Electronic Journal of Statistics

Exchangeable trait allocations

Trevor Campbell, Diana Cai, and Tamara Broderick

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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.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 2290-2322.

Received: September 2016
First available in Project Euclid: 25 July 2018

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Trait allocation exchangeability paintbox probability function partition feature allocation graph vertex allocation edge exchangeability

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Campbell, Trevor; Cai, Diana; Broderick, Tamara. Exchangeable trait allocations. Electron. J. Statist. 12 (2018), no. 2, 2290--2322. doi:10.1214/18-EJS1455. https://projecteuclid.org/euclid.ejs/1532484331

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