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.