Marginal AMP Chain Graphs

TL;DR

This paper introduces MAMP chain graphs, a new family of models combining directed, undirected, and bidirected edges, which generalize existing models and are suitable for representing complex dependencies in Gaussian data.

Contribution

The paper defines MAMP chain graphs, establishes their Markov properties, characterizes their equivalence, and shows their relation to DAGs with deterministic nodes, extending the modeling capabilities.

Findings

MAMP chain graphs unify AMP and multivariate regression chain graphs.

They are equivalent to certain DAGs with deterministic nodes under marginalization.

MAMP chain graphs are closed under marginalization for Gaussian distributions.

Abstract

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent. For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.