  Variational autoencoders (VAEs) are a popular framework for modeling complex
data distributions; they can be efficiently trained via variational inference
by maximizing the evidence lower bound (ELBO), at the expense of a gap to the
exact (log-)marginal likelihood. While VAEs are commonly used for
representation learning, it is unclear why ELBO maximization would yield useful
representations, since unregularized maximum likelihood estimation cannot
invert the data-generating process. Yet, VAEs often succeed at this task. We
seek to elucidate this apparent paradox by studying nonlinear VAEs in the limit
of near-deterministic decoders. We first prove that, in this regime, the
optimal encoder approximately inverts the decoder -- a commonly used but
unproven conjecture -- which we refer to as {\em self-consistency}. Leveraging
self-consistency, we show that the ELBO converges to a regularized
log-likelihood. This allows VAEs to perform what has recently been termed
independent mechanism analysis (IMA): it adds an inductive bias towards
decoders with column-orthogonal Jacobians, which helps recovering the true
latent factors. The gap between ELBO and log-likelihood is therefore welcome,
since it bears unanticipated benefits for nonlinear representation learning. In
experiments on synthetic and image data, we show that VAEs uncover the true
latent factors when the data generating process satisfies the IMA assumption.
