结构方程模型的单联系法则
One-connection rule for structural equation models
摘要：
线性结构方程模型是由混合图编码的多元统计模型。特别地，属于固定混合图$G=(V, D,B)$的线性结构方程模型的分布的协方差阵的集合由具有$G$中的每个顶点和边的参数的有理函数参数化。这种合理的参数化自然允许从代数和组合的角度来研究这些模型。事实上，这一观点导致了文献中的一系列结果，主要集中在与可识别性相关的问题和确定协方差之间的关系（即，在高斯消失理想中找到多项式）。到目前为止，这些结果的很大一部分集中在$D$的情况下，有向部分的混合图$G$，是非循环的。这是由于在非循环的情况下，参数化变成多项式，并且存在根据有限和对协方差阵的条目的描述。我们超越了非循环的情况，并给出了一个封闭的形式表达的协方差阵的条目在一个图中的一个连接从$D$通过一些小的操作。这个封闭形式的表达式允许我们证明，如果$G$是简单的，那么参数化映射一般是有限对一的。最后，具有协方差阵的封闭形式表达式允许开发用于系统地探索高斯消失理想中的可能多项式的算法。
英文摘要：
Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph $G=(V, D,B)$ is parameterized by a rational function with parameters for each vertex and edge in $G$. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when $D$, the directed part of the mixed graph $G$, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from $D$ through some small operations. This closed form expression then allows us to show that if $G$ is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.
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