  Markov chain Monte Carlo methods have become popular in statistics as
versatile techniques to sample from complicated probability distributions. In
this work, we propose a method to parameterize and train transition kernels of
Markov chains to achieve efficient sampling and good mixing. This training
procedure minimizes the total variation distance between the stationary
distribution of the chain and the empirical distribution of the data. Our
approach leverages involutive Metropolis-Hastings kernels constructed from
reversible neural networks that ensure detailed balance by construction. We
find that reversibility also implies $C_2$-equivariance of the discriminator
function which can be used to restrict its function space.
