Expansion of marginal correlations in terms of partial correlations

The marginal correlation between two variables is a measure of the linear dependence of one variable on the other. The two original variables need not interact directly, because marginal correlation may arise from the mediation of other variables in the system. The underlying network of direct interactions can be captured by a weighted graphical model. The connection between two variables can be weighted by their partial correlation, defined as the residual correlation left after accounting for the linear effects of mediating variables. While matrix inversion can be used to obtain marginal correlations from partial correlations, in large systems this approach does not reveal how the former emerge from the latter. Here we present an expansion of marginal correlations in terms of partial correlations, which shows that the effect of mediating variables can be quantified by the weight of the paths in the graphical model that connect the original pair of variables. The expansion is proved to converge for arbitrary probability distributions. The graphical interpretation reveals a close connection between the topology of the graph and the marginal correlations. Moreover, the expansion shows how marginal correlations change when some variables are severed from the graph, and how partial correlations change when some variables are marginalized out from the description. It also establishes the minimum number of latent variables required to replicate the exact effect of a collection of variables that are marginalized out, ensuring that the partial and marginal correlations of the remaining variables remain unchanged. Notably, the number of latent variables may be significantly smaller than the number of variables that they effectively replicate. Finally, for Gaussian variables, marginal correlations are shown to be related to the efficacy with which information propagates along the paths in the graph.