ISSN: 0162-8828

Lawrence Carin , Duke University, Durham

Mingyuan Zhou , University of Texas at Austin, Austin

ABSTRACT

The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to a NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions distributions, and construct a Poisson-logarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gamma-NB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural and computational advantages of the NB process. A variety of NB processes, including the beta-geometric, beta-NB, marked-beta-NB, marked-gamma-NB and zero-inflated-NB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters.

INDEX TERMS

Niobium, Analytical models, Data models, Random variables, Atomic measurements, Bayes methods, Joints, Normalized Random Measures, Negative Binomial Process, Mixture Modeling, Count Modeling, Completely Random Measures, Poisson Process, Gamma Process, Dirichlet Process, Hierarchical Dirichlet Process, Chinese Restaurant Process, Poisson Factor Analysis, Topic Modeling, Mixed-Membership Modeling, Bayesian Nonparametrics, Beta Process

CITATION

Lawrence Carin, Mingyuan Zhou, "Negative Binomial Process Count and Mixture Modeling",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol. , no. , pp. 0, 5555, doi:10.1109/TPAMI.2013.211