  Normalizing flows, diffusion normalizing flows and variational autoencoders
are powerful generative models. This chapter provides a unified framework to
handle these approaches via Markov chains. We consider stochastic normalizing
flows as a pair of Markov chains fulfilling some properties and show how many
state-of-the-art models for data generation fit into this framework. Indeed
numerical simulations show that including stochastic layers improves the
expressivity of the network and allows for generating multimodal distributions
from unimodal ones. The Markov chains point of view enables us to couple both
deterministic layers as invertible neural networks and stochastic layers as
Metropolis-Hasting layers, Langevin layers, variational autoencoders and
diffusion normalizing flows in a mathematically sound way. Our framework
establishes a useful mathematical tool to combine the various approaches.

Machine Learning;Probability