Combinatorial Clustering and the Beta Negative Binomial Process

We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the
negative binomial process
(
${\rm NBP}$
) as an infinite-dimensional prior appropriate for such problems. We show that the
${\rm NBP}$
is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process (
${\rm BNBP}$
) and hierarchical models based on the
${\rm BNBP}$
(the
${\rm HBNBP}$
). We study the asymptotic properties of the
${\rm BNBP}$
and develop a three-parameter extension of the
${\rm BNBP}$
that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the
${\rm HBNBP}$
, and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis.