Exchangeable Trait Allocations

TL;DR

This paper extends the theory of exchangeability from clustering to trait allocations, providing new probabilistic representations and characterizations, including for models of sparse graphs, with full treatment of dust groups.

Contribution

It develops the de Finetti representation and exchangeable trait probability function (ETPF) for trait allocations, capturing dust groups and connecting to graph models.

Findings

Derived the de Finetti representation for trait allocations.

Characterized all trait allocations with an ETPF.

Applied theory to edge-exchangeable graph distributions.

Abstract

Trait allocations are a class of combinatorial structures in which data may belong to multiple groups and may have different levels of belonging in each group. Often the data are also exchangeable, i.e., their joint distribution is invariant to reordering. In clustering---a special case of trait allocation---exchangeability implies the existence of both a de Finetti representation and an exchangeable partition probability function (EPPF), distributional representations useful for computational and theoretical purposes. In this work, we develop the analogous de Finetti representation and exchangeable trait probability function (ETPF) for trait allocations, along with a characterization of all trait allocations with an ETPF. Unlike previous feature allocation characterizations, our proofs fully capture single-occurrence "dust" groups. We further introduce a novel constrained version of the ETPF that we use to establish an intuitive connection between the probability functions for clustering, feature allocations, and trait allocations. As an application of our general theory, we characterize the distribution of all edge-exchangeable graphs, a class of recently-developed models that captures realistic sparse graph sequences.