What makes the posterior intractable?
It's usually the denominator $p(x)$ (the &quot;evidence&quot;) 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.)