A Bayesian Characterization of Relative Entropy

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 → Y together with a stochastic right inverse s : Y → 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,∞] 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.