Expansion of net correlations in terms of partial correlations

Graphical models are usually employed to represent statistical relationships between pairs of variables when all the remaining variables are fixed. In this picture, conditionally independent pairs are disconnected. In the real world, however, strict conditional independence is almost impossible to prove. Here we use a weaker version of the concept of graphical models, in which only the linear component of the conditional dependencies is represented. This notion enables us to relate the marginal Pearson correlation coefficient (a measure of linear marginal dependence) with the partial correlations (a measure of linear conditional dependence). Specifically, we use the graphical model to express the marginal Pearson correlation $\rho_{ij}$ between variables $X_i$ and $X_j$ as a sum of the efficacies with which messages propagate along all the paths connecting the variables in the graph. The expansion is convergent, and provides a mechanistic interpretation of how global correlations arise from local interactions. Moreover, by weighing the relevance of each path and of each intermediate node, an intuitive way to imagine interventions is enabled, revealing for example what happens when a given edge is pruned, or the weight of an edge is modified. The expansion is also useful to construct minimal equivalent models, in which latent variables are introduced to replace a larger number of marginalised variables. In addition, the expansion yields an alternative algorithm to calculate marginal Pearson correlations, particularly beneficial when partial correlation matrix inversion is difficult. Finally, for Gaussian variables, the mutual information is also related to message-passing efficacies along paths in the graph.