Complete Diagrammatic Axiomatisations of Relative Entropy
Ralph Sarkis, Fabio Zanasi
TL;DR
This work considers two natural monoidal structures on stochastic matrices, given by the Kronecker product and the direct sum, and proposes complete axiomatisations of Kullback-Leibler divergence and, more generally, of R\'enyi divergences of arbitrary order for each such structure.
Abstract
Relative entropy is a fundamental class of distances between probability distributions, with widespread applications in probability theory, statistics, and machine learning. In this work, we study relative entropy from a categorical perspective, viewing it as a quantitative enrichment of categories of stochastic matrices. We consider two natural monoidal structures on stochastic matrices, given by the Kronecker product and the direct sum. Our main results are complete axiomatisations of Kullback-Leibler divergence and, more generally, of Rényi divergences of arbitrary order, for each such structure. Our axiomatic theories are formulated within the framework of quantitative monoidal algebra, using a graphical language of string diagrams enriched with quantitative equations.