Partial Information Decomposition via Deficiency for Multivariate Gaussians.

We consider the problem of decomposing the information content of three jointly Gaussian random vectors using the partial information decomposition (PID) framework. Barrett previously characterized the Gaussian PID for a scalar "source" or "message" in closed form - we extend this to the case where the message is a vector. Specifically, we revisit a connection between the notions of Blackwell sufficiency of statistical experiments and stochastic degradedness of broadcast channels, to provide a necessary and sufficient condition for the existence of unique information in the fully multivariate Gaussian PID. The condition we identify indicates that the closed form PID for the scalar case rarely extends to the vector case. We also provide a convex optimization approach for approximating a PID in the vector case, analyze its properties, and evaluate it empirically on randomly generated Gaussian systems.