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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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

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

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

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Zentralblatt MATH identifier

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

Indian buffet process exchangeable feature allocations Gibbs-type partitions


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.

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  • [1] Berti, P., Crimaldi, I., Pratelli, L. and Rigo, P. (2015). Central limit theorems for an Indian buffet model with random weights. Ann. Appl. Probab. 25 523–547.
  • [2] Broderick, T., Pitman, J. and Jordan, M. I. (2013). Feature allocations, probability functions, and paintboxes. Bayesian Anal. 8 801–836.
  • [3] Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probab. 6 705–730.
  • [4] Ghahramani, Z., Griffiths, T. L. and Sollich, P. (2007). Bayesian nonparametric latent feature models. In Bayesian Statistics 8. Oxford Univ. Press, Oxford.
  • [5] Gnedin, A. and Pitman, J. (2006). Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. (N.Y.) 138 5674–5685.
  • [6] Griffiths, T. L. and Ghahramani, Z. (2006). Infinite latent feature models and the Indian buffet process. Adv. Neural Inf. Process. Syst. 18 475–482.
  • [7] Griffiths, T. L. and Ghahramani, Z. (2011). The Indian buffet process: An introduction and review. J. Mach. Learn. Res. 12 1185–1224.
  • [8] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145–158.
  • [9] Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin.
  • [10] Teh, Y. and Görür, D. (2010). Indian buffet processes with power-law behavior. Adv. Neural Inf. Process. Syst. 22 1838–1846.