## The Annals of Applied Probability

### A characterization of product-form exchangeable feature probability functions

#### 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

#### 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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