  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.
