the problem of approximate inference in deep learning

Suppose we have a latent variable model

$$p(x,z)=p(x|z)p(z),$$

where $z$ are latent, and $x$ is the data. A core problem in Bayesian statistics is to compute the conditional distribution $p(z|x)$, i.e., to solve the problem: what are the hidden parameters given the data?

Writing,

$$p(z|x)=\frac{p(x,z)}{p(x)}=p(x,z){\left(\int p(x,z)\text{d}z\right)}^{-1},$$

we start to see the trouble. The integral on the rhs is usually intractable, since it involves integrating across all latent variables. So we typically approximate $p(z|x)$ in some way. This is the problem of approximate inference. Noet that $p(x)$ is usually called the “evidence”.

Common solutions to the problem include MCMC and variational inference.