The evolution of the global Markov property for multivariate regression chain graphs: differences and conflicts

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 chronologically, and for the first time we show that two different and incompatible interpretations have been proposed in the literature. Differences and inconsistencies between them will be discussed by several examples. The older (original) interpretation has no factorization associated to it in the published literature. We derive such a factorization. For the newer (alternative) interpretation we provide an explicit global Markov property, which implies the other Markov properties published in the literature for this interpretation. We provide a summary table comparing different features of LWF, AMP, and the two kinds of MVR chain graph interpretations, which we call MVR and Alternative MVR (AMVR) respectively.