Covariance decomposition in undirected Gaussian graphical models

The covariance between two variables in a multivariate Gaussian distribution is decomposed into a sum of path weights for all paths connecting the two variables in an undirected independence graph. These weights are useful in determining which variables are important in mediating correlation between the two path endpoints. The decomposition arises in undirected Gaussian graphical models and does not require or involve any assumptions of causality. This covariance decomposition is derived using basic linear algebra. The decomposition is feasible for very large numbers of variables if the corresponding precision matrix is sparse, a circumstance that arises in examples such as gene expression studies in functional genomics. Additional computational efficiencies are possible when the undirected graph is derived from an acyclic directed graph.