# What makes the posterior intractable?

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

$p_{\mathbf{\theta}}(\mathbf{z}|\mathbf{x})&space;=&space;p_{\mathbf{\theta}}(\mathbf{x}|\mathbf{z})p_{\mathbf{\theta}}(\mathbf{z})/p_{\mathbf{\theta}}(\mathbf{x})$

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

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.)
• $z$ will usually be high-dimensional and to accurately sample from the large" volume of the space" you need exponentially many more samples for each additional dimension. ("curse of dimensionality"). even if there is a closed-form solution to the integral the computation can still be exponential see e.g. the example here