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

TL;DR

The paper develops a synthetic, category-theoretic version of the Aldous–Hoover theorem within Markov categories, using the Cauchy–Schwarz axiom to replace representability and to derive a robust four-way decomposition of row–column exchangeable arrays. It introduces plate notation and extends conditional independence to multivariate and infinite-index settings, culminating in a synthetic AH decomposition through three independence lemmas and a shift-covariance argument. An alternative proof via the ordered Markov property reframes the problem as compatibility with a generalized causal model, yielding a structured, Bayesian-network–like derivation of the same representation. The results unify and extend de Finetti–type theorems in a categorical framework, enabling parametric versions, dilations, and omnirandomness-based functional forms, with direct measure-theoretic corollaries in standard spaces.

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.

Figures (7)

• Figure 1: All the Americans play against all Europeans and we simulate the results of each game. $A$ means that the American won, $D$ means a draw, and $E$ means that the European won. The predicted result of a game (blue part) depends only on the orange part, i.e. on the simulation of the asymptotic Elo ratings of the two players involved and of the external factors.
• Figure 2: Graphical description of some subproducts of $X^{\mathbb{N},\mathbb{N}}$ as per \ref{['nota:ah']}. The first index in the exponent is the row index, the second index the column index, and $\mathcal{A}$ denotes $\{1, \dots, n\}$.
• Figure 3: The Aldous--Hoover statement for $n=2$. The only purpose of colors is to aid the reader's eye. This highlights that the output $X^{i,j}$ only depend on the tail of its row $X^{\overline{\mathcal{A}},j}$, the tail of its column $X^{i,\overline{\mathcal{A}}}$, and the tail of the whole infinite matrix $X^{\overline{\mathcal{A}},\overline{\mathcal{A}}}$.
• Figure 4: Overall strategy of the proof of \ref{['thm:AldousHoover']}. The specific conditional independence relations established by shift covariance and the three lemmas in the second row are depicted in \ref{['fig:shift_covariance', 'fig:IndependenceLemmas']} respectively. Orange statements establish conditional independence relations for 1D sequences of random variables while the yellow and green ones are for 2D arrays. The arrows are used to indicate the dependence structure in our proof.
• Figure 5: Visualization of \ref{['cor:shift_covariance']}, which expresses the conditional independence relations resulting from shift covariance. The upper and lower row represent the sequences $X^{\mathbb{N}}$ and $Y^{\mathbb{N}}$ respectively where each $X$ and each $Y$ is an output of a joint state $q$ on the full array $(X \otimes Y)^\mathbb{N}$. The index set $\mathcal{A}$ is the subset of the first $n$ natural numbers (\ref{['nota:ah']}). When conditioned on the gray region ($X^{\overline{\mathcal{A}}}$), the outputs in the red ($X^{\mathcal{A}}$) and blue ($Y^{\overline{\mathcal{A}}}$) regions are independent of each other.

Theorems & Definitions (73)

• Theorem
• Example 2.2: Markov categories satisfying the assumptions
• Definition 2.3
• Lemma 2.5: Semigraphoid properties
• Definition 2.6
• Lemma 2.7
• proof
• Remark 2.8
• Definition 2.9
• Proposition 2.10: Permutation invariance is equivalent to shift invariance