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 (NBP) as an infinite-dimensional prior appropriate for such problems.
We show that the NBP is conjugate to the beta process, and we characterize the
posterior distribution under the beta-negative binomial process (BNBP) and
hierarchical models based on the BNBP (the HBNBP). We study the asymptotic
properties of the BNBP and develop a three-parameter extension of the BNBP that
exhibits power-law behavior. We derive MCMC algorithms for posterior inference
under the HBNBP, and we present experiments using these algorithms in the
domains of image segmentation, object recognition, and document analysis.Comment: 56 pages, 4 figures, 6 table