A characterization of moral transitive acyclic directed graph Markov models as labeled trees

It follows from the known relationships among the different classes of graphical Markov models for conditional independence that the intersection of the classes of moral acyclic directed graph Markov models (or decomposable ≡ DEC Markov models), and transitive acyclic directed graph ≡ TDAG Markov models (or lattice conditional independence ≡ LCI Markov models) is non-empty. This paper shows that the conditional independence models in the intersection can be characterized as labeled trees. This fact leads to the definition of a specific Markov property for labeled trees and therefore to the introduction of labeled trees as part of the family of graphical Markov models.