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 $Y|X$...