What makes the posterior intractable?

Question

In the setting of Variational AutoEncoders, i.e. when we want to find the posterior distribution

over the data generating, latent variable z, given some observations x, what exactly (which part of the equation) makes this posterior distribution intractable and why?

Cheers

Answer 1

It's usually the denominator $p(x)$ (the "evidence") which is intractable. You could attempt to compute it by marginalizing over the latent variable $p(x) = \int p(x|z)p(z)dz$. However, you would need to evaluate all possible values of $z$ which would require exponential time. (That's why in maximium-likelihood estimation you have no intractability problem because you can treat the evidence as a constant.)