Can posterior become tractable if we know p(x)?

Question

In the VAE framework where x is an input data (a vector) and z is a vector of continuous latent variables,

the posterior p(z|x) is intractable because p(x) is intractable, so we approximate it using an approximate posterior q(z|x).

But, what happens if we know p(x)?

1. e.g., if p(x) = N(mu, covariance mtx) is known, does this make the posterior tractable?

2. If yes, does this also apply even if the covariance mtx has off-diagonal terms (i.e., elements in the vector x are correlated) - or does it not matter?

Answer 1

Bayes theorem is

$$ p(z|x) = \frac{p(x|z)\, p(z)}{p(x)} $$

so if you know all the components, it's just a multiplication and division. If you don't, you usually need to take the integral

$$ p(x) = \int \, p(x|z)\, p(z) \,dz $$

and the integral is often intractable. Correlated elements have nothing to do with that.