  Estimating a distribution given access to its unnormalized density is pivotal
in Bayesian inference, where the posterior is generally known only up to an
unknown normalizing constant. Variational inference and Markov chain Monte
Carlo methods are the predominant tools for this task; however, both methods
are often challenging to apply reliably, particularly when the posterior has
complex geometry. Here, we introduce Soft Contrastive Variational Inference
(SoftCVI), which allows a family of variational objectives to be derived
through a contrastive estimation framework. These objectives have zero variance
gradient when the variational approximation is exact, without the need for
specialized gradient estimators. The approach involves parameterizing a
classifier in terms of the variational distribution, which allows the inference
task to be reframed as a contrastive estimation problem, aiming to identify a
single true posterior sample among a set of samples. Despite this framing, we
do not require positive or negative samples, but rather learn by sampling the
variational distribution and computing ground truth soft classification labels
from the unnormalized posterior itself. We empirically investigate the
performance on a variety of Bayesian inference tasks, using both using both
simple (e.g. normal) and expressive (normalizing flow) variational
distributions. We find that SoftCVI objectives often outperform other commonly
used variational objectives.
