Alternative Markov properties for chain graphs

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 represe nt both structural and associative dependences simultaneously and that incl ude both undirected graphs (UG) and acyclic directed graphs (ADG) as specia l cases. Here an alternative Markov property (AMP) for CGs is introduced an d shown to be the Markov property satisfied by a block-recursive linear sys tem with multivariate normal errors. This model can be decomposed into a co llection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, f acilitating the estimation of its parameters. In the general case, necessar y and sufficient conditions are given for the equivalence of the LWF and AM P 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 direc t extension of the ADG Markov property than is the LWP property.