Covariance decomposition in multivariate analysis

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 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. The novel theory underlying this covariance decomposition is derived using basic linear algebra. The resulting weights can be interpreted as involving two components: a measure of the strengths of the relationships between the variables along the path, conditioning on all other variables; and a function of the entropy lost in that conditioning. The decomposition is feasible for very large numbers of variables if the corresponding precision matrix is sparse, a circumstance that arises in currently topical 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.