An Extremal Inequality for Long Markov Chains

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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 4 2 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.