Combinatorial clustering and the beta negative binomial process

In this work, we establish novel connections between the Bayesian nonparametric clustering and featural paradigms by considering the problem of admixture modeling. We examine the Dirichlet process and its representation in terms of a gamma process on the traditional clustering side of Bayesian nonparametrics. On the featural side, we examine the beta process and introduce a new model, the beta negative binomial process (BNBP), for admixture modeling. We prove theoretical connections between the BNBP and the more familiar gamma-Poisson process and further develop and compare asymptotic behavior for these processes, theoretically and via simulation. Finally, we demonstrate the applicability of the BNBP on data relevant to national security.