Meaning of expectation with respect to a function?

This might be a trivial question, but I've come across a paper where some expectation is said to be taken with respect to some pdf. See example:

How am I to interpret this, and is there some notation to indicate that it should be taken with respect to that specific pdf? I assume that "with respect to" means that it's the function to use within the product inside the summation, but I'm not sure.

For example I interpret $${E}[\log q(z)]$$ as $${E}[\log X]$$ where $$X$$ is a random variable distributed according to q, and then make use of the law of the unconcious statitician, where $$g(X)$$ is $$log(X)$$, and similarly with $${E}[\log p(z|x)]$$, given that x are observed variables. So given my assumption above, the first case should be with respect to $$q(z)$$, but in the second case it should be with respect to $$p(z | X = x)$$, so my assumption seems to fail.

Bonus question: Can they break $$\log p(x)$$ out of the expectation because it is treated as a constant?

This isn’t quite right, $$\mathbb{E}[\log p(X)]$$ is the expected value of the log of the pdf of the random variable. So if you have a random variable, work out what the density of that value for $$p$$ is and log it, that’s the expectation you’re considering. Here it explicitly says the expectation is with respect to $$q$$ but often you will see it in a subscript after the $$\mathbb{E}$$ when it’s not clear ($$\mathbb{E}_q$$). And by “respect to” we’re saying that’s how the random variable is distributed and so when we integrate we use $$q$$.
• So, another super trivial question, what is the difference of saying $q(z)$ and that Z is distributed according to q(z) i.e (Z ~ some pdf that is q(z)), because I'm assuming that the lowercase z is not set? Furthermore, what you're saying is that $E[\log q(z)]=\int \log q(z)*q(z) \mathrm{dz}$ and $E[\log p(z|x)]=\int \log p(z|x)*q(z) \mathrm{dz}$ – NotoriousFunk Aug 10 at 8:55