
    "To determine which model is better we compute the ratio of the posterior probabilities or the **odds ratio** for two models as\n",
    "$$O_{21} \\equiv \\frac{p(M_2|D,I)}{p(M_1|D,I)}.$$\n",
    "\n",
    "The posterior probability that the model $M$ is correct given data $D$ (a number between 0 and 1) is\n",
    "$$p(M|D,I) = \\frac{p(D|M,I)p(M|I)}{p(D|I)},$$\n",
    "the odds ratio can ignore $p(D|I)$ since it  will be the same for both models.  \n",
    "\n",
    "We get \n",
    "$$O_{21} = \\frac{p(D\\,|\\,M_2,I)\\,p(M_2\\,|\\,I)}{p(D\\,|\\,M_1,I)\\,p(M_1\\,|\\,I)} \\equiv B_{21} \\, \\frac{p(M_2\\,|\\,I)}{p(M_1\\,|\\,I)},$$\n",
    "\n",
    "where $B_{21}$ is called the Bayes factor. \n",
    "\n",
    "The Bayes factor compares how well the models fit the data: it is a ratio of data likelihoods averaged over \n",
    "all allowed values of the model parameters; if two models fit the data equally well, the \"winner\" is decided based on priors. \n",
    "\n",
    "For example, consider a noisy image of a source which is equally likely to be a star or a galaxy. \n",
    "The posterior probability that the source is a star will greatly depend on whether we are looking \n",
    "at the Galactic plane or not. \n",
    "\n",
    "\n",
    "We can compute \n",
    "$$ E(M) \\equiv p(D\\,|\\,M, I) = \\int p(D\\,|\\,M, \\theta, I) \\, p(\\theta\\,|\\,M, I) \\, d\\theta, $$\n",
    "where $E(M)$ is called **the marginal likelihood for model $M$** and it quantifies the probability that the \n",
    "data $D$ would be observed if the model $M$ were the correct model. \n",
    "\n",
    "In the physics literature, the marginal likelihood $E(M)$ is often called **evidence** (despite the fact that to \n",
    "scientists, evidence and data mean essentially the same thing). \n",
    "Since the evidence $E(M)$ involves integration of the data likelihood $p(D|M,\\theta,I)$, it is also \n",
    "called **the global likelihood for model M**. The global likelihood, or evidence, is a weighted average \n",
    "of the likelihood function, with the prior for model parameters acting as the weighting function.\n",
    "\n",
    "The following plot from the last time shows the data likelihood $p(D|M, \\theta, I)$ for a sample drawn\n",
    "from Cauchy distribution. \n",
    "<img src=\"figures/Lcauchy.tiff\" width=\"600\">\n",
    "\n",
    "The best $\\mu$ and $\\sigma$ are found as the maximum of this distribution. \n",
    "The marginal likelihood for model $M=Gauss$ is found by **integrating** this image over $\\mu$ and $\\sigma$.\n",
    "\n",
    "If we think that the sample might have been drawn from some other distribution, we can produce an analogous\n",
    "$p(D|M, \\theta, I)$, say for $M=Gauss$ ([here is an example](http://www.astroml.org/book_figures/chapter5/fig_likelihood_gaussian.html))\n",
    "integrate it and compute the Bayes factor and the models odds\n",
    "ratio (with models assigned equal prior probabilities unless there is a reason that one model is more \n",
    "likely than another, e.g. recall the star/galaxy case) \n",
    "\n",
    "\n",
    "**How do we interpret the values of the odds ratio in practice?**\n",
    "\n",
    "Jeffreys proposed a five-step scale for \n",
    "interpreting the odds ratio, where $O_{21} > 10$ represents “strong” evidence in favor of $M_2$ ($M_2$ \n",
    "is ten times more probable than $M_1$), and $O_{21} > 100$ is “decisive” evidence ($M_2$ is one hundred \n",
    "times more probable than $M_1$). When $O_{21} < 3$, the evidence is “not worth more than a bare mention.”\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Hypothesis Testing\n",
    "\n",
    "In *hypothesis testing* we are essentially comparing a model, $M_1$, to its complement.  That is $p(M_1) + p(M_2) = 1$.  \n",
    "\n",
    "If we take $M_1$ to be the \"null\" (default) hypothesis (which is generally that, for example, a correlation does *not* exist), then we are asking whether or not the data reject the null hypothesis.\n",
    "\n",
    "In classical hypothesis testing we can ask whether or not a single model provides a good description of the data.  \n",
    "\n",
    "** In Bayesian hypothesis testing, we must have an alternative comparison model! **\n",
    "\n",
    "\n",
    "Let's draw sample from a Cauchy distribution and perform Bayesian Model Comparison for \n",
    "Cauchy and Gaussian models. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results for first 10 points:\n",
      "  L(M = Cauchy) = 1.18e-12 +/- 5.01e-13\n",
