Bayes' rule in application

Question

I am reading the material about variational autoencoders of stanford's cs231n course. In the equations below, I am not sure how they got from the first step to the second using Bayes' rule. From what I know about Bayes' rule, it gives you a formula to compute conditional probabilities. However, this does not look like the same. Can someone give some clarification?

Answer 1

See if this helps for the second step? \begin{align*} p_\theta(X|Z)p_\theta(Z) &= p_\theta(Z|X)p_\theta(X)\\ p_\theta(X) &= \frac{p_\theta(X|Z)p_\theta(Z)}{p_\theta(Z|X)}\\ log(p_\theta(X)) &= log\left(\frac{p_\theta(X|Z)p_\theta(Z)}{p_\theta(Z|X)}\right)\\ \mathbb{E}_Z\left[log(p_\theta(X))\right] &= \mathbb{E}_Z\left[log\left(\frac{p_\theta(X|Z)p_\theta(Z)}{p_\theta(Z|X)}\right)\right] \end{align*}