A Generalization of Hierarchical Exchangeability on Trees 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 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.