      "  L(M = Gauss)  = 8.09e-13 +/- 4.78e-13\n",
      "  O_{CG} = 1.45 +/- 0.859\n"
     ]
    },
    {
     "data": {
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//x2uumoBbrnlO1i79kc4/fQzTNiaxPX29uL999+DLPvh9x9Ad3cYCxY4cfnl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\nIiKiLMGkTkRElCX+P0Kc5n86cYluAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113b90f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Author: Jake VanderPlas\n",
    "# License: BSD\n",
    "#   The figure produced by this code is published in the textbook\n",
    "#   \"Statistics, Data Mining, and Machine Learning in Astronomy\" (2013)\n",
    "#   For more information, see http://astroML.github.com\n",
    "#   To report a bug or issue, use the following forum:\n",
    "#    https://groups.google.com/forum/#!forum/astroml-general\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy.stats import cauchy, norm\n",
    "from scipy import integrate\n",
    "\n",
    "#----------------------------------------------------------------------\n",
    "# This function adjusts matplotlib settings for a uniform feel in the textbook.\n",
    "# Note that with usetex=True, fonts are rendered with LaTeX.  This may\n",
    "# result in an error if LaTeX is not installed on your system.  In that case,\n",
    "# you can set usetex to False.\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=8, usetex=True)\n",
    "\n",
    "\n",
    "def logL_cauchy(xi, gamma, mu,\n",
    "                mu_min=-10, mu_max=10, sigma_min=0.01, sigma_max=100):\n",
    "    \"\"\"Equation 5.74: cauchy likelihood\"\"\"\n",
    "    xi = np.asarray(xi)\n",
    "    n = xi.size\n",
    "    shape = np.broadcast(gamma, mu).shape\n",
    "\n",
    "    xi = xi.reshape(xi.shape + tuple([1 for s in shape]))\n",
    "\n",
    "    prior_normalization = - (np.log(mu_max - mu_min)\n",
    "                             + np.log(np.log(sigma_max / sigma_min)))\n",
    "\n",
    "    return (prior_normalization\n",
    "            - n * np.log(np.pi)\n",
    "            + (n - 1) * np.log(gamma)\n",
    "            - np.sum(np.log(gamma ** 2 + (xi - mu) ** 2), 0))\n",
    "\n",
    "\n",
    "def logL_gaussian(xi, sigma, mu,\n",
    "                  mu_min=-10, mu_max=10, sigma_min=0.01, sigma_max=100):\n",
    "    \"\"\"Equation 5.57: gaussian likelihood\"\"\"\n",
    "    xi = np.asarray(xi)\n",
    "    n = xi.size\n",
    "    shape = np.broadcast(sigma, mu).shape\n",
    "\n",
    "    xi = xi.reshape(xi.shape + tuple([1 for s in shape]))\n",
    "\n",
    "    prior_normalization = - (np.log(mu_max - mu_min)\n",
    "                             + np.log(np.log(sigma_max / sigma_min)))\n",
    "\n",
    "    return (prior_normalization\n",
    "            - 0.5 * n * np.log(2 * np.pi)\n",
    "            - (n + 1) * np.log(sigma)\n",
    "            - np.sum(0.5 * ((xi - mu) / sigma) ** 2, 0))\n",
    "\n",
    "\n",
    "def calculate_odds_ratio(xi, epsrel=1E-8, epsabs=1E-15):\n",
    "    \"\"\"\n",
    "    Compute the odds ratio by perfoming a double integral\n",
    "    over the likelihood space.\n",
    "    \"\"\"\n",
    "    gauss_Ifunc = lambda mu, sigma: np.exp(logL_gaussian(xi, mu, sigma))\n",
    "    cauchy_Ifunc = lambda mu, gamma: np.exp(logL_cauchy(xi, mu, gamma))\n",
    "\n",
    "    I_gauss, err_gauss = integrate.dblquad(gauss_Ifunc, -np.inf, np.inf,\n",
    "                                           lambda x: 0, lambda x: np.inf,\n",
    "                                           epsabs=epsabs, epsrel=epsrel)\n",
    "    I_cauchy, err_cauchy = integrate.dblquad(cauchy_Ifunc, -np.inf, np.inf,\n",
    "                                             lambda x: 0, lambda x: np.inf,\n",
    "                                             epsabs=epsabs, epsrel=epsrel)\n",
    "\n",
    "    if I_gauss == 0:\n",
    "        O_CG = np.inf\n",
    "        err_O_CG = np.inf\n",
    "    else:\n",
    "        O_CG = I_cauchy / I_gauss\n",
    "        err_O_CG = O_CG * np.sqrt((err_gauss / I_gauss) ** 2)\n",
    "\n",
    "    return (I_gauss, err_gauss), (I_cauchy, err_cauchy), (O_CG, err_O_CG)\n",
    "\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Draw points from a Cauchy distribution\n",
    "np.random.seed(44)\n",
    "mu = 0\n",
    "gamma = 2\n",
    "xi = cauchy(mu, gamma).rvs(100)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# compute the odds ratio for the first 10 points\n",
    "((I_gauss, err_gauss),\n",
    " (I_cauchy, err_cauchy),\n",
