Gaussian Graphical Models

Abstract

This chapter describes graphical models for multivariate continuous data based on the Gaussian (normal) distribution. We gently introduce the undirected models by examining the partial correlation structure of two sets of data, one relating to meat composition of pig carcasses and the other to body fat measurements. We then give a concise exposition of the model theory, covering topics such as maximum likelihood estimation using the IPS algorithm, hypothesis testing, and decomposability. We also explain the close relation between the models and linear regression models. We describe various approaches to model selection, including stepwise selection, the glasso algorithm and the SIN algorithm and apply these to the example datasets. We then turn to directed Gaussian graphical models that can be represented as DAGs. We explain a key concept, Markov equivalence, and describe how certain mixed graphs called pDAGS and essential graphs are used to represent equivalence classes of models. We describe various model selection algorithms for directed Gaussian models, including PC algorithm, the hill-climbing algorithm, and the max-min hill-climbing algorithm and apply them to the example datasets. Finally, we briefly describe Gaussian chain graph models and illustrate use of a model selection algorithm for these models.