If you look at this paper Variational Inference: A Review for Statisticians (written by renowned statisticians), they use the notation, like $\mathbb{E}\left[ p(x \mid z) \right]$, even though $p(x \mid z)$ is supposed to be a likelihood, so $x$ is given, thus $p(x \mid z)$ should be interpreted as a function that is evaluated at $x$ and varies as a function of $z$.
Recently, I have asked multiple questions because of this notation that I don't understand why it is correct. In particular,
- When does it make sense to use $p(X)$ where $p$ is a pdf and $X$ a random variable, and $p(X)$ is the composition of $p$ and $X$?
- Can expectations be defined for something other than random variables?
- Is the codomain of random variable $X$ always equal to the domain of the associated p.d.f. (or p.m.f.)?
- Why is the exact relationship between a Gaussian p.d.f. and its associated probability measure and random variable?
- What is the definition of a Gaussian random variable?
- Can we really compose random variables and probability density functions?
So, why does this notation $\mathbb{E}\left[ p(x \mid z) \right]$, when $x$ is given, make sense? It cannot be interpreted as the composition of $p$ and the random variables $x$ and $z$, because $x$ is given there, so we are evaluating $p$ at $x$, but then we are also using $z$ there (compositing it with $p$?), i.e. we are taking the expectation of a likelihood, but we need to take expectations of random variables!!
So, please, can someone explain to me which assumptions are being made in order for that notation in that paper to make sense MATHEMATICALLY? Why exactly can we use that notation in the paper? And what is the equivalent rigorous notation of that notation?
Moreover, note that this notation is used ALL THE TIME in machine learning and statistics in all the papers I have read and I have read many.