    " (O_CG, err_O_CG)) = calculate_odds_ratio(xi[:10])\n",
    "\n",
    "print \"Results for first 10 points:\"\n",
    "print \"  L(M = Cauchy) = %.2e +/- %.2e\" % (I_cauchy, err_cauchy)\n",
    "print \"  L(M = Gauss)  = %.2e +/- %.2e\" % (I_gauss, err_gauss)\n",
    "print \"  O_{CG} = %.3g +/- %.3g\" % (O_CG, err_O_CG)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# calculate the results as a function of number of points\n",
    "Nrange = np.arange(10, 101, 2)\n",
    "Odds = np.zeros(Nrange.shape)\n",
    "for i, N in enumerate(Nrange):\n",
    "    res = calculate_odds_ratio(xi[:N])\n",
    "    Odds[i] = res[2][0]\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# plot the results\n",
    "fig = plt.figure(figsize=(8, 5))\n",
    "fig.subplots_adjust(hspace=0.1)\n",
    "\n",
    "ax1 = fig.add_subplot(211, yscale='log')\n",
    "ax1.plot(Nrange, Odds, '-k')\n",
    "ax1.set_ylabel(r'$O_{CG}$ for $N$ points')\n",
    "ax1.set_xlim(0, 100)\n",
    "ax1.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "ax1.yaxis.set_major_locator(plt.LogLocator(base=10000.0))\n",
    "\n",
    "ax2 = fig.add_subplot(212)\n",
    "ax2.scatter(np.arange(1, len(xi) + 1), xi, lw=0, s=16, c='k')\n",
    "ax2.set_xlim(0, 100)\n",
    "ax2.set_xlabel('Sample Size $N$')\n",
    "ax2.set_ylabel('Sample Value')\n",
    "\n",
    "plt.show() \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
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   "source": [
    "The odds ratio for the first 10 points is very close to unity \n",
    "($O_{CG} = 1.45$) and is therefore inconclusive.\n",
    " \n",
    "Why do we get an inconclusive odds ratio? With only 10 points, the probability of \n",
    "drawing at least one point far away from the mean, which would strongly argue against \n",
    "the Gaussian model, is fairly small. As the number of data values is increased, the \n",
    "ability to discriminate between the models increases, too. As the above figure shows,\n",
    "when we increase the size of the observed sample, the odds ratio quickly favors the \n",
    "Cauchy over the Gaussian model.\n",
    "\n",
    "Going back to our polynomial example, we could use the same method and \n",
    "integrate (marginalize) the posterior over model parameter space $\\theta$.\n",
    "However, the data likelihood in that example can become cumbersome to track\n",
    "(for the cubic fit, it's a 4-dimensional space to integrate) and it's much\n",
    "easier to compare the fits using numerical methods such as Markov Chain\n",
    "Monte Carlo (MCMC). We shall introduce MCMC next week (and apply it to\n",
    "this example, too). \n",
    "\n",
    "Meanwhile, let's introduce an approximate method for model comparison."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Approximate Bayesian Model Comparison\n",
    "\n",
    "We can always increase the number of parameters in our model and improve the fit.  \n",
    "So we need some kind of \"scoring\" system that accounts for the complexity of the \n",
    "model and penalizes models that are too complex (given the data): this principle\n",
    "of selecting the simplest model that is in fair agreement with the data is known \n",
    "as Occam’s razor. \n",
    "\n",
    "Hidden in the above expression for the odds ratio is its ability to penalize complex \n",
    "models with many free parameters; that is, Occam’s razor is naturally included into \n",
    "the Bayesian model comparison! \n",
    "\n",
    "A common scoring system is the **Bayesian information criterion (BIC)**. It can \n",
    "be derived from the model odds ratio by assuming that the data likelihood is \n",
    "Gaussian (see $\\S$5.4.2 in textbook for details). The BIC is easier to compute \n",
    "than the model odds ratio because it is based on the maximum value of the data \n",
    "likelihood, $L_0(M)$, rather than on the integration of the data likelihood over \n",
    "the full parameter space (i.e. evidence $E(M)$). For $N$ data points and a model \n",
    "with $k$ parameters,\n",
    "$${\\rm BIC} \\equiv -2 \\ln [L_0(M)] + k \\ln N.$$ \n",
    "where the 1st term is equal to model's $\\chi^2$ (under the assumption of normality;\n",
    "note that this is not $\\chi^2$ per degree of freedom!)\n",
    "and the 2nd term on the RHS penalizes complex models relative to simple ones.\n",
    "\n",
    "Another well-known scoring system, which can be derived using information theory \n",
    "is the **Aikake information criterion (AIC)**,  computed as\n",
