A Generalization of Hierarchical Exchangeability on Trees to Directed Acyclic Graphs

A random array indexed by the paths of an infinitely-branching rooted tree of finite depth is hierarchically exchangeable if its joint distribution is invariant under rearrangements that preserve the tree structure underlying the index set. Austin and Panchenko (2014) prove that such arrays have de Finetti-type representations, and moreover, that an array indexed by a finite collection of such trees has an Aldous-Hoover-type representation. Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we generalize hierarchical exchangeability to a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting 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.