On the Properties of MVR Chain Graphs

Depending on the interpretation of the type of edges, a chain graph can represent different relations between variables and thereby independence models. Three interpretations, known by the acronyms LWF, MVR, and AMP, are prevalent. Multivariate regression (MVR) chain graphs were introduced by Cox and Wermuth in 1993. We review Markov properties for MVR chain graphs and propose an alternative global and local Markov property for them. Also, we prove equivalence of all proposed Markov properties in the literature for compositional graphoids. We derive a new factorization formula for MVR chain graphs which is more explicit than and different from the proposed factorizations for MVR chain graphs in the literature. Finally, we provide a summary table comparing different features of LWF, AMP, and MVR chain graphs.