Graphs and Conditional Independence

Abstract

In this chapter we introduce graphs as mathematical objects, show how to work with them using R and explain how they are related to statistical models. We focus mainly on undirected graphs and directed acyclic graphs (DAGs), but also briefly treat chain graphs, that have both undirected and directed edges. Key concepts such as clique, path, separation, ancestral set, triangulated graph and perfect vertex ordering, and operations such as moralization and triangulation, are described and illustrated through examples using R. Certain models for multivariate data give rise to patterns of conditional independences that can be represented as a graph, the so-called dependence graph. For such a model, the conditional independences that hold can be directly read off the dependence graph. For undirected graphs, we show how this may be done using the graphical property of separation: for DAGS and chain graphs, analogous properties called d-separation and c-separation are used.