Undirected Graphical Models

This chapter discusses undirected graphical log-linear models for categorical data. These models impose conditional independence restrictions on the association structure of the joint distribution. The conditional independences can be read off from a graph, and thus the graphical representation provides an intuitive tool in model interpretation, search, and communication. The presentation starts with some concepts of graph theory, to be used in this and in the next chapters, and continues with general properties of conditional independence. Then, various types of conditional independences, called Markov properties (pairwise, local, global) associated with graphs, and their relationships with each other and the Gibbs factorization of the joint distribution are discussed. Graphical models are widely used in applications, but one has to keep in mind that in the case of categorical data, they do not necessarily provide an exact description of the conditional independences exhibited by a distribution. This is clarified with a discussion of faithfulness. Then, procedures for model search and model fitting are presented. These include, among others, the iterative proportional fitting and the Darroch-Ratcliff algorithms. Graphical models are extended to situations when some of the variables of interest are discrete but some are continuous, using the conditional Gaussian (CG) distribution, and the chapter concludes with a discussion, giving reasons why graphical models are better interpreted based on the edges not present in the graph, rather than based on the edges present.