The Aldous-Hoover Theorem in Categorical Probability

2024-11-19

arXiv.org

AI Summary

Aldous–Hoover theorem characterizes the structure of row–column exchangeable infinite random matrices—each entry being expressible as a measurable function of four variables: a global, a row-specific, a column-specific, and an entry-specific variable. Its original proof lacks conceptual clarity and categorical coherence. Method: We provide the first categorical reconstruction and proof of the theorem within the framework of Markov categories. We introduce a novel categorical axiom—the Cauchy–Schwarz axiom—which systematically generalizes exchangeability, ordered Markov properties, and d-separation to the categorical probability setting, yielding a compositional version of de Finetti’s theorem. Contribution: Our approach significantly enhances the abstraction, transparency, and intuition of the original proof; establishes a unified categorical pathway for hierarchical exchangeability and related advanced results; and advances the categorical semantics of probability theory and the categorical formalization of Bayesian networks.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Action, Change, and Causality
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Extends Aldous-Hoover theorem to categorical probability using Markov categories
• Proves the theorem more transparently via the Cauchy-Schwarz axiom
• Generalizes Bayesian network concepts like d-separation to Markov categories

Innovation

• Markov categories framework for probability theory
• Cauchy-Schwarz axiom enabling synthetic proofs
• Ordered Markov property with d-separation generalization