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 non-negative 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. In Theorem 1.1, we prove that the only members of this class satisfying the consistency condition are mixtures of the Indian Buffet Process over its mass parameter $\gamma$ and mixtures of the Beta--Bernoulli model over its dimensionality parameter $N$. Hence, we provide a characterization of these two models as the only, up to randomization of the parameters, consistent exchangeable feature allocations having the required product form.