A generalization of hierarchical exchangeability on trees to directed acyclic graphs
Annales Henri Lebesgue, Volume 4 (2021) , pp. 325-368.

KeywordsBayesian nonparametrics, exchangeability, hierarchical exchangeability, Aldous–Hoover representation, de Finetti representation

### 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 $G=\left(V,E\right)$, 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.

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