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.
翻译:线形结构方程式模型是多变量统计模型,由混合图形编码。特别是,固定混合图形$G=(V,D,B)$(V,D,B)美元,属于线性结构方程式分布模型的共变量矩阵,通过一个理性函数参数参数进行参数参数的参数参数,每个脊椎和边缘参数参数的参数为$G$(G$,V,D,B)$(G)美元。这种理性的对称自然允许从代数和组合角度研究这些模型。事实上,这一观点导致文献结果的集合,主要侧重于与可识别性和确定共性关系有关的问题(即,在Gaussian 混合图形=(V,D,D,B) 等值分布在线性结构矩阵中找到多数值的多数值矩阵。我们从一个系统化的变数矩阵表达式开始, 等值的变数的变数的变数, 等值表示式的变数在最小化中, 等值的变数的变数的变数, 以最小化为最小的变数的变数。