Discrete graphical models and their parameterization

This chapter is devoted to graphical models in which the observed variables are categorical, that is, whose state space consists of a finite number of values. The focus is on regression graph models, because this family of models allows us to approach discrete graphical models with a sufficient degree of generality. In particular, we cover as special cases both undirected and bidirected graphical models, as well as acyclic directed graphical models. Special attention is given to the problem of specifying suitable parameterizations and, from this viewpoint, we provide a unified approach to the parameterization of all models considered. This relies on a few general properties of conditional independence and Moebius inversion which are applied, almost identically, to all the models considered to derive suitable parameterizations and show their properties.