De Finetti's Theorem in Categorical Probability

TL;DR

This paper introduces a new, category-theoretic proof of de Finetti's Theorem for exchangeable measures, using Markov categories to provide an abstract and intuitive framework that generalizes the classical measure-theoretic version.

Contribution

It offers the first proof of de Finetti's Theorem within the language of Markov categories, broadening the theorem's applicability and understanding.

Findings

Proof is diagrammatic and intuitive

Connects categorical framework with classical measure theory

Generalizes de Finetti's Theorem to abstract settings

Abstract

We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finetti's Theorem for standard Borel spaces is an instance of this result.