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June 2018 A characterization of product-form exchangeable feature probability functions
Marco Battiston, Stefano Favaro, Daniel M. Roy, Yee Whye Teh
Ann. Appl. Probab. 28(3): 1423-1448 (June 2018). DOI: 10.1214/17-AAP1333

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.

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Marco Battiston. Stefano Favaro. Daniel M. Roy. Yee Whye Teh. "A characterization of product-form exchangeable feature probability functions." Ann. Appl. Probab. 28 (3) 1423 - 1448, June 2018. https://doi.org/10.1214/17-AAP1333

Information

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

zbMATH: 06919729
MathSciNet: MR3809468
Digital Object Identifier: 10.1214/17-AAP1333

Subjects:
Primary: 60K99
Secondary: 60C99

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 3 • June 2018
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