The combinatorial structure of beta negative binomial processes
This work addresses a theoretical gap in Bayesian nonparametrics for modeling latent multisets, offering a new combinatorial framework that is incremental but specific to this domain.
The paper characterizes the combinatorial structure of sequences of negative binomial processes with a beta process base measure, introducing a count analogue of the Indian buffet process called the negative binomial Indian buffet process, and provides a construction that avoids representing the base measure for use in MCMC algorithms.
We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for latent multisets of features underlying data. Analogously, random subsets arise from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in which 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 a construction for the beta negative binomial process that avoids a representation of the underlying beta process base measure. We describe the key Markov kernels needed to use a NB-IBP representation in a Markov Chain Monte Carlo algorithm targeting a posterior distribution.