Review of Probability Distributions for Modeling Count Data

TL;DR

This paper reviews various probability distributions used for modeling count data, highlighting their theoretical connections and applications in fields like text analysis and genomics.

Contribution

It provides a comprehensive review of count data models, including proofs of their relationships, aiding in the development of methods for applications like topic modeling and genomics.

Findings

Detailed proofs of relationships between multinomial and count models

Insights into the suitability of different distributions for count data

Guidance for method development in applications like text and genomics

Abstract

Count data take on non-negative integer values and are challenging to properly analyze using standard linear-Gaussian methods such as linear regression and principal components analysis. Generalized linear models enable direct modeling of counts in a regression context using distributions such as the Poisson and negative binomial. When counts contain only relative information, multinomial or Dirichlet-multinomial models can be more appropriate. We review some of the fundamental connections between multinomial and count models from probability theory, providing detailed proofs. These relationships are useful for methods development in applications such as topic modeling of text data and genomics.