| Summary: | Conditional independence models associated with directed acyclic
graphs (DAGs) may be characterized in at least three different ways:
via a factorization, the global Markov property (given by the dseparation criterion), and the local Markov property. 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 are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination.
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
ordered local and global Markov properties, and a factorization, a
graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in
this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that
the fixing operation used to define the model leads to a particularly
simple characterization of identifiable causal effects in hidden variable
causal DAG models.
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