The Annals of Applied Probability

A characterization of product-form exchangeable feature probability functions

Marco Battiston, Stefano Favaro, Daniel M. Roy, and Yee Whye Teh

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Abstract

We characterize the class of exchangeable feature allocations assigning probability $V_{n,k}\prod_{l=1}^{k}W_{m_{l}}U_{n-m_{l}}$ to a feature allocation of $n$ individuals, displaying $k$ features with counts $(m_{1},\ldots,m_{k})$ for these features. Each element of this class is parametrized by a countable matrix $V$ and two sequences $U$ and $W$ of nonnegative weights. Moreover, a consistency condition is imposed to guarantee that the distribution for feature allocations of $(n-1)$ individuals is recovered from that of $n$ individuals, when the last individual is integrated out. We prove that the only members of this class satisfying the consistency condition are mixtures of three-parameter Indian buffet Processes over the mass parameter $\gamma$, mixtures of $N$-dimensional Beta–Bernoulli models over the dimension $N$, or degenerate limits thereof. Hence, we provide a characterization of these two models as the only consistent exchangeable feature allocations having the required product form, up to randomization of the parameters.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1423-1448.

Dates
Received: July 2016
Revised: June 2017
First available in Project Euclid: 1 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1527840023

Digital Object Identifier
doi:10.1214/17-AAP1333

Mathematical Reviews number (MathSciNet)
MR3809468

Zentralblatt MATH identifier
06919729

Subjects
Primary: 60K99: None of the above, but in this section
Secondary: 60C99: None of the above, but in this section

Keywords
Indian buffet process exchangeable feature allocations Gibbs-type partitions

Citation

Battiston, Marco; Favaro, Stefano; Roy, Daniel M.; Teh, Yee Whye. A characterization of product-form exchangeable feature probability functions. Ann. Appl. Probab. 28 (2018), no. 3, 1423--1448. doi:10.1214/17-AAP1333. https://projecteuclid.org/euclid.aoap/1527840023


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