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An Aldous-Hoover type representation for row exchangeable arrays

Evan Donald, Jason Swanson

TL;DR

This work addresses modeling and inference for row-exchangeable arrays $\xi=\{\xi_{ij}\}$ with entries in a space $S$ by introducing an Aldous–Hoover type representation and a corresponding de Finetti theorem. It shows that $\xi$ can be generated from a measurable $f:\mathbb{R}^3\to S$ driven by i.i.d. uniforms, and it identifies a hierarchical structure of row measures $\mu_i$ and a row-distribution generator $\varpi$ such that $\mu|\varpi \sim \varpi^\infty$ and $\xi_i|\mu_i \sim \mu_i^\infty$, enabling Bayesian inference. The posterior distribution for unseen entries is shown to be determined by the posteriors of the $\mu_i$ and obeys a conditional independence structure that reduces the computational complexity by focusing on the first $m$ rows. Overall, the results provide a principled nonparametric Bayesian framework for row-exchangeable data using random measures, kernels, and de Finetti-type representations, with practical implications for inference in populations of exchangeable sequences.

Abstract

In an array of random variables, each row can be regarded as a single, sequence-valued random variable. In this way, the array is seen as a sequence of sequences. Such an array is said to be row exchangeable if each row is an exchangeable sequence, and the entire array, viewed as a sequence of sequences, is exchangeable. We give a representation theorem, analogous to those of Aldous and Hoover, which characterizes row exchangeable arrays. We then use this representation theorem to address the problem of performing Bayesian inference on row exchangeable arrays.

An Aldous-Hoover type representation for row exchangeable arrays

TL;DR

This work addresses modeling and inference for row-exchangeable arrays with entries in a space by introducing an Aldous–Hoover type representation and a corresponding de Finetti theorem. It shows that can be generated from a measurable driven by i.i.d. uniforms, and it identifies a hierarchical structure of row measures and a row-distribution generator such that and , enabling Bayesian inference. The posterior distribution for unseen entries is shown to be determined by the posteriors of the and obeys a conditional independence structure that reduces the computational complexity by focusing on the first rows. Overall, the results provide a principled nonparametric Bayesian framework for row-exchangeable data using random measures, kernels, and de Finetti-type representations, with practical implications for inference in populations of exchangeable sequences.

Abstract

In an array of random variables, each row can be regarded as a single, sequence-valued random variable. In this way, the array is seen as a sequence of sequences. Such an array is said to be row exchangeable if each row is an exchangeable sequence, and the entire array, viewed as a sequence of sequences, is exchangeable. We give a representation theorem, analogous to those of Aldous and Hoover, which characterizes row exchangeable arrays. We then use this representation theorem to address the problem of performing Bayesian inference on row exchangeable arrays.
Paper Structure (11 sections, 12 theorems, 45 equations)

This paper contains 11 sections, 12 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Random measures and kernels
  4. Regular conditional distributions
  5. Conditional independence
  6. Exchangeability and empirical measures
  7. Conditional law of large numbers
  8. Bayesian inference for row exchangeable arrays
  9. An Aldous-Hoover type representation theorem
  10. A de Finetti theorem for row exchangeability
  11. Computing the posterior distribution

Key Result

Theorem 1.1

Let $\xi = \{\xi_{ij}: i, j \in \mathbb{N}\}$ be an array of $S$-valued random variables. Then $\xi$ is separately exchangeable if and only if there exists a measurable function $g: \mathbb{R}^4 \to S$ and an i.i.d. collection of random variables $\{\alpha, \beta_i, \eta_j, \lambda_ {ij}: i, j \in \

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 13 more