A Bayesian Characterization of Relative Entropy

TL;DR

This paper provides a new Bayesian-inspired categorical characterization of relative entropy, showing it as the unique measure satisfying certain functorial and optimality conditions within a probability-based category.

Contribution

It introduces a novel categorical framework for relative entropy, independent of previous characterizations, inspired by Petz's work, and highlights its uniqueness under specific functorial properties.

Findings

Relative entropy is characterized as a unique functor under certain conditions.

The framework uses a category with probability distributions and measure-preserving functions.

The characterization is independent of earlier approaches and based on categorical and Bayesian concepts.

Abstract

We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence. We use a number of interesting categories related to probability theory. In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function $f: X \to Y$ together with a stochastic right inverse $s: Y \to X$. The function $f$ can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement. Given this data we can define the entropy of the probability distribution on $X$ relative to the "prior" given by pushing the probability distribution on $Y$ forwards along $s$. We say that $s$ is "optimal" if these distributions agree. We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,\infty]$ which vanishes when $s$ is optimal must be a scalar multiple of this relative entropy. Our proof is independent of all earlier characterizations, but inspired by the work of Petz.