A GENERALIZATION OF HIERARCHICAL EXCHANGEABILITY ON TREES TO DIRECTED ACYCLIC GRAPHS

Abstract

— Motivated by the problem of designing inference-friendly Bayesian non-parametric models in probabilistic programming languages, we introduce a general class of partially exchangeable random arrays which generalizes the notion of hierarchical exchange-ability 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.