ABSTRACT
This chapter considers models based on some of the more common varieties of the mixed graphs, specifically those introduced for two purposes: the unification of directed and undirected graphs; and models that are closed under marginalization. There is much more to say about the nested Markov property, and the nested Markov model it induces. The classes of models defined by undirected and directed graphs are distinct, but they have a non-empty intersection. The set of models in this intersection are the decomposable models, and they are represented by decomposable graphs. The chain graph model is most easily defined via its factorization property, which combines those properties from the directed acyclic and undirected graph cases. A chain graph is a simple mixed graph with directed and undirected edges, such that there are no semi-directed cycles. Variables in a particular component are considered to be ‘on an equal footing’ and an undirected graphical structure is used to model relationships between them.
