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：与定向环绕图(DAGs)相关的条件独立模型至少可以用三种不同的方式定性:通过一种因数化,全球Markov属性(由d-分离标准制成)和本地Markov属性。DAG模型的边际还意味着不附带条件独立的平等限制;众所周知的“Verma限制”就是一个例子。这种类型的限制用于测试边缘,并通过变量消除在计算上有效的边缘化计划。我们表明,“Verma限制”等平等制约因素可被视为通过一般化调控和边缘化的固定操作从联合分布中获得的内核物体的有条件独立。我们利用这些制约因素,通过订购的当地和全球Markov属性和一种因数化,界定与环绕方向混合图(ADMGs)相关的图形模型。我们证明,DAG模型的边际分布是这一模型的边际分布,而Tian给出的一组这些制约因素提供了模型的替代定义。最后,我们证明,确定模型的固定操作方法导致模型的易变的因果关系。