Graphical models of marginal independencies for categorical variables

Undirected graphical models for categorical data represent a set of conditional independencies between pairs of variables given all the remaining variables. These models can be fitted by classical log-linear models. In this paper we discuss instead models of marginal independence between categorical variables. These models have a graphical representation as bi-directed graphs called also covariance graphs (Cox and Wermuth, 1993; Richardson, 2003). We discuss a parameterization of marginal independence models for discrete variables based on the marginal log-linear models by Bergsma and Rudas (2002). This allows the encoding of any marginal independence model for a set of categorical variables associated with a given bi-directed graph. We develop an algorithm for maximum likelihood estimation of bi-directed graph models presenting a simple illustration.