Abstract: Undirected graphical models, such as Gaussian graphical models, Ising, and
multinomial/categorical graphical models, are widely used in a variety of applications
for modeling distributions over a large number of variables. These standard
instances, however, are ill-suited to modeling count data, which are increasingly
ubiquitous in big-data settings such as genomic sequencing data, user-ratings data,
spatial incidence data, climate studies, and site visits. Existing classes of Poisson
graphical models, which arise as the joint distributions that correspond to Poisson
distributed node-conditional distributions, have a major drawback: they can
only model negative conditional dependencies for reasons of normalizability given
its infinite domain. In this paper, our objective is to modify the Poisson graphical
model distribution so that it can capture a rich dependence structure between
count-valued variables. We begin by discussing two strategies for truncating the
Poisson distribution and show that only one of these leads to a valid joint distribution.
While this model can accommodate a wider range of conditional dependencies,
some limitations still remain. To address this, we investigate two additional
novel variants of the Poisson distribution and their corresponding joint graphical
model distributions. Our three novel approaches provide classes of Poisson-like
graphical models that can capture both positive and negative conditional dependencies
between count-valued variables. One can learn the graph structure of
our models via penalized neighborhood selection, and we demonstrate the performance
of our methods by learning simulated networks as well as a network from
microRNA-sequencing data.
Download: pdf
Citation
- On Poisson Graphical Models (pdf, software)
E. Yang, P. Ravikumar, G. Allen, Z. Liu.
In Neural Information Processing Systems (NIPS), December 2013.
Bibtex: