Markov Categories and Entropy

TL;DR

This paper introduces an enriched Markov category framework that unifies probability, information theory, and diversity measures, providing new categorical definitions for entropy and mutual information.

Contribution

It combines categorical formalism with entropy and divergence measures, offering a novel way to define and analyze information-theoretic quantities within Markov categories.

Findings

Defines entropy as a measure of non-determinism in sources and channels.

Recovers Shannon, Rénnyi, and Gini-Simpson indices within the categorical framework.

Provides a conceptual basis for generalized entropy measures.

Abstract

Markov categories are a novel framework to describe and treat problems in probability and information theory. In this work we combine the categorical formalism with the traditional quantitative notions of entropy, mutual information, and data processing inequalities. We show that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. As it is customary in information theory, mutual information can be defined as a measure of how far a joint source is from displaying independence of its components. More strikingly, Markov categories give a notion of determinism for sources and channels, and we can define entropy exactly by measuring how far a source or channel is from being deterministic. This recovers Shannon and R\'enyi entropies, as well as the Gini-Simpson index used in ecology to quantify diversity, and it can be used to give a conceptual definition of generalized entropy.