Markov Categories and Entropy

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. Following standard practices of information theory, we get measures of mutual information by quantifying, with a chosen divergence, 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 quantifying how far a source or channel is from being deterministic. This recovers Shannon and Rényi 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. No previous knowledge of category theory is assumed.