Nested Markov Properties for Acyclic Directed Mixed Graphs
Abstract
Directed acyclic graph (DAG) models may be characterized in at least four different ways: via a factorization, the dseparation criterion, the moralization criterion, and the local Markov property. As pointed out by Robins (1986, 1999), Verma and Pearl (1990), and Tian and Pearl (2002b), marginals of DAG models also imply equality constraints that are not conditional independences. The wellknown `Verma constraint' is an example. Constraints of this type were used for testing edges (Shpitser et al., 2009), and an efficient marginalization scheme via variable elimination (Shpitser et al., 2011). We show that equality constraints like the `Verma constraint' can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via Markov properties and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We show that marginal distributions of DAG models lie in this model, prove that a characterization of these constraints given in (Tian and Pearl, 2002b) gives an alternative definition of the model, and finally show that the fixing operation we used to define the model can be used to give a particularly simple characterization of identifiable causal effects in hidden variable graphical causal models.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.06686
 Bibcode:
 2017arXiv170106686R
 Keywords:

 Statistics  Methodology
 EPrint:
 67 pages (not including appendix and references), 8 figures