[Paper Review] On the Properties of MVR Chain Graphs

Mohammad Ali Javidian, Marco Valtorta|arXiv (Cornell University)|Mar 9, 2018

Complex Network Analysis Techniques31 references3 citations

TL;DR

This paper proposes an alternative local Markov property for multivariate regression chain graphs (MVR CGs) and establishes its equivalence to existing Markov properties under compositional semi-graphoid conditions. It introduces a new explicit factorization formula for MVR CGs, differing from prior work, and demonstrates that all standard Markov properties are equivalent for semi-graphoids, significantly advancing the theoretical foundation of MVR CGs in graphical models.

ABSTRACT

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.

Motivation & Objective

• To develop an alternative local Markov property for MVR chain graphs that is equivalent to existing Markov properties for compositional semi-graphoids.
• To derive a new, more explicit factorization criterion for MVR chain graphs, distinct from previous formulations in the literature.
• To compare and unify various Markov properties (global, pairwise, block recursive, etc.) for MVR CGs and identify conditions under which they are equivalent.
• To provide a comprehensive comparison of LWF, AMP, and MVR chain graph features, emphasizing the causal interpretability of bidirected edges in MVR CGs.

Proposed method

• Proposes a new local Markov property for MVR CGs based on ancestral and anterior relationships, using the concept of predecessors and parent sets in chain components.
• Applies properties of compositional graphoids—decomposition, weak union, contraction, and symmetry—to derive equivalence among Markov properties under semi-graphoid assumptions.
• Derives a new factorization formula for MVR CGs inspired by Evans and Richardson’s (2014) factorization for acyclic directed mixed graphs (ADMGs), adapted to MVR CG structure.
• Uses path-based separation criteria (m-separation) and ancestral ordering to formalize the Markov properties and prove their equivalence.
• Employs case analysis on connected and disconnected subsets of chain components to validate the proposed local Markov property under different structural configurations.
• Compares LWF, AMP, and MVR chain graphs via a summary table, highlighting differences in edge interpretation, Markov properties, and conditional independence behavior.

Experimental results

Research questions

• RQ1 Is there a new local Markov property for MVR chain graphs that is equivalent to existing ones under compositional semi-graphoid assumptions?
• RQ2 Can a more explicit and distinct factorization formula be derived for MVR chain graphs compared to prior work?
• RQ3 Under what conditions are the various Markov properties (global, pairwise, block recursive) equivalent for MVR CGs?
• RQ4 How do the structural and independence properties of MVR CGs compare to those of LWF and AMP CGs?

Key findings

• An alternative local Markov property for MVR chain graphs is proven equivalent to other Markov properties in the literature for compositional semi-graphoids.
• All standard Markov properties (global, pairwise, block recursive, etc.) are equivalent for MVR CGs when the independence model is a compositional semi-graphoid.
• A new explicit factorization formula for MVR CGs is derived, differing from previous formulations and providing a clearer decomposition of the joint distribution.
• The bidirected edges in MVR CGs have a strong causal interpretation as representing unobserved common causes, which is not as directly interpretable in LWF or AMP models.
• The proposed factorization is shown to be more explicit and structurally aligned with the underlying graph structure than earlier factorizations.
• A comprehensive comparison table is provided, highlighting key differences in edge interpretation, Markov properties, and conditional independence behavior across LWF, AMP, and MVR chain graphs.