  The framework of variational autoencoders allows us to efficiently learn deep
latent-variable models, such that the model's marginal distribution over
observed variables fits the data. Often, we're interested in going a step
further, and want to approximate the true joint distribution over observed and
latent variables, including the true prior and posterior distributions over
latent variables. This is known to be generally impossible due to
unidentifiability of the model. We address this issue by showing that for a
broad family of deep latent-variable models, identification of the true joint
distribution over observed and latent variables is actually possible up to very
simple transformations, thus achieving a principled and powerful form of
disentanglement. Our result requires a factorized prior distribution over the
latent variables that is conditioned on an additionally observed variable, such
as a class label or almost any other observation. We build on recent
developments in nonlinear ICA, which we extend to the case with noisy,
undercomplete or discrete observations, integrated in a maximum likelihood
framework. The result also trivially contains identifiable flow-based
generative models as a special case.
