Graphical Models and Exponential Families

TL;DR

This paper classifies various graphical models within the exponential family framework, detailing their types and properties, and explores automatic generation of constraints for model selection.

Contribution

It provides a comprehensive classification of graphical models as exponential families and introduces methods to generate constraints for model selection.

Findings

Undirected models are linear exponential families.

Directed models are curved exponential families.

Hidden variable models are stratified exponential families.

Abstract

We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined.