Redundancy and synergy in multivariate Gaussians via the Blackwell order

The goal of the partial information decomposition (PID) is to quantify the redundant and synergistic information that multiple sources provide about a target. PID has many applications in machine learning, neuroscience, and other fields, but defining and computing it for high-dimensional continuous systems remains challenging. Here, we define a PID for multivariate Gaussian systems using the Blackwell order, which formalizes when one channel is more informative than another. We show that, among existing redundancy measures that apply to any number of sources, Blackwell redundancy is the only one that satisfies a natural set of desiderata. We also prove that Gaussian channels are optimal for extracting both redundant and union information, yielding an exact and efficient PID as well as an intuitive geometric interpretation in terms of nested ellipsoids. Our union information and synergy coincide with the well-known BROJA measures, and we derive closed-form expressions for both in the case of two sources. We demonstrate the scalability of our method on systems with up to a thousand dimensions or sources. Finally, we illustrate the approach on an optimal control problem, uncovering redundant and synergistic interactions between sensor and memory.