  The representation of the approximate posterior is a critical aspect of
effective variational autoencoders (VAEs). Poor choices for the approximate
posterior have a detrimental impact on the generative performance of VAEs due
to the mismatch with the true posterior. We extend the class of posterior
models that may be learned by using undirected graphical models. We develop an
efficient method to train undirected approximate posteriors by showing that the
gradient of the training objective with respect to the parameters of the
undirected posterior can be computed by backpropagation through Markov chain
Monte Carlo updates. We apply these gradient estimators for training discrete
VAEs with Boltzmann machines as approximate posteriors and demonstrate that
undirected models outperform previous results obtained using directed graphical
models. Our implementation is available at https://github.com/QuadrantAI/dvaess .
