Priors for Random Count Matrices Derived from a Family of Negative Binomial Processes
Abstract
We define a family of probability distributions for random count matrices with a potentially unbounded number of rows and columns. The three distributions we consider are derived from the gammaPoisson, gammanegative binomial, and betanegative binomial processes. Because the models lead to closedform Gibbs sampling update equations, they are natural candidates for nonparametric Bayesian priors over count matrices. A key aspect of our analysis is the recognition that, although the random count matrices within the family are defined by a rowwise construction, their columns can be shown to be i.i.d. This fact is used to derive explicit formulas for drawing all the columns at once. Moreover, by analyzing these matrices' combinatorial structure, we describe how to sequentially construct a columni.i.d. random count matrix one row at a time, and derive the predictive distribution of a new row count vector with previously unseen features. We describe the similarities and differences between the three priors, and argue that the greater flexibility of the gamma and beta negative binomial processes, especially their ability to model overdispersed, heavytailed count data, makes these well suited to a wide variety of realworld applications. As an example of our framework, we construct a naiveBayes text classifier to categorize a count vector to one of several existing random count matrices of different categories. The classifier supports an unbounded number of features, and unlike most existing methods, it does not require a predefined finite vocabulary to be shared by all the categories, and needs neither feature selection nor parameter tuning. Both the gamma and beta negative binomial processes are shown to significantly outperform the gammaPoisson process for document categorization, with comparable performance to other stateoftheart supervised text classification algorithms.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.3331
 Bibcode:
 2014arXiv1404.3331Z
 Keywords:

 Statistics  Methodology;
 Statistics  Machine Learning
 EPrint:
 To appear in Journal of the American Statistical Association (Theory and Methods). 31 pages + 11 page supplement, 5 figures