    "$${\\rm AIC} \\equiv -2 \\ln [L_0(M)] + 2k + \\frac{2k(k+1)}{N-k-1}.$$\n",
    "\n",
    "When two models are compared, their BIC (or AIC) are compared analogously to the \n",
    "odds ratio, that is, the model with the smaller value wins. If the models are \n",
    "equally successful in describing the data (i.e., they have the same value of $L_0(M)$), \n",
    "then the model with fewer free parameters wins. \n",
    "\n",
    "Both the BIC and AIC are approximations and might not be valid if the underlying \n",
    "assumptions are not met. Furthermore, unlike the odds ratio, both of them penalize \n",
    "unconstrained parameters. In general, it is better to compute the odds ratio when \n",
    "computationally feasible. \n",
    "\n",
    "Let's us BIC to choose the best polynomial from the earlier line fitting example. \n",
    "Starting with\n",
    "$${\\rm BIC} = \\chi^2 + k \\ln N,$$ \n",
    "with $N=20$ we have:\n",
    "\n",
    "<pre>\n",
    "     model     chi2    k   k*ln(N)   BIC       \n",
    "    linear    11.32    2    5.99    17.31   \n",
    "  quadratic    9.46    3    8.99    18.45   \n",
    "     cubic     8.99    4   11.98    20.97  \n",
    "</pre>\n",
    "\n",
    "Therefore, the improvement in $\\chi^2$ for the quadratic model, relative \n",
    "to the linear model, ** is not large enough to justify another free model\n",
    "parameter.** \n",
    "\n",
    "\n",
    "Another useful method for model selection is the **cross-validation method**.\n",
    "Its disadvantage is that it may be computationally intensive for complex\n",
    "problems, but its great advantage is that it is conceptually simple and\n",
    "easy to implement (for details see textbook, $\\S$8.11.1). The basic idea of \n",
    "the cross-validation method is not to use a fraction of the sample when\n",
    "obtaining the best-fit parameters and then compute the goodness-of-fit \n",
    "(e.g. $\\chi^2$, but any metric can be implemented) with this unused subsample. \n",
    "The \"leave-one-out\" implementation repeats model fitting $N$ times, using\n",
    "$N-1$ data points. Each time the missing point is used to compute the \n",
    "goodness-of-fit, and the $N$ values are averaged. \n",
    "\n",
    "When the model is too simple, it will have high bias (under-fitting)\n",
    "and when the model is too complex, the variance will be high (over-fitting).\n",
    "\n",
    "\n",
    "![BayesSlide1](figures/fig8-13.tiff)\n",
    "\n"
   ]
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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    "\n",
    "  \n",
    "### Approximate Bayesian Model Comparison\n",
    "\n",
    "The data likelihood, required to compute the models odds ratio, and the commonly \n",
    "used $\\chi^2$ goodness-of-fit parameter are related. By introducting a few \n",
    "assumptions and approximations, the computations of the odds ratio can be\n",
    "greatly simplified and expressed as the sum of $\\chi^2$ and a term that penalizes\n",
    "models for their parameters (when models achieve similar values of $\\chi^2$,\n",
    "the one with the smallest number of free parameters wins). \n",
    " \n",
    "We consider a one dimensional case with unknown parameter $\\mu$ and\n",
    "start with an approximate computation of the evidence $E(M)$,\n",
    "$$ E(M) = \\int p(\\{x_i\\}, \\,|\\,M, \\mu, I) \\, p(\\mu \\,|\\,M, I) \\, d \\mu .$$\n",
    "\n",
    "Our first assumption is that the prior is uniform \n",
    "$$  p(\\mu\\,|\\, M, I) = \\frac{1}{\\Delta} \\,\\,\\,\\,\\,\\, {\\rm for} \\,\\,\\, -\\frac{\\Delta}{2} < \\mu < \\frac{\\Delta}{2}$$\n",
    "and 0 otherwise. \n",
    "\n",
    "The second assumption is that the data likelihood can\n",
    "be approximated as a Gaussian around its maximum at $\\mu=\\mu_0$\n",
    "$$ p(\\{x_i\\} \\,|\\,M, \\mu, I) \\approx L(\\mu_0) \\, \\exp \\left( - \\frac{(\\mu - \\mu_0)^2}{2 \\sigma_\\mu^2} \\right). $$\n",
    "\n",
    "These assumptions lead to \n",
    "$$ E(M) \\approx \\frac{L(\\mu_0)}{\\Delta} \\, \\int_{-\\Delta/2}^{\\Delta/2} \\exp \\left( - \\frac{(\\mu - \\mu_0)^2}{2 \\sigma_\\mu^2}\\right) d\\mu, $$\n",
    "and with an additional assumption $\\sigma_\\mu \\ll \\Delta$ (data overcomes\n",
    "the prior), we get\n",
    "$$ E(M) \\approx \\frac{\\sigma_\\mu}{\\Delta} \\, L(\\mu_0) \\, \\sqrt{2\\pi}. $$\n",
    "\n",
    "Note that $E(M) \\ll L(\\mu_0)$, because $\\sigma_\\mu \\ll \\Delta$. In multi-dimensional\n",
