  To make decisions based on a model fit with auto-encoding variational Bayes
(AEVB), practitioners often let the variational distribution serve as a
surrogate for the posterior distribution. This approach yields biased estimates
of the expected risk, and therefore leads to poor decisions for two reasons.
First, the model fit with AEVB may not equal the underlying data distribution.
Second, the variational distribution may not equal the posterior distribution
under the fitted model. We explore how fitting the variational distribution
based on several objective functions other than the ELBO, while continuing to
fit the generative model based on the ELBO, affects the quality of downstream
decisions. For the probabilistic principal component analysis model, we
investigate how importance sampling error, as well as the bias of the model
parameter estimates, varies across several approximate posteriors when used as
proposal distributions. Our theoretical results suggest that a posterior
approximation distinct from the variational distribution should be used for
making decisions. Motivated by these theoretical results, we propose learning
several approximate proposals for the best model and combining them using
multiple importance sampling for decision-making. In addition to toy examples,
we present a full-fledged case study of single-cell RNA sequencing. In this
challenging instance of multiple hypothesis testing, our proposed approach
surpasses the current state of the art.
