  We prove that the evidence lower bound (ELBO) employed by variational
auto-encoders (VAEs) admits non-trivial solutions having constant posterior
variances under certain mild conditions, removing the need to learn variances
in the encoder. The proof follows from an unexpected journey through an array
of topics: the closed form optimal decoder for Gaussian VAEs, a proof that the
decoder is always smooth, a proof that the ELBO at its stationary points is
equal to the exact log evidence, and the posterior variance is merely part of a
stochastic estimator of the decoder Hessian. The penalty incurred from using a
constant posterior variance is small under mild conditions, and otherwise
discourages large variations in the decoder Hessian. From here we derive a
simplified formulation of the ELBO as an expectation over a batch, which we
call the Batch Information Lower Bound (BILBO). Despite the use of Gaussians,
our analysis is broadly applicable -- it extends to any likelihood function
that induces a Riemannian metric. Regarding learned likelihoods, we show that
the ELBO is optimal in the limit as the likelihood variances approach zero,
where it is equivalent to the change of variables formulation employed in
normalizing flow networks. Standard optimization procedures are unstable in
this limit, so we propose a bounded Gaussian likelihood that is invariant to
the scale of the data using a measure of the aggregate information in a batch,
which we call Bounded Aggregate Information Sampling (BAGGINS). Combining the
two formulations, we construct VAE networks with only half the outputs of
ordinary VAEs (no learned variances), yielding improved ELBO scores and scale
invariance in experiments. As we perform our analyses irrespective of any
particular network architecture, our reformulations may apply to any VAE
implementation.
