Markov Properties for Loopless Mixed Graphs

In this paper we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs (LMG), and show that all of the corresponding independence models induced by $m$-separation are compositional graphoids. Special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as maximal ancestral graphs. We define maximality of such graphs as well as a pairwise and a global Markov property. We prove for a maximal LMG that the global and pairwise Markov properties are equivalent for any independence model which is a compositional graphoid.