Graphical Gaussian models have proven to be useful tools for exploring
network structures based on multivariate data. Applications to studies of gene
expression have generated substantial interest in these models, and resulting
recent progress includes the development of fitting methodology involving
penalization of the likelihood function. In this paper we advocate the use of
multivariate t-distributions for more robust inference of graphs.

We study a class of parametrizations of convex cones of positive semidefinite
matrices with prescribed zeros. Each such cone corresponds to a graph whose
non-edges determine the prescribed zeros. Each parametrization in this class is
a polynomial map associated with a simplicial complex supported on cliques of
the graph. The images of the maps are convex cones, and the maps can only be
surjective onto the cone of zero-constrained positive semidefinite matrices
when the associated graph is chordal and the simplicial complex is the clique
complex of the graph.

The statistical literature discusses different types of Markov properties for
chain graphs that lead to four possible classes of chain graph Markov models.
The different models are rather well understood when the observations are
continuous and multivariate normal, and it is also known that one model class,
referred to as models of LWF (Lauritzen--Wermuth--Frydenberg) or block
concentration type, yields discrete models for categorical data that are
smooth. This paper considers the structural properties of the discrete models
based on the three alternative Markov properties.