fixed some minor typos in anchors

manuscript/05.8.1-agnostic-Anchors.Rmd

@@ -64,13 +64,13 @@ Wherein:
 
 - $x$ represents the instance being explained (e.g. one row in a tabular data set).
 - $A$ is a set of predicates, i.e., the resulting rule or anchor, such that $A(x)=1$ when all feature predicates defined by $A$ correspond to $x$’s feature values.
-- $f$ denotes the classification model to be explained (e.g. an artificial neural network model). It can be queried to predict a label for $x$ and its perturbations.
-- $D_x (\cdot|A)$ indicates the distribution of neighbors of $x$, matching $A$.
+- $\hat{f}$ denotes the classification model to be explained (e.g. an artificial neural network model). It can be queried to predict a label for $x$ and its perturbations.
+- $\mathcal{D}_x (\cdot|A)$ indicates the distribution of neighbors of $x$, matching $A$.
 - $0 \leq \tau \leq 1$ specifies a precision threshold. Only rules that achieve a local fidelity of at least $\tau$ are considered a valid result.
 
 The formal description may be intimidating and can be put in words:
 
-> Given an instance $x$ to be explained, a rule or an anchor $A$ is to be found, such that it applies to $x$, while the same class as for $x$ gets predicted for a fraction of at least $\tau$ of $x$’s neighbors where the same $A$ is applicable. A rule’s precision results from evaluating neighbors or perturbations (following $D_x (z|A)$) using the provided machine learning model (denoted by the indicator function $1_{\hat{f}(x) = \hat{f}(z)}$).
+> Given an instance $x$ to be explained, a rule or an anchor $A$ is to be found, such that it applies to $x$, while the same class as for $x$ gets predicted for a fraction of at least $\tau$ of $x$’s neighbors where the same $A$ is applicable. A rule’s precision results from evaluating neighbors or perturbations (following $\mathcal{D}_x (z|A)$) using the provided machine learning model (denoted by the indicator function $1_{\hat{f}(x) = \hat{f}(z)}$).
 
 
 ### Finding Anchors
@@ -82,7 +82,7 @@ $$P(prec(A)\geq\tau)\geq{}1-\delta\quad\textrm{with}\quad{}prec(A)=\mathbb{E}_{\
 
 The previous two definitions are combined and extended by the notion of coverage. Its rationale consists of finding rules that apply to a preferably large part of the model’s input space. Coverage is formally defined as an anchor's probability of applying to its neighbors, i.e. its perturbation space: 
 
-$$cov(A)=\mathbb{E}_{\mathcal{D}_{(z)}}[A(z)]$$ 
+$$cov(A)=\mathbb{E}_{\mathcal{D}_x(z)}[A(z)]$$ 
 
 Including this element leads to anchor's final definition taking into account the maximization of coverage: