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We define a family of probability distributions for random count matriceswith a potentially unbounded number of rows and columns. The threedistributions we consider are derived from the gamma-Poisson, gamma-negativebinomial, and beta-negative binomial processes. Because the models lead toclosed-form Gibbs sampling update equations, they are natural candidates fornonparametric Bayesian priors over count matrices. A key aspect of our analysisis the recognition that, although the random count matrices within the familyare defined by a row-wise construction, their columns can be shown to be i.i.d.This fact is used to derive explicit formulas for drawing all the columns atonce. Moreover, by analyzing these matrices' combinatorial structure, wedescribe how to sequentially construct a column-i.i.d. random count matrix onerow at a time, and derive the predictive distribution of a new row count vectorwith previously unseen features. We describe the similarities and differencesbetween the three priors, and argue that the greater flexibility of the gamma-and beta- negative binomial processes, especially their ability to modelover-dispersed, heavy-tailed count data, makes these well suited to a widevariety of real-world applications. As an example of our framework, weconstruct a naive-Bayes text classifier to categorize a count vector to one ofseveral existing random count matrices of different categories. The classifiersupports an unbounded number of features, and unlike most existing methods, itdoes not require a predefined finite vocabulary to be shared by all thecategories, and needs neither feature selection nor parameter tuning. Both thegamma- and beta- negative binomial processes are shown to significantlyoutperform the gamma-Poisson process for document categorization, withcomparable performance to other state-of-the-art supervised text classificationalgorithms.

Priors for Random Count Matrices Derived from a Family of Negative Binomial Processes