An Equivalence between Bayesian Priors and Penalties in Variational Inference

TL;DR

This paper establishes a formal connection between Bayesian priors and regularization penalties in variational inference, providing a systematic method to identify the prior from a given penalty function.

Contribution

It characterizes which regularizers correspond to Bayesian priors in variational inference and offers a way to compute the prior from a specified penalty.

Findings

Characterization of regularizers that correspond to Bayesian priors.

Method to compute the prior from a given penalty function.

Guidelines for designing penalties that preserve Bayesian properties.

Abstract

In machine learning, it is common to optimize the parameters of a probabilistic model, modulated by an ad hoc regularization term that penalizes some values of the parameters. Regularization terms appear naturally in Variational Inference, a tractable way to approximate Bayesian posteriors: the loss to optimize contains a Kullback--Leibler divergence term between the approximate posterior and a Bayesian prior. We fully characterize the regularizers that can arise according to this procedure, and provide a systematic way to compute the prior corresponding to a given penalty. Such a characterization can be used to discover constraints over the penalty function, so that the overall procedure remains Bayesian.