In VAE, the goal is to maximize $\log p(x)$, where $x$ is the data and $p(\cdot)$ is a parameterized distribution.
For any distribution $q(z)$, the following identity holds, $$ \log p(x)=D[q(z)||p(z|x)]+E_{z\sim q(z)}[\log p(x|z)]-D[q(z)||p(z)]. $$
The usual approach is to learn $q_\phi(z)=p(z|x)$ so that $D[q(z)||p(z|x)]\rightarrow0$ and maximize the ELBO $E_{z\sim q(z)}[\log p(x|z)]-D[q(z)||p(z)]$.
My question is that if all we want is maximize $\log p(x)$, why not maximize $D[q(z)||p(z|x)]$ instead?
