# What kind of distributions can be used to model discrete latent variables?

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 $$p(z|x) \approx q(z|x)$$?