A de Finetti-type representation of joint hierarchically exchangeable arrays on directed acyclic graphs

TL;DR

This paper introduces a new form of joint exchangeability for arrays indexed by DAG vertices, extending hierarchical exchangeability, and proves a representation theorem using independent uniform variables.

Contribution

It defines a novel joint exchangeability concept on DAG-indexed arrays and establishes a de Finetti-type representation theorem for this setting.

Findings

Defines joint exchangeability on DAG-indexed arrays

Proves a representation theorem using independent uniform variables

Extends hierarchical exchangeability to a new joint version

Abstract

We define joint exchangeability on arrays indexed by a vector of natural numbers with coordinates being the vertices of directed acyclic graphs (DAGs) using local isomorphisms. The notion provides a new version of exchangeability, which is a joint version of hierarchical exchangeability defined in Jung, L., Staton, Yang (2020). We also prove the existence of a generic representation by independent uniform random variables.