    "case, each model parameter constrained by the model carries a similar \n",
    "multiplicative penalty, proportional to $\\sigma_\\mu/\\Delta$, when computing the \n",
    "Bayes factor. If a parameter, or a degenerate parameter combination, is unconstrained \n",
    "by the data (i.e., $\\sigma_\\mu \\approx \\Delta$), there is no penalty.  \n",
    "The odds ratio can justify an additional model parameter **only if this penalty is offset** by either \n",
    "an increase of the maximum value of the data likelihood, $L(\\mu_0$), or by the ratio \n",
    "of prior model probabilities, $p(M_2|I)/p(M_1|I)$. If both of these quantities are \n",
    "similar for the two models, the one with fewer parameters typically wins.\n",
    "\n",
    "If a model is well constrained by the data, for each of $k$ constrained\n",
    "parameters, $\\sigma_\\mu \\propto 1/\\sqrt{N}$, where $N$ is the number of data \n",
    "points. Therefore,\n",
    "$$ E(M) \\propto L(\\mu_0) \\, \\left(\\sqrt{N}\\right)^{-k}. $$\n",
    " \n",
    "In order to establish connection with $\\chi^2$ via $L(\\mu_0) =\n",
    "\\exp\\left(-\\frac{\\chi^2}{2}\\right)$, we define for a model the **Bayesian information \n",
    "criterion (BIC)** as $BIC \\equiv -2\\ln{E(M)}$, and finally obtain  \n",
    "$${\\rm BIC} = -2 \\ln [L_0(M)] + k \\ln N.$$ \n",
    "The 1st term on the RHS is equal to model's $\\chi^2$ (under the assumption \n",
    "of normality; note that this is *not* $\\chi^2$ per degree of freedom!) and \n",
    "the 2nd term on the RHS penalizes complex models relative to simple ones.\n",
    "\n",
    "In summary, when multiple models are considered, their BIC values, computed as\n",
    "$${\\rm BIC} = \\chi^2 + k \\ln N,$$ \n",
    "where $N$ is the number of data points and $k$ is the number of constrained\n",
    "model parameters, are compared and the model with the smallest BIC value wins.   \n",
    " \n",
    "**N.B.** BIC is an approximation and might not be valid if the underlying \n",
    "assumptions (data ovecoming prior and Gaussian likelihood) are not met! In general, it is better to compute the odds ratio when computationally feasible.\n",
    "\n",
    "   \n",
    " "
   ]
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     "slide_type": "slide"
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   "source": [
    "####   \n",
    "### Homework #3 \n",
    "\n",
    "Due on Thursday, February 15. \n",
    "\n",
    "Last time we discussed fitting a burst + background model, with two types \n",
    "of burst profile: exponential decay and a gaussian profile. Let's now\n",
    "assume that we do not know which profile fits best and fit both profiles. \n",
    "\n",
    "With Group 1 generating data using the exponential decay profile, and \n",
    "Group 2 using the gaussian profile (with the same parameters as in the\n",
    "lecture): \n",
    "\n",
    "1) Can you tell which model is correct using global likelihood computation?\n",
    "\n",
    "2) Can you tell which model is correct using BIC?\n",
    "\n",
    "3) What happens when you increase the number of data points by a factor of 2\n",
    "    (using BIC)?\n",
    "\n",
    "4) What happens when you decrease the number of data points by a factor of 2\n",
    "     (using BIC)?\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "## Coin Flip as an Example of Bayesian Model Comparison\n",
    "\n",
    "Let's look at an example we already discussed: the coin flip. Let's assume we have\n",
    "N draws and k are success (say, heads). \n",
    "\n",
    "We will compare two hypotheses; \n",
    "\n",
    "**M1**: the coin has a known heads probability $b_\\ast$ (say, a fair coin with $b_\\ast=0.5$), and \n",
    "\n",
    "**M2**: the heads probability $b$ is unknown, with a uniform prior in the range 0–1. \n",
    "Note that the prior for model M1 is a delta function, $\\delta(b-b_\\ast)$. \n",
    "\n",
    "Given a model parametrized by the probability of success $b$, the likelihood that the data set \n",
    "contains k outcomes equal to 1 is given by \n",
    "  $$    p(k\\,|\\,b, N) = \\frac{N!}{k! \\, (N-k)!} \\, b^k \\, (1-b)^{N-k} $$\n",
    "\n",
    "For model M2 the prior for $b$ is flat in the range 0-1 and the product of the \n",
    "data likelihood and prior is same as above. However, for model M1 the prior is a \n",
    "delta function $\\delta(b-b_\\ast)$ and we get for the product of the \n",
    "data likelihood and prior  \n",
