A characterization of product-form exchangeable feature probability functions

We characterize the class of exchangeable feature allocations assigning probability Vn,k.kl=1WmlUn.ml to a feature allocation of n individuals, displaying k features with counts (m1,.,mk) 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 ., 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.