# A Generalization of Hierarchical Exchangeability on Trees to Directed Acyclic Graphs

## TL;DR

This paper introduces a new class of partially exchangeable random arrays governed by directed acyclic graphs, extending hierarchical exchangeability concepts to more general DAG structures, with a representation theorem generalizing classical results.

## Contribution

It generalizes hierarchical exchangeability to DAG-exchangeability and provides a new representation theorem for these arrays.

## Key findings

- Introduces DAG-exchangeable arrays indexed by $
^{|V|}$
- Proves a representation theorem generalizing Aldous-Hoover and Austin-Panchenko
- Extends the framework of hierarchical exchangeability to directed acyclic graphs

## Abstract

Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we introduce a general class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs. More specifically, such a random array is indexed by $\mathbb{N}^{|V|}$ for some DAG $G=(V,E)$, and its exchangeability structure is governed by the edge set $E$. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems.