A hierarchical version of the de Finetti and Aldous-Hoover representations

TL;DR

This paper extends classical exchangeability theorems to hierarchically structured arrays, providing new representation results for arrays indexed by trees, which generalize de Finetti and Aldous-Hoover theorems.

Contribution

It introduces hierarchical versions of de Finetti and Aldous-Hoover representations for arrays indexed by trees, broadening the scope of exchangeability theory.

Findings

Hierarchically exchangeable arrays satisfy an analogue of de Finetti's theorem.

A hierarchical version of the Aldous-Hoover representation is established.

The results apply to arrays indexed by multiple trees, generalizing existing frameworks.

Abstract

We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and prove that they satisfy an analogue of de Finetti's theorem. We also prove a more general result for arrays indexed by several trees, which includes a hierarchical version of the Aldous-Hoover representation.