Expectation of ELBO in Variational Autoencoder

I am working with VAEs. My input is x, which is a product of two variables $x_1$ and $x_2$.

The objective (ELBO) of VAE in terms of x is:

$E_{z\sim Q}[\log P(x|z)] - \mathcal{D}[Q(z|x)||P(z)]$.

I want to compute the expected value of ELBO w.r.t $x_1$, i.e.

$E_{x_1}[E_{z\sim Q}[\log P(x|z)] - \mathcal{D}[Q(z|x)||P(z)]]$.

Given: I know the quantity: $E_{x_1}[\log P(x|z)]$.

My questions are:

  1. Can I move the first expectation, I,e, $E_{x_1}$ inside expectation w.r.t $z$, i.e. $E_{z\sim Q}$? Given $z$ is sampled from $Q(z|x)$.

  2. Is the expectation $E_{x_1}[\mathcal{D}[Q(z|x)||P(z)]]$ independent of $x_1$? Given x is the conditional variable for posterior?

Topic vae bayesian autoencoder loss-function

Category Data Science

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