On Graphical Models via Univariate Exponential Family Distributions

TL;DR

This paper introduces a flexible class of graphical models based on univariate exponential family distributions, providing new estimation methods and theoretical guarantees for structure recovery, with applications in genomics and proteomics.

Contribution

It proposes a novel subclass of graphical models derived from univariate exponential families, along with M-estimators and rigorous analysis ensuring accurate structure recovery.

Findings

M-estimators effectively recover true graphical structures

Models successfully applied to genomic and proteomic data

High-probability guarantees for structure estimation

Abstract

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential families. This allows us to derive multivariate graphical model distributions from univariate exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributions.