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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?

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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.

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