    "$$    p(k\\,|\\,b_\\ast, N, M1)\\,p(b|M1, I) = \\frac{N!}{k! \\, (N-k)!} \\, b_\\ast^k \\, (1-b_\\ast)^{N-k}. $$\n",
    "\n",
    "Consequently, the odds ratio is given by \n",
    "$$ O_{21} = \\int_0^1 \\left(\\frac{b}{b_\\ast}\\right)^k \\left(\\frac{1-b}{1-b_\\ast}\\right)^{N-k} db, $$\n",
    "as illustrated in the following figure. \n",
    "\n",
    " \n",
    "![OddsRatio](figures/odds.tiff)\n",
    "\n",
    "This figure (from the textbook) illustrates the behavior of $O_{21}$ as a function of $k$\n",
    "for two different values of N and for two different values of $b_\\ast$: $b_\\ast = 0.5$ \n",
    "(M1: the coin is fair) and $b_\\ast = 0.1$. As the figure shows, the ability to distinguish \n",
    "the two hypothesis **improves** with the sample size. For example, when $b_\\ast= 0.5$ and \n",
    "k/N = 0.1, the odds ratio in favor of M2 increases from about 9 for N = 10 to about \n",
    "263 for N = 20. When k = $b_\\ast N$, the odds ratio is 0.37 for N = 10 and 0.27 for N = 20. \n",
    "In other words, **the simpler model is favored by the data**, and the support strengthens \n",
    "with the sample size. \n",
    "\n",
    "It is easy to show by integration of the above equation for $O_{21}$ that \n",
    "$O_{21}= \\sqrt{\\pi/(2N)}$ when k = $b_\\ast N$ and $b_\\ast = 0.5$. For example, to build strong \n",
    "evidence that a coin is fair, $O_{21} < 0.1$, it takes as many as N $>$ 157 tosses. With \n",
    "N = 10,000, the heads probability of a fair coin is measured with a precision of 1% and\n",
    "the corresponding odds ratio is $O_{21} \\approx 1/80$, approaching Jeffreys’ decisive \n",
    "evidence level. \n",
    " \n",
    "In **frequentist approach**, we would ask whether we can reject the null hypothesis that \n",
    "our coin is fair. We would ask whether a given $k$ is a very unusual outcome (at some \n",
    "significance level $\\alpha$, say $\\alpha=0.05$, which corresponds to about \"2$\\sigma$\"\n",
    "deviation) for a fair coin with $b_\\ast = 0.5$ and with\n",
    "a given N. In the **Bayesian approach**, we offer an alternative hypothesis that the coin \n",
    "has an unknown heads probability. While this probability can be estimated from provided \n",
    "data ($b_0$), **we consider all the possible values** of $b_0$ when comparing the two proposed \n",
    "hypotheses. \n",
    "\n",
    "As a numerical example, let's consider N=20 and k=16. Using the results discussed earlier, \n",
    "we find that the scatter around the expected value $k_0 = b_\\ast N$ = 10 is $\\sigma_k = 2.24$. \n",
    "Therefore, k = 16 is about 2.7$\\sigma_k$ away from $k_0$, and at the adopted significance \n",
    "level $\\alpha=0.05$ we **reject the null hypothesis** (this rejection means that it is unlikely that k = 16 would have arisen by chance). Of course, k = 16 does **not** imply \n",
    "that it is impossible that the coin is fair (infrequent events happen, too!).\n",
    "\n",
    "As shown in the above figure, the chosen parameters (N=20 and k=16) correspond to the \n",
    "Bayesian **odds ratio** of about 10 in favor of hypothesis M2.  \n",
    "\n",
    "\n",
    "### Approximate Bayesian Model Comparison\n",
    "\n",
    "Let's now use BIC to address the same problem and illustrate how model M2 gets \n",
    "penalized for its free parameter. Essentially, we want an approximation for the\n",
    "integral expression for $O_{21}$\n",
    "$$ O_{21} = \\int_0^1 \\left(\\frac{b}{b_\\ast}\\right)^k \\left(\\frac{1-b}{1-b_\\ast}\\right)^{N-k} db, $$\n",
    " \n",
    "We can approximate \n",
    "$$ E(M2) \\approx \\sqrt{2\\pi} \\, L(b_0) \\, \\sigma_b, $$\n",
    "where $b_0 = k/N$ and for largish N we have $\\sigma_b \\approx \\sqrt{b_0(1-b_0)/N}$. \n",
    "\n",
    "For M1, we have exact result because the prior is $\\delta$ function\n",
    "$$ E(M1) = \\frac{N!}{k! \\,(N-k)!} b_\\ast^k (1-b_\\ast)^{N-k}.$$\n",
    "\n",
    "And the odds ratio becomes\n",
    "$$ O_{21} \\approx \\sqrt{2\\pi} \\, \\sigma_b \\,\n",
    "\\left(\\frac{b_0}{b_\\ast}\\right)^k \\left(\\frac{1-b_0}{1-b_\\ast}\\right)^{N-k} =\n",
    "\\sqrt{2\\pi}\\, \\sqrt{\\frac{b_0(1-b_0)}{N}} \\, \\left(\\frac{b_0}{b_\\ast}\\right)^k \\left(\\frac{1-b_0}{1-b_\\ast}\\right)^{N-k} .$$\n",
    "\n",
    "Now we can explicitly see that the evidence in favor of model M2 decreases (the model is “penalized”) proportionally to the posterior pdf width of its free parameter, $\\sigma_b$. \n",
    "\n",
    "If indeed $b_0 \\approx b_\\ast$, model M1 wins because it explained the data without\n",
