Alternative Markov Properties for Chain Graphs

TLDR

Graphical Markov models use graphs to represent possible dependences among statistical variables, and Lauritzen, Wermuth, and Frydenberg introduced the LWF Markov property for chain graphs, which unify undirected and acyclic directed graph dependencies. The authors introduce an alternative Markov property (AMP) for chain graphs and demonstrate that it is satisfied by a block‑recursive linear system with multivariate normal errors. The AMP model decomposes into conditional normal models that blend multivariate linear regression and covariance selection, enabling parameter estimation, and the authors provide necessary and sufficient conditions for equivalence between LWF and AMP, between CGs, and between a CG and an ADG or decomposable UG. The AMP property serves as a more direct extension of the acyclic directed graph Markov property than the LWF property for chain graphs.

Abstract

Graphical Markov models use graphs to represent possible dependences among statistical variables. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs (CG): graphs that can be used to represent both structural and associative dependences simultaneously and that include both undirected graphs (UG) and acyclic directed graphs (ADG) as special cases. Here an alternative Markov property (AMP) for CGs is introduced and shown to be the Markov property satisfied by a block‐recursive linear system with multivariate normal errors. This model can be decomposed into a collection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, facilitating the estimation of its parameters. In the general case, necessary and sufficient conditions are given for the equivalence of the LWF and AMP Markov properties of a CG, for the AMP Markov equivalence of two CGs, for the AMP Markov equivalence of a CG to some ADG or decomposable UG, and for other equivalences. For CGs, in some ways the AMP property is a more direct extension of the ADG Markov property than is the LWF property.