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 d-separation 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.

翻译：与有向无环图（DAG）相关的条件独立模型可通过至少三种不同方式刻画：基于因子分解、全局马尔可夫性质（由d-分离准则定义）和局部马尔可夫性质。DAG模型的边际分布也会产生非条件独立的等式约束，著名的"Verma约束"即为一例。此类约束可用于检验边结构，并借助变量消除法实现计算高效边际化。我们证明，诸如"Verma约束"的等式约束可被视作核对象中的条件独立性——这些核对象通过一种泛化条件化与边际化的"固定操作"从联合分布中导出。利用这些约束，我们通过有序局部与全局马尔可夫性质及因子分解，定义了与有向无环混合图（ADMG）关联的图模型。我们证明DAG模型的边际分布包含于此模型，且Tian给出的约束集为该模型提供了另一种等价定义。最后，我们证明用于定义模型的固定操作可导出隐藏变量因果DAG模型中因果效应的简洁可识别性刻画。