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 $\left(\genfrac{}{}{0px}{}{4}{2}\right)$ 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 L"{o}wner-John ellipsoids are approximated based on rate-constrained descriptions of the data.