Notation: What does the tilde below of the expectation mean?

Question

I am reading about variational auto encoders, and there is the below loss function:

$$l_i(\Theta,\phi) = - {\mathbb{E}}_{z\sim q} \left[\log p_\phi(x_i|z)\right] + KL(q_{\phi}(z_i|x)||p(z))$$

What does the notation $z\sim q$ under $\mathbb{E}$ mean? I just have seen notations like $E(X)$ or $ \langle X\rangle $ for expected value, $\mathbb{E}$.

What does this notation generally mean when using $\mathbb{E}_{x\sim y}$ for some $x$ and some $y$?

Answer 1

$z\sim q$ means that RV $Z$ is distributed with respect to $q$ function, i.e. $q(z)$, where $q(z)$ is a valid PDF/PMF. So, the expectation can be unfold as (assuming $z$ being continuous) $$\mathbb{E}_{z\sim q}[\log_{\phi}(x_i|z)]=\int_{-\infty}^\infty \log_\phi (x_i|z) q(z) dz$$