A Generalization of Hierarchical Exchangeability on Trees to Directed Acyclic Graphs
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Motivated by problems in Bayesian nonparametrics and probabilistic programming discussed in Staton et al. (2018), we present a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting, a given random array is indexed by certain subgraphs of a directed acyclic graph (DAG) of finite depth, where each nonterminal vertex has infinitely many outgoing edges. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover representation theorem. In the case that the DAGs are finite collections of certain rooted trees, our arrays are hierarchically exchangeable in the sense of Austin and Panchenko (2014), and we recover the representation theorem proved by them. Additionally, our representation is fine-grained in the sense that representations at lower levels of the hierarchy are also available. This latter feature is important in applications to probabilistic programming, thus offering an improvement over the Austin-Panchenko representation even for hierarchical exchangeability.
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