SummaryA graphical model provides a compact and efficient representation of the association structure in a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and higher-order graphical structures such as cliques or paths. Typically, paths play a central role in these models because they determine the dependence relationships between variables. However, while a theory of path coefficients is available for directed graph models, little research exists on the strength of the association represented by a path in an undirected graph. Essentially, it has been shown that the covariance between two variables can be decomposed into a sum of weights associated with each of the paths connecting the two variables in the corresponding concentration graph. In this context, we consider concentration graph models and provide an extensive analysis of the properties of path weights and their interpretation. Specifically, we give an interpretation of covariance weights through their factorization into a partial covariance and an inflation factor. We then extend the covariance decomposition over the paths of an undirected graph to other measures of association, such as the marginal correlation coefficient and a quantity that we call the inflated correlation. Application of these results is illustrated with an analysis of dietary intake networks.

中文翻译：

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浓度图中的路径权重

小结图形模型通过图形提供了多元分布中关联结构的紧凑而有效的表示。分布的相关特征由顶点、边和高阶图形结构（例如团或路径）表示。通常，路径在这些模型中起着核心作用，因为它们决定了变量之间的依赖关系。然而，虽然路径系数理论可用于有向图模型，但关于无向图中路径表示的关联强度的研究很少。本质上，已经表明两个变量之间的协方差可以分解为与相应浓度图中连接两个变量的每个路径相关联的权重总和。在这种情况下，我们考虑了集中图模型，并对路径权重的属性及其解释进行了广泛的分析。具体来说，我们通过将协方差权重分解为部分协方差和膨胀因子来解释协方差权重。然后，我们将无向图路径上的协方差分解扩展到其他关联度量，例如边际相关系数和我们称之为膨胀相关的数量。通过对膳食摄入网络的分析来说明这些结果的应用。然后，我们将无向图路径上的协方差分解扩展到其他关联度量，例如边际相关系数和我们称之为膨胀相关的数量。通过对膳食摄入网络的分析来说明这些结果的应用。然后，我们将无向图路径上的协方差分解扩展到其他关联度量，例如边际相关系数和我们称之为膨胀相关的数量。通过对膳食摄入网络的分析来说明这些结果的应用。