A review of Gaussian Markov models for conditional independence

TL;DR

This paper reviews Gaussian Markov models, comparing directed and undirected types, discussing estimation, model selection, modern techniques, and applications, highlighting their theoretical foundations and practical relevance.

Contribution

It provides a comprehensive overview of Gaussian Markov models, emphasizing their similarities, differences, and place within the broader hierarchy of Markov models, including recent methodological advances.

Findings

Comparison of directed and undirected Gaussian Markov models

Discussion of modern estimation and regularization techniques

Overview of applications and relaxations of Gaussian assumptions

Abstract

Markov models lie at the interface between statistical independence in a probability distribution and graph separation properties. We review model selection and estimation in directed and undirected Markov models with Gaussian parametrization, emphasizing the main similarities and differences. These two model classes are similar but not equivalent, although they share a common intersection. We present the existing results from a historical perspective, taking into account the amount of literature existing from both the artificial intelligence and statistics research communities, where these models were originated. We cover classical topics such as maximum likelihood estimation and model selection via hypothesis testing, but also more modern approaches like regularization and Bayesian methods. We also discuss how the Markov models reviewed fit in the rich hierarchy of other, higher level Markov model classes. Finally, we close the paper overviewing relaxations of the Gaussian assumption and pointing out the main areas of application where these Markov models are nowadays used.