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If we take the vanilla variational auto-encoder (VAE), we $p(z)$ is a Gaussian distribution with zero mean and unit variance and we approximate $p(z|x) \approx q(z|x)$ to be a Gaussian distribution as well, for each latent variable $z$.

But what if $z$ is a discrete variable? What kind of distributions can be used to model discrete latent variables? For example, what kind of distribution can be used to model $p(z)$ and $p(z|x) \approx q(z|x)$?