A Review of Multivariate Distributions for Count Data Derived from the Poisson Distribution

TL;DR

This paper reviews multivariate Poisson-based distributions for count data, categorizes existing models, compares their interpretability and theory, and empirically evaluates them on diverse real-world datasets to guide future research.

Contribution

It provides a comprehensive categorization and comparison of multivariate Poisson models, along with empirical insights across different data domains.

Findings

Different models have varying interpretability and theoretical properties.

Empirical results highlight trade-offs among model classes across datasets.

The review suggests promising directions for future research in multivariate count modeling.

Abstract

The Poisson distribution has been widely studied and used for modeling univariate count-valued data. Multivariate generalizations of the Poisson distribution that permit dependencies, however, have been far less popular. Yet, real-world high-dimensional count-valued data found in word counts, genomics, and crime statistics, for example, exhibit rich dependencies, and motivate the need for multivariate distributions that can appropriately model this data. We review multivariate distributions derived from the univariate Poisson, categorizing these models into three main classes: 1) where the marginal distributions are Poisson, 2) where the joint distribution is a mixture of independent multivariate Poisson distributions, and 3) where the node-conditional distributions are derived from the Poisson. We discuss the development of multiple instances of these classes and compare the models in terms of interpretability and theory. Then, we empirically compare multiple models from each class on three real-world datasets that have varying data characteristics from different domains, namely traffic accident data, biological next generation sequencing data, and text data. These empirical experiments develop intuition about the comparative advantages and disadvantages of each class of multivariate distribution that was derived from the Poisson. Finally, we suggest new research directions as explored in the subsequent discussion section.