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.
Citation
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
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