Markov Fields and Log-Linear Interaction Models for Contingency Tables

TLDR

The authors introduce and study a new class of graphical models for contingency tables that link Markov field theory with log‑linear interaction models, and outline estimation issues and future research directions. These models are hierarchical, represented by an undirected graph whose vertices correspond to table dimensions, and their conditional independence structure is read directly from the graph as a Markov property. They show that this class subsumes decomposable models and provide a simple criterion to determine when a graphical model is decomposable.

Abstract

We use a close connection between the theory of Markov fields and that of log-linear interaction models for contingency tables to define and investigate a new class of models for such tables, graphical models. These models are hierarchical models that can be represented by a simple, undirected graph on as many vertices as the dimension of the corresponding table. Further all these models can be given an interpretation in terms of conditional independence and the interpretation can be read directly off the graph in the form of a Markov property. The class of graphical models contains that of decomposable models and we give a simple criterion for decomposability of a given graphical model. To some extent we discuss estimation problems and give suggestions for further work.