Priors for Random Count Matrices Derived from a Family of Negative Binomial Processes
This provides a flexible nonparametric Bayesian prior for count matrices, useful in applications like text classification where handling unbounded features and over-dispersed data is crucial, though it is incremental as it builds on existing negative binomial process frameworks.
The authors tackled the problem of modeling random count matrices with unbounded rows and columns by defining a family of distributions derived from negative binomial processes, which enable closed-form Gibbs sampling and support over-dispersed data. They demonstrated that the gamma- and beta-negative binomial processes significantly outperform the gamma-Poisson process in document categorization, achieving comparable performance to state-of-the-art supervised text classification algorithms.
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 gamma-Poisson, gamma-negative binomial, and beta-negative binomial processes. Because the models lead to closed-form 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 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 at once. Moreover, by analyzing these matrices' combinatorial structure, we describe how to sequentially construct a column-i.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 over-dispersed, heavy-tailed count data, makes these well suited to a wide variety of real-world applications. As an example of our framework, we construct a naive-Bayes 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 gamma-Poisson process for document categorization, with comparable performance to other state-of-the-art supervised text classification algorithms.