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表征：通过因子分解、全局马尔可夫性质（由d-分离准则给出）以及局部马尔可夫性质。DAG模型的边缘分布还蕴含非条件独立的等式约束，著名的“Verma约束”即为其中一例。此类约束被用于检验边的存在性，以及通过变量消去法实现计算高效的边缘化方案。我们证明，“Verma约束”这类等式约束可被视为核对象中的条件独立性，这些核对象通过一种推广了条件化与边缘化的固定操作从联合分布中导出。我们利用这些约束，通过有序局部与全局马尔可夫性质以及因子分解，定义了与有向无环混合图（ADMG）关联的图模型。我们证明DAG模型的边缘分布属于该模型，且由Tian给出的一组此类约束为该模型提供了另一种等价定义。最后，我们展示用于定义模型的固定操作可导出隐变量因果DAG模型中可识别因果效应的一种极简表征。