Shpitser Ilya, Evans Robin J, Richardson Thomas S
Department of Computer Science, Johns Hopkins University,
Department of Statistics, University of Oxford,
Uncertain Artif Intell. 2018 Aug;2018.
The conditional independence structure induced on the observed marginal distribution by a hidden variable directed acyclic graph (DAG) may be represented by a graphical model represented by mixed graphs called maximal ancestral graphs (MAGs). This model has a number of desirable properties, in particular the set of Gaussian distributions can be parameterized by viewing the graph as a path diagram. Models represented by MAGs have been used for causal discovery [22], and identification theory for causal effects [28]. In addition to ordinary conditional independence constraints, hidden variable DAGs also induce generalized independence constraints. These constraints form the nested Markov property [20]. We first show that acyclic linear SEMs obey this property. Further we show that a natural parameterization for all Gaussian distributions obeying the nested Markov property arises from a generalization of maximal ancestral graphs that we call maximal arid graphs (MArG). We show that every nested Markov model can be associated with a MArG; viewed as a path diagram this MArG parametrizes the Gaussian nested Markov model. This leads directly to methods for ML fitting and computing BIC scores for Gaussian nested models.
由隐藏变量有向无环图（DAG）在观测边际分布上诱导出的条件独立结构，可以由一种称为最大祖先图（MAG）的混合图所表示的图形模型来表示。该模型具有许多理想的性质，特别是通过将图视为路径图，可以对高斯分布集进行参数化。由MAG表示的模型已被用于因果发现[22]以及因果效应的识别理论[28]。除了普通的条件独立约束外，隐藏变量DAG还会诱导出广义独立约束。这些约束构成了嵌套马尔可夫性质[20]。我们首先表明无环线性结构方程模型服从该性质。进一步地，我们表明，对于所有服从嵌套马尔可夫性质的高斯分布，一种自然的参数化源于我们称为最大干旱图（MArG）的最大祖先图的推广。我们表明每个嵌套马尔可夫模型都可以与一个MArG相关联；将其视为路径图时，这个MArG为高斯嵌套马尔可夫模型进行参数化。这直接引出了用于高斯嵌套模型的最大似然拟合和计算贝叶斯信息准则（BIC）分数的方法。
