An extremal inequality for long Markov chains

Let X, Y be jointly Gaussian vectors, and consider random variables U, V that satisfy the Markov constraint U - X - Y - V. We prove an extremal inequality relating the mutual informations between all (⁴₂) pairs of random variables from the set (U, X, Y, V). As a first application, we show that the rate region for the two-encoder quadratic Gaussian source coding problem follows as an immediate corollary of the the extremal inequality. In a second application, we establish the rate region for a vector-Gaussian source coding problem where Löwner-John ellipsoids are approximated based on rate-constrained descriptions of the data.