Variational Autoencoder: Theory

In simple terms, a variational autoencoder is a probabilistic version of autoencoders.

Why?

Because we want to be able to sample from the latent vector ($z$) space to generate new data, which is not possible with vanilla autoencoders.

Each latent variable $z$ that is generated from the input will now represent a probability distribution (or what we call the posterior distribution denoted as $p(z|x)$).

All we need to do is find the posterior $p(z|x)$ or solve the inference problem.

In fact, the encoder will try to approximate the posterior by computing another distribution $q(z|x)$, known as the variational posterior.

Note that a probability distribution is fully characterized by its parameters. In the case of the Gaussian, these are the mean $\mu$ ...