On Graphical Models via Univariate Exponential Family Distributions
This provides a flexible modeling framework for researchers in fields like genomics and proteomics dealing with diverse data types, though it is incremental as it extends existing graphical model concepts.
The paper tackles the problem of selecting appropriate graphical models for non-Gaussian and non-categorical data by proposing a general subclass derived from univariate exponential family distributions, such as Poisson and exponential, and shows that their M-estimators recover the true graphical model structure exactly with high probability.
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.