    "any free parameter. On the other hand, the evidence in favor of M2 increases as the \n",
    "data-based value $b_0$ becomes very different from the prior $b_\\ast$ claimed by model M1.\n",
    "** Model M1 becomes disfavored because it is unable to explain the observed data.**\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "### Knuth's Histograms \n",
    "\n",
    "Details are in textbook, $\\S$5.7.2.\n",
    " \n",
    "The Scott’s rule and the Freedman–Diaconis rule for estimating optimal bin width produce the same answer for multimodal and unimodal distributions as long as their data set size and scale parameter are the same. \n",
    "\n",
    "This undesired result is avoided when using a method developed by Knuth. \n",
    "\n",
    "\n",
    "\n",
    "![KnuthSlide](figures/knuth.tiff)\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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lYIxmZjKZ6scPH24gkRgFAJw69TjroRNRW2g8omnBti1Eoz0CWkRhxkCThAvKaObWikGa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\n553N9u16bGTkqqNtcsLMzIzbTTiTcLjy3oyPD57o5/x4nfvxnE+jtn5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      "text/plain": [
       "<matplotlib.figure.Figure at 0x113d69ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Author: Jake VanderPlas\n",
    "# License: BSD\n",
    "#   The figure produced by this code is published in the textbook\n",
    "#   \"Statistics, Data Mining, and Machine Learning in Astronomy\" (2013)\n",
    "#   For more information, see http://astroML.github.com\n",
    "#   To report a bug or issue, use the following forum:\n",
    "#    https://groups.google.com/forum/#!forum/astroml-general\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy import stats\n",
    "from astroML.plotting import hist as astroMLhist\n",
    "\n",
    "#----------------------------------------------------------------------\n",
    "# This function adjusts matplotlib settings for a uniform feel in the textbook.\n",
    "# Note that with usetex=True, fonts are rendered with LaTeX.  This may\n",
    "# result in an error if LaTeX is not installed on your system.  In that case,\n",
    "# you can set usetex to False.\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=8, usetex=True)\n",
    "\n",
    "\n",
    "def plot_labeled_histogram(style, data, name,\n",
    "                           x, pdf_true, ax=None,\n",
    "                           hide_x=False,\n",
    "                           hide_y=False):\n",
    "    if ax is not None:\n",
    "        ax = plt.axes(ax)\n",
    "\n",
    "    counts, bins, patches = astroMLhist(data, bins=style, ax=ax,\n",
    "                                 color='k', histtype='step', normed=True)\n",
    "    ax.text(0.95, 0.93, '%s:\\n%i bins' % (name, len(counts)),\n",
    "            transform=ax.transAxes,\n",
    "            ha='right', va='top')\n",
    "\n",
    "    ax.fill(x, pdf_true, '-', color='#CCCCCC', zorder=0)\n",
    "\n",
    "    if hide_x:\n",
    "        ax.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    if hide_y:\n",
    "        ax.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "\n",
    "    ax.set_xlim(-5, 5)\n",
    "\n",
    "    return ax\n",
    "\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Set up distributions:\n",
    "Npts = 5000\n",
    "np.random.seed(0)\n",
    "x = np.linspace(-6, 6, 1000)\n",
    "\n",
    "# Gaussian distribution\n",
    "data_G = stats.norm(0, 1).rvs(Npts)\n",
    "pdf_G = stats.norm(0, 1).pdf(x)\n",
    "\n",
    "# Non-Gaussian distribution\n",
    "distributions = [stats.laplace(0, 0.4),\n",
    "                 stats.norm(-4.0, 0.2),\n",
    "                 stats.norm(4.0, 0.2)]\n",
    "\n",
    "weights = np.array([0.8, 0.1, 0.1])\n",
    "weights /= weights.sum()\n",
    "\n",
    "data_NG = np.hstack(d.rvs(int(w * Npts))\n",
    "                    for (d, w) in zip(distributions, weights))\n",
    "pdf_NG = sum(w * d.pdf(x)\n",
    "             for (d, w) in zip(distributions, weights))\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# Plot results\n",
    "fig = plt.figure(figsize=(10, 8))\n",
    "fig.subplots_adjust(hspace=0, left=0.07, right=0.95, wspace=0.05, bottom=0.15)\n",
    "ax = [fig.add_subplot(3, 2, i + 1) for i in range(6)]\n",
    "\n",
    "# first column: Gaussian distribution\n",
    "plot_labeled_histogram('scotts', data_G, 'Scott\\'s Rule', x, pdf_G,\n",
    "                       ax=ax[0], hide_x=True, hide_y=True)\n",
