Nested Markov Properties for Acyclic Directed Mixed Graphs
Thomas Richardson, James Robins, Ilya Shpitser
Directed acyclic graph (DAG) models may be characterized in four different ways: via a factorization, the dseparation criterion, the moralization criterion, and the local Markov property. As pointed out by Robins [2, 1], Verma and Pearl , and Tian and Pearl , marginals of DAG models also imply equality constraints that are not conditional independences. The well-known `Verma constraint' is an example. Constraints of this type were used for testing edges , and an efficient variable elimination scheme . Using acyclic directed mixed graphs (ADMGs) we provide a graphical characterization of the constraints given in  via a nested Markov property that uses a `fixing' transformation on graphs. We give four characterizations of our nested model that are analogous to those given for DAGs. We show that marginal distributions of DAG models obey this property.
PDF Link: /papers/12/p13-richardson.pdf
AUTHOR = "Thomas Richardson
and James Robins and Ilya Shpitser",
TITLE = "Nested Markov Properties for Acyclic Directed Mixed Graphs",
BOOKTITLE = "Proceedings of the Twenty-Eighth Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-12)",
PUBLISHER = "AUAI Press",
ADDRESS = "Corvallis, Oregon",
YEAR = "2012",
PAGES = "13--13"