  In this work, we advocate for the importance of singular learning theory
(SLT) as it pertains to the theory and practice of variational inference in
Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of
the confusion surrounding discrepancies between downstream predictive
performance measured via e.g., the test log predictive density, and the
variational objective. Next, we use the SLT-corrected asymptotic form for
singular posterior distributions to inform the design of the variational family
itself. Specifically, we build upon the idealized variational family introduced
in \citet{bhattacharya_evidence_2020} which is theoretically appealing but
practically intractable. Our proposal takes shape as a normalizing flow where
the base distribution is a carefully-initialized generalized gamma. We conduct
experiments comparing this to the canonical Gaussian base distribution and show
improvements in terms of variational free energy and variational generalization
error.
