  The recently proposed Thermodynamic Variational Objective (TVO) leverages
thermodynamic integration to provide a family of variational inference
objectives, which both tighten and generalize the ubiquitous Evidence Lower
Bound (ELBO). However, the tightness of TVO bounds was not previously known, an
expensive grid search was used to choose a "schedule" of intermediate
distributions, and model learning suffered with ostensibly tighter bounds. In
this work, we propose an exponential family interpretation of the geometric
mixture curve underlying the TVO and various path sampling methods, which
allows us to characterize the gap in TVO likelihood bounds as a sum of KL
divergences. We propose to choose intermediate distributions using equal
spacing in the moment parameters of our exponential family, which matches grid
search performance and allows the schedule to adaptively update over the course
of training. Finally, we derive a doubly reparameterized gradient estimator
which improves model learning and allows the TVO to benefit from more refined
bounds. To further contextualize our contributions, we provide a unified
framework for understanding thermodynamic integration and the TVO using Taylor
series remainders.
