I am reading about variational auto encoders, and there is the below loss function:
$$l_i(Theta,phi) = – {mathbb{E}}_{zsim q} left[log p_phi(x_i|z)right] + KL(q_{phi}(z_i|x)||p(z))$$
What does the notation $zsim q$ under $mathbb{E}$ mean? I just have
seen notations like $E(X)$ or $ langle Xrangle $ for expected value, $mathbb{E}$.
What does this notation generally mean when using $mathbb{E}_{xsim y}$ for some $x$ and some $y$?
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Best Answer
$zsim q$ 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) $$mathbb{E}_{zsim q}[log_{phi}(x_i|z)]=int_{-infty}^infty log_phi (x_i|z) q(z) dz$$
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