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Cover and Thomas ("Elements of information theory") define relative entropy as:

$$H(Y|X)=-E[ \log p(Y|X)]$$

where $\log p(Y|X)$ is therefore a random variable defined, at least as I understand it, by composition with the function $p(y|x)$: $\Omega \rightarrow (X,Y) \rightarrow p(Y|X)$.

My question is if there is a probabilistic way to interpret the symbol $Y|X$ in this expression in a different way. I know for example that a meaning can be attached to the symbol $E[Y|X]$ as a random variable, but I do not know just for $Y|X$...

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    $\begingroup$ Much to the disgust of many readers of this forum, there is no random variable (or any other entity) that is denoted $Y \mid X$. There are things like the conditional distribution of $Y$ given $X=x$ but that is denoted by something like $p_{Y\mid X}(y \mid X = x)$ $\endgroup$ – Dilip Sarwate Aug 23 '17 at 19:20
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In probability and statistics $Y \mid X$ is a symbol used for conditional relation between two random variables, $Y$ given $X$, e.g. conditional probability $\Pr(Y\mid X)$, conditional distribution $f_Y(y \mid X=x)$, conditional expectation $E(Y \mid X)$, or conditional entropy $H(Y \mid X)$ as in your example.

For example, $\Pr(Y = y \mid X = x )$ is a probability that $Y=y$ when $X=x$, while $E(Y | X=x)$ is the expected value for $Y$ in the cases where $X=x$, etc.

Formally, the definition of conditional probability is

$$ \Pr(Y = y \mid X = x) = \frac{\Pr( Y=y\ \cap X=x)}{\Pr(X=x)} $$

So it's an operation on the "slice" of the joint distribution of $Y$ and $X$ for fixed $X=x$. For more details and multiple worked examples check the linked Wikipedia articles.

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    $\begingroup$ The classical definition of conditional probability (which is this one) comes with the asterisk that it only works when $Pr(X = x) > 0$. $\endgroup$ – Kodiologist Aug 23 '17 at 19:54

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