Parameterization and fitting of a class of discrete graphical models

Graphical Markov models are multivariate statistical models in which the joint distribution satis¯es independence statements that are captured by a graph. We consider models for discrete variables, that are analogous to multivariate regressions in the linear case, when the variables can be arranged in sequences of joint response, intermediate and purely explanatory variables. In the case of one single group of variables, these models specify marginal independencies of pairs of variables. We show that the models admit a proper marginal log-linear parame- terization that can accommodate all the marginal and conditional independence constraints involved, and can be ¯tted, using maximum likelihood under a multi- nomial assumption, by a general iterative gradient-based algorithm. We discuss a technique for determining fast approximate estimates, that can also be used for initializing the general algorithm and we present an illustration based on data from the U.S. General Social Survey.