Acyclic Linear SEMs Obey the Nested Markov Property.

Area of Science:

• Causal inference and graphical models
• Statistical modeling and machine learning

Background:

• Hidden variable directed acyclic graphs (DAGs) induce conditional independence structures representable by maximal ancestral graphs (MAGs).
• MAGs are valuable for causal discovery and causal effect identification.
• Hidden variable DAGs also impose generalized independence constraints, forming the nested Markov property.

Purpose of the Study:

• To demonstrate that acyclic linear structural equation models (SEMs) adhere to the nested Markov property.
• To introduce maximal arid graphs (MArGs) as a generalization of MAGs for parameterizing Gaussian distributions obeying the nested Markov property.
• To establish a framework for Maximum Likelihood (ML) fitting and Bayesian Information Criterion (BIC) score computation for Gaussian nested models.

Main Methods:

• We prove that acyclic linear SEMs satisfy the nested Markov property.
• We define maximal arid graphs (MArGs) as a generalization of MAGs.
• We associate every nested Markov model with a MArG and demonstrate its use as a path diagram for parameterization.

Main Results:

• Acyclic linear SEMs are shown to obey the nested Markov property.
• Maximal arid graphs (MArGs) provide a natural parameterization for Gaussian distributions satisfying the nested Markov property.
• MArGs enable direct methods for ML fitting and BIC score computation for Gaussian nested models.

Conclusions:

• The nested Markov property is fundamental for understanding hidden variable DAGs.
• Maximal arid graphs (MArGs) offer a powerful graphical tool for Gaussian nested Markov models.
• This work facilitates advanced statistical analysis and causal inference in the presence of latent variables.