A characterization of moral transitive directed acyclic graph Markov models as trees and its properties

Abstract

It follows from the known relationships among the dierent classes of graphical Markov models for conditional independence that the intersection of the classes of moral directed acyclic graph models (or decomposable {DEC models), and transitive directed acyclic graph {TDAG models (or lattice conditional independence {LCI models) is non-empty. This paper shows that the conditional independence models in the intersection can be characterized as labeled trees, where every vertex on the tree corresponds to a single random variable. This fact leads to the denition of a specic Markov property for trees and therefore to the introduction of trees as part of the family of graphical Markov Models.