Determinantal Generalizations of Instrumental Variables
Determinantal Generalizations of Instrumental Variables
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DOI：
10.1515/jci-2017-0009
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发表时间：
2017-02
影响因子：
1.4
通讯作者：
Luca Weihs;B. Robinson;E. Dufresne;J. Kenkel;Kaie Kubjas Reginald McGee II;McGee II Reginald;Nhan Nguyen;Elina Robeva;M. Drton
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文献类型：
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作者：
Luca Weihs;B. Robinson;E. Dufresne;J. Kenkel;Kaie Kubjas Reginald McGee II;McGee II Reginald;Nhan Nguyen;Elina Robeva;M. Drton
摘要线性结构方程模型利用线性相关性和高斯噪声将随机向量的分量联系起来。每个这样的模型可以自然地与其顶点对应于随机向量的分量的混合图相关联。该图包含表示组件之间的线性关系的有向边，以及对未观察到的混淆进行编码的双向边。我们研究了通用可识别性的问题，也就是说，一个通用的线性和混杂效应的选择是否可以唯一地从所观察到的随机向量的联合协方差矩阵恢复。一个现有的组合标准，建立一般的可识别性是半跋涉标准（HTC），它使用的混合图中的跋涉系统的存在性，迭代发现一般可逆的线性方程组在多项式时间。通过集中在边缘的时间，我们建立了新的充分和新的必要条件，通用的边缘效应扩展的HTC的可识别性。特别是，我们展示了如何边缘系数可以恢复作为子行列式的协方差矩阵，这构成了一个行列式推广的公式时，使用工具变量进行识别。虽然我们的结果并没有完全关闭现有的充分和必要条件之间的差距差距，我们发现，经验上，我们的结果使我们能够证明比现有的最先进的混合图的通用可识别性。
Abstract Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and new necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification. While our results do not completely close the gap between existing sufficient and necessary conditions we find, empirically, that our results allow us to prove the generic identifiability of many more mixed graphs than the prior state-of-the-art.