The combinatorial structure of beta negative binomial processes

We characterize the combinatorial structure of conditionally-iid sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for unknown multisets of a measurable space. Previous work has characterized random subsets arising from conditionally-iid sequences of Bernoulli processes with a common beta process base measure. In this case, the combinatorial structure is described by the Indian buffet process. Our results give a count analogue of the Indian buffet process, which we call a negative binomial Indian buffet process. As an intermediate step toward this goal, we provide constructions for the beta negative binomial process that avoid a representation of the underlying beta process base measure.