Mismatched estimation and relative entropy in vector Gaussian channels

We derive a novel relation between mismatched estimation and relative entropy (KL divergence) in vector Gaussian channels under the mean squared estimation criterion. This relation includes as special cases several previous results connecting estimation theory and information theory. A direct proof is provided, together with a verification using Gaussian inputs. An interesting relationship between the KL divergence and Fisher divergence is derived as a direct consequence of our work. The relations established here are potentially useful for inference in graphical models and the design of information systems.