A Tutorial on VAEs: From Bayes' Rule to Lossless Compression

The Variational Auto-Encoder (VAE) belongs to a class of models, which we will refer to as deep maximum likelihood models, that uses a deep neural network to learn a maximum likelihood model for some input data. They are perhaps the most simple and efficient deep maximum likelihood model available, and have thus gained popularity in representation learning and generative image modeling. Unfortunately, in my opinion, in some circles the term "VAE" has become somewhat synonymous with "an auto-encoder with stochastic regularization that generates useful or beautiful samples", which has led to various misconceptions about VAEs. In this tutorial, we will return to the probabilistic and information theoretic roots of VAEs, clarify common misconceptions about VAEs, and look at a toy example on 2D data that will illustrate the capabilities and limitations of VAEs. In Section 2, we will give an overview of what is a maximum likelihood model and what a VAE looks like.