Relative exchangeability with equivalence relations

TL;DR

This paper provides a comprehensive characterization of exchangeable random relational structures relative to fixed structures with equivalence relations, unifying previous results on relative and hierarchical exchangeability.

Contribution

It generalizes existing theorems on relative exchangeability and hierarchical exchangeability within a unified Aldous--Hoover-type framework.

Findings

Unified characterization of relative exchangeability.

Generalization of prior theorems by Ackerman, Crane, and Towsner.

Connections between exchangeability and equivalence relations.

Abstract

We describe an Aldous--Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \cite{Ackerman2015} and Crane and Towsner \cite{CraneTowsner2015} and hierarchical exchangeability results due to Austin and Panchenko \cite{AustinPanchenko2014}.