    "plot_labeled_histogram('freedman', data_G, 'Freed.-Diac.', x, pdf_G,\n",
    "                       ax=ax[2], hide_x=True, hide_y=True)\n",
    "plot_labeled_histogram('knuth', data_G, 'Knuth\\'s Rule', x, pdf_G,\n",
    "                       ax=ax[4], hide_x=False, hide_y=True)\n",
    "\n",
    "ax[0].set_title('Gaussian distribution')\n",
    "ax[2].set_ylabel('$p(x)$')\n",
    "ax[4].set_xlabel('$x$')\n",
    "\n",
    "# second column: non-gaussian distribution\n",
    "plot_labeled_histogram('scotts', data_NG, 'Scott\\'s Rule', x, pdf_NG,\n",
    "                       ax=ax[1], hide_x=True, hide_y=True)\n",
    "plot_labeled_histogram('freedman', data_NG, 'Freed.-Diac.', x, pdf_NG,\n",
    "                       ax=ax[3], hide_x=True, hide_y=True)\n",
    "plot_labeled_histogram('knuth', data_NG, 'Knuth\\'s Rule', x, pdf_NG,\n",
    "                       ax=ax[5], hide_x=False, hide_y=True)\n",
    "\n",
    "ax[1].set_title('non-Gaussian distribution')\n",
    "ax[5].set_xlabel('$x$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Blocks algorithm\n",
    "Though Knuth’s Bayesian method for choosing histogram bin width is \n",
    "an improvement over the Scott’s rule and the Freedman–Diaconis rule, \n",
    "it still has a distinct weakness: it assumes a **uniform** width for \n",
    "the optimal histogram bins.  \n",
    "\n",
    "This limitation can be lifted by maximizing a well-designed likelihood function over bins of varying width. This approach has been dubbed Bayesian blocks. The method was first developed in the field of \n",
    "time-domain analysis, but is readily applicable to histogram data as well. \n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9iIDhusTZs2fj+++ddwlOlcryFDhSKV2TSAgh1hA1SZTL5ZDJZEhLS8O+ffv4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B5T0LAsixUrVmD58uV4++23IZfLAQBr167FunXret1vf3vmDDZUTsJRWQlD5SQMlZMwVE7CuaKs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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113da63d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate sample: a mix of several Cauchy distributions\n",
    "np.random.seed(0)\n",
    "N = 10000\n",
    "mu_gamma_f = [(5, 1.0, 0.1),\n",
    "              (7, 0.5, 0.5),\n",
    "              (9, 0.1, 0.1),\n",
    "              (12, 0.5, 0.2),\n",
    "              (14, 1.0, 0.1)]\n",
    "true_pdf = lambda x: sum([f * stats.cauchy(mu, gamma).pdf(x)\n",
    "                          for (mu, gamma, f) in mu_gamma_f])\n",
    "x = np.concatenate([stats.cauchy(mu, gamma).rvs(int(f * N))\n",
    "                    for (mu, gamma, f) in mu_gamma_f])\n",
    "np.random.shuffle(x)\n",
    "x = x[x > -10]\n",
    "x = x[x < 30]\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# plot the results\n",
    "fig = plt.figure(figsize=(10, 8))\n",
    "fig.subplots_adjust(bottom=0.08, top=0.95, right=0.95, hspace=0.1)\n",
    "N_values = (100, 1000, 10000)\n",
    "subplots = (311, 312, 313)\n",
    "\n",
    "for N, subplot in zip(N_values, subplots):\n",
    "    ax = fig.add_subplot(subplot)\n",
    "    xN = x[:N]\n",
    "    t = np.linspace(-10, 30, 1000)\n",
    "\n",
    "    # plot the results\n",
    "    ax.errorbar(xN, -0.01 + 0*xN, xerr=0.0, yerr=0.005, ecolor='black', lw=1)\n",
    "    ax.plot(t, true_pdf(t), '-', color='black', lw=1,\n",
    "            label=\"Generating Distribution\")\n",
    "    hist(xN, bins='knuth', ax=ax, normed=True,\n",
    "         histtype='stepfilled', alpha=0.3,\n",
    "         label='Knuth Histogram')\n",
    "    hist(xN, bins='blocks', ax=ax, normed=True,\n",
    "         histtype='step', color='red', alpha=0.99,\n",
    "         label=\"Bayesian Blocks\")\n",
    "  \n",
    "    # label the plot\n",
    "    ax.text(0.02, 0.95, \"%i points\" % N, ha='left', va='top',\n",
    "            transform=ax.transAxes)\n",
    "    ax.set_ylabel('$p(x)$')\n",
    "    ax.legend(loc='upper right', prop=dict(size=8))\n",
    "\n",
    "    if subplot == 212:\n",
    "        ax.set_xlabel('$x$')\n",
    "\n",
    "    ax.set_xlim(0, 20)\n",
    "    ax.set_ylim(-0.02, 0.4001)\n",
    "\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Blocks algorithm\n",
    "\n",
    "Note that for largest sample (N=10,000), both methods give similar results (e.g. three\n",
    "components are found in the data). However, for the middle dataset (N=1,000), the\n",
    "fixed bin width method misses the middle peak - this could be your missed Nobel prize!"
   ]
  }
 ],
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