  We introduce a new algorithm for approximate inference that combines
reparametrization, Markov chain Monte Carlo and variational methods. We
construct a very flexible implicit variational distribution synthesized by an
arbitrary Markov chain Monte Carlo operation and a deterministic transformation
that can be optimized using the reparametrization trick. Unlike current methods
for implicit variational inference, our method avoids the computation of log
density ratios and therefore it is easily applicable to arbitrary continuous
and differentiable models. We demonstrate the proposed algorithm for fitting
banana-shaped distributions and for training variational autoencoders.
