Generalized Negative Binomial Processes and the Representation of Cluster Structures

The paper introduces the concept of a cluster structure to define a joint distribution of the sample size and its exchangeable random partitions. The cluster structure allows the probability distribution of the random partitions of a subset of the sample to be dependent on the sample size, a feature not presented in a partition structure. A generalized negative binomial process count-mixture model is proposed to generate a cluster structure, where in the prior the number of clusters is finite and Poisson distributed and the cluster sizes follow a truncated negative binomial distribution. The number and sizes of clusters can be controlled to exhibit distinct asymptotic behaviors. Unique model properties are illustrated with example clustering results using a generalized Polya urn sampling scheme. The paper provides new methods to generate exchangeable random partitions and to control both the cluster-number and cluster-size distributions.Comment: 30 pages, 8 figure

Beta-Negative Binomial Process and Exchangeable Random Partitions for Mixed-Membership Modeling

The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clusters the data points of each group into a random number of exchangeable partitions, which are shared across all the groups. A fully collapsed Gibbs sampler is developed for the BNBP, leading to a novel nonparametric Bayesian topic model that is distinct from existing ones, with simple implementation, fast convergence, good mixing, and state-of-the-art predictive performance.Comment: in Neural Information Processing Systems (NIPS) 2014. 9 pages + 3 page appendi

Augment-and-Conquer Negative Binomial Processes

By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gamma-NB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its unique theoretical, structural and computational advantages. A variety of NB processes with distinct sharing mechanisms are constructed and applied to topic modeling, with connections to existing algorithms, showing the importance of inferring both the NB dispersion and probability parameters.Comment: Neural Information Processing Systems, NIPS 201