Note on the equivalence of hierarchical variational models and auxiliary deep generative models

This is a short note to mathematically compare two recently published methods for constructing flexible, but tractable families of variational posteriors. The first method, called "hierarchical variational models" enriches the inference model with an extra variable, while the other, called "auxiliary deep generative models", enriches the generative model instead. We conclude that the two methods are mathematically equivalent.