Error AMP Chain Graphs

TL;DR

This paper introduces EAMP chain graphs, explicitly representing errors in AMP CGs, and demonstrates their equivalence to other graphical models, enabling better understanding of data generating processes with hidden variables.

Contribution

The paper proposes adding deterministic nodes to AMP CGs to explicitly represent errors, establishing their equivalence to other models and their closure under marginalization.

Findings

EAMP CGs are Markov equivalent to AMP CGs after marginalizing error nodes.

EAMP CGs are equivalent to certain LWF CGs and DAGs under marginalization and conditioning.

EAMP CGs are closed under marginalization, ensuring model simplicity.

Abstract

Any regular Gaussian probability distribution that can be represented by an AMP chain graph (CG) can be expressed as a system of linear equations with correlated errors whose structure depends on the CG. However, the CG represents the errors implicitly, as no nodes in the CG correspond to the errors. We propose in this paper to add some deterministic nodes to the CG in order to represent the errors explicitly. We call the result an EAMP CG. We will show that, as desired, every AMP CG is Markov equivalent to its corresponding EAMP CG under marginalization of the error nodes. We will also show that every EAMP CG under marginalization of the error nodes is Markov equivalent to some LWF CG under marginalization of the error nodes, and that the latter is Markov equivalent to some directed and acyclic graph (DAG) under marginalization of the error nodes and conditioning on some selection nodes. This is important because it implies that the independence model represented by an AMP CG can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we will show that EAMP CGs are closed under marginalization. This is a desirable feature because it guarantees parsimonious models under marginalization.