  Recent work in unsupervised representation learning has focused on learning
deep directed latent-variable models. Fitting these models by maximizing the
marginal likelihood or evidence is typically intractable, thus a common
approximation is to maximize the evidence lower bound (ELBO) instead. However,
maximum likelihood training (whether exact or approximate) does not necessarily
result in a good latent representation, as we demonstrate both theoretically
and empirically. In particular, we derive variational lower and upper bounds on
the mutual information between the input and the latent variable, and use these
bounds to derive a rate-distortion curve that characterizes the tradeoff
between compression and reconstruction accuracy. Using this framework, we
demonstrate that there is a family of models with identical ELBO, but different
quantitative and qualitative characteristics. Our framework also suggests a
simple new method to ensure that latent variable models with powerful
stochastic decoders do not ignore their latent code.
