On the Properties of MVR Chain Graphs
This work provides theoretical clarifications for researchers in graphical models, but it is incremental as it builds on established interpretations like LWF, MVR, and AMP.
The paper reviews Markov properties for MVR chain graphs, proposes alternative global and local Markov properties, and derives a new factorization formula that is more explicit than existing ones.
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 chain graphs (MVR CGs) 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. Except for pairwise Markov properties, we show that for MVR chain graphs all Markov properties in the literature are equivalent for semi-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.