Acyclic Linear SEMs Obey the Nested Markov Property.
Acyclic Linear SEMs Obey the Nested Markov Property.
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发表时间：
2018-08
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通讯作者：
I. Shpitser;R. Evans;T. Richardson
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作者：
I. Shpitser;R. Evans;T. Richardson
隐变量有向无环图（DAG）对观察到的边缘分布所产生的条件独立结构可以用一种称为极大祖先图（MAGs）的混合图表示的图模型来表示。该模型具有许多理想的特性，特别是高斯分布集可以通过将图视为路径图来参数化。以mag为代表的模型已用于因果发现[22]，并用于因果效应识别理论[28]。除了普通的条件独立约束外，隐变量dag还会产生广义独立约束。这些约束形成嵌套马尔可夫属性[20]。我们首先证明了非循环线性sem服从这个性质。进一步，我们证明了服从嵌套马尔可夫性质的所有高斯分布的自然参数化是由极大祖先图的推广产生的，我们称之为极大arid图（MArG）。我们证明了每个嵌套马尔可夫模型都可以与MArG相关联；作为一个路径图，这个MArG参数化了高斯嵌套马尔可夫模型。这直接导致了高斯嵌套模型的ML拟合和计算BIC分数的方法。
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.