change notation A, B, C to X, Z, Y so it is consistent with the figure (

#209)

* change A, B, C to X, Z, Y

* change notation so consistent with figure

representation/directed/index.md

@@ -71,13 +71,13 @@ Let us use the notation $$I(p)$$ to denote the set of all independencies that ho
 
 It turns out that a Bayesian network $$p$$ very elegantly describes many independencies in $$I(p)$$; these independencies can be recovered from the graph by looking at three types of structures.
 
-For simplicity, let's start by looking at a Bayes net $$G$$ with three nodes: $$A$$, $$B$$, and $$C$$. In this case, $$G$$ essentially has only three possible structures, each of which leads to different independence assumptions. The interested reader can easily prove these results using a bit of algebra.
+For simplicity, let's start by looking at a Bayes net $$G$$ with three nodes: $$X$$, $$Y$$, and $$Z$$. In this case, $$G$$ essentially has only three possible structures, each of which leads to different independence assumptions. The interested reader can easily prove these results using a bit of algebra.
 
-- {% include marginfigure.html id="bn" url="assets/img/3node-bayesnets.png" description="Bayesian networks over three variables, encoding different types of dependencies: cascade (a,b), common parent (c), and v-structure (d)." %}*Common parent.* If $$G$$ is of the form $$A \leftarrow B \rightarrow C$$, and $$B$$ is observed, then $$A \perp C \mid B$$. However, if $$B$$ is unobserved, then $$A \not\perp C$$. Intuitively this stems from the fact that $$B$$ contains all the information that determines the outcomes of $$A$$ and $$C$$; once it is observed, there is nothing else that affects these variables' outcomes.
-- *Cascade*: If $$G$$ equals $$A \rightarrow B \rightarrow C$$, and $$B$$ is again observed, then, again $$A \perp C \mid B$$. However, if $$B$$ is unobserved, then $$A \not\perp C$$. Here, the intuition is again that $$B$$ holds all the information that determines the outcome of $$C$$; thus, it does not matter what value $$A$$ takes.
-- *V-structure* (also known as *explaining away*): If $$G$$ is $$A \rightarrow C \leftarrow B$$, then knowing $$C$$ couples $$A$$ and $$B$$. In other words, $$A \perp B$$ if $$C$$ is unobserved, but $$A \not\perp B \mid C$$ if $$C$$ is observed.
+- {% include marginfigure.html id="bn" url="assets/img/3node-bayesnets.png" description="Bayesian networks over three variables, encoding different types of dependencies: cascade (a,b), common parent (c), and v-structure (d)." %}*Common parent.* If $$G$$ is of the form $$X \leftarrow Z \rightarrow Y$$, and $$Z$$ is observed, then $$X \perp Y \mid Z$$. However, if $$Z$$ is unobserved, then $$X \not\perp Y$$. Intuitively this stems from the fact that $$Z$$ contains all the information that determines the outcomes of $$X$$ and $$Y$$; once it is observed, there is nothing else that affects these variables' outcomes.
+- *Cascade*: If $$G$$ equals $$X \rightarrow Z \rightarrow Y$$, and $$Z$$ is again observed, then, again $$X \perp Y \mid Z$$. However, if $$Z$$ is unobserved, then $$X \not\perp Y$$. Here, the intuition is again that $$Z$$ holds all the information that determines the outcome of $$Y$$; thus, it does not matter what value $$X$$ takes.
+- *V-structure* (also known as *explaining away*): If $$G$$ is $$X \rightarrow Z \leftarrow Y$$, then knowing $$Z$$ couples $$X$$ and $$Y$$. In other words, $$X \perp Y$$ if $$Z$$ is unobserved, but $$X \not\perp Y \mid Z$$ if $$Z$$ is observed.
 
-The latter case requires additional explanation. Suppose that $$C$$ is a Boolean variable that indicates whether our lawn is wet one morning; $$A$$ and $$B$$ are two explanations for it being wet: either it rained (indicated by $$A$$), or the sprinkler turned on (indicated by $$B$$). If we know that the grass is wet ($$C$$ is true) and the sprinkler didn't go on ($$B$$ is false), then the probability that $$A$$ is true must be one, because that is the only other possible explanation. Hence, $$A$$ and $$B$$ are not independent given $$C$$.
+The latter case requires additional explanation. Suppose that $$Z$$ is a Boolean variable that indicates whether our lawn is wet one morning; $$X$$ and $$Y$$ are two explanations for it being wet: either it rained (indicated by $$X$$), or the sprinkler turned on (indicated by $$Y$$). If we know that the grass is wet ($$Z$$ is true) and the sprinkler didn't go on ($$Y$$ is false), then the probability that $$X$$ is true must be one, because that is the only other possible explanation. Hence, $$X$$ and $$Y$$ are not independent given $$Z$$.
 
 These structures clearly describe the independencies encoded by a three-variable Bayesian net. We can extend them to general networks by applying them recursively over any larger graph. This leads to a notion called $$d$$-separation (where $$d$$ stands for directed).