  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 an
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 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.
