Learning Model Reparametrizations: Implicit Variational Inference by Fitting MCMC distributions

Consider a probabilistic model with joint distribution p(x, z) where x are data and z are latent variables and/or random parameters. Suppose that exact inference in p(x, z) is intractable which means that the posterior distribution p(z x) p(x, z) p(x, z)dz, is difficult to compute due to the normalizing constant p(x) p(x, z)dz that represents the probability of the data and it is known as evidence or marginal likelihood. The marginal likelihood is essential for estimation of any extra parameters in p(x) or for model comparison. Approximate inference algorithms target to approximate p(z x) and/or p(x). Two general frameworks, that we briefly review next, are based on Markov chain Monte Carlo (MCMC) [33, 2] and variational inference (VI) [17, 40].