  The variational auto-encoder (VAE) is a deep latent variable model that has
two neural networks in an autoencoder-like architecture; one of them
parameterizes the model's likelihood. Fitting its parameters via maximum
likelihood (ML) is challenging since the computation of the marginal likelihood
involves an intractable integral over the latent space; thus the VAE is trained
instead by maximizing a variational lower bound. Here, we develop a ML training
scheme for VAEs by introducing unbiased estimators of the log-likelihood
gradient. We obtain the estimators by augmenting the latent space with a set of
importance samples, similarly to the importance weighted auto-encoder (IWAE),
and then constructing a Markov chain Monte Carlo coupling procedure on this
augmented space. We provide the conditions under which the estimators can be
computed in finite time and with finite variance. We show experimentally that
VAEs fitted with unbiased estimators exhibit better predictive performance.
