The Aldous$\unicode{x2013}$Hoover Theorem in Categorical Probability

TL;DR

This paper presents a category-theoretic proof of the Aldous-Hoover Theorem, simplifying the understanding of exchangeability in infinite random matrices and introducing new categorical properties like the Cauchy--Schwarz axiom.

Contribution

It reformulates the Aldous-Hoover Theorem within Markov categories, providing a more transparent proof and introducing the Cauchy--Schwarz axiom for categorical probability.

Findings

Categorical proof of the Aldous-Hoover Theorem

Introduction of the Cauchy--Schwarz axiom in Markov categories

Development of a synthetic de Finetti Theorem

Abstract

The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability.