Chain graph models of multivariate regression type for categorical data

We discuss a class of graphical models for discrete data defined by what we call a multivariate regression chain graph Markov property. We propose a parameterization based on a sequence of generalized linear models with a multivariate logistic link function. We show the relationship with a chain graph model recently defined in the literature, and we prove that the proposed parametrization for the joint distribution defines a complete and hierarchical marginal log-linear model.