高斯图模型的Trek分离
Trek separation for Gaussian graphical models
摘要：
高斯图模型是正定协方差阵锥的半代数子集。低秩子矩阵对应于随机变量集合上条件独立约束的推广。我们给出了一个精确的图论表征时的协方差阵的子矩阵有小秩的一般类的混合图，其中包括有向无环和无向图作为特殊情况。我们新的Trek分离准则推广了熟悉的$d$ -分离准则。证明是基于跋涉规则，由此产生的矩阵分解和经典的代数组合定理的道路多项式的行列式的扩展。
英文摘要：
Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.
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