Optimal Belief Approximation

TL;DR

This paper discusses the optimal way to approximate Bayesian beliefs using a loss function derived from an old proof, clarifying the correct order of arguments in the Kullback-Leibler divergence to minimize information loss.

Contribution

It provides an elementary derivation of the unique loss function for belief approximation, clarifying the correct argument order in the KL divergence.

Findings

The derived loss function is the normalized Kullback-Leibler divergence.

Correct argument order ensures minimal information loss in communication.

Gaussian belief approximation is optimally achieved by moment matching.

Abstract

In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a loss function that quantifies how "embarrassing" it is to communicate a given approximation. We reproduce and discuss an old proof showing that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. The loss function that is obtained in the derivation is equal to the Kullback-Leibler divergence when normalized. This loss function is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments, the approximated and non-approximated beliefs, should be used. The correct order ensures that the recipient of a communication is only deprived of the minimal amount of information. We hope that the elementary derivation settles the apparent confusion. For example when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many suggested computational schemes.