Mismatched Estimation and Relative Entropy

A random variable with distribution P is observed in Gaussian noise and is estimated by a mismatched minimum mean-square estimator that assumes that the distribution is Q, instead of P . This paper shows that the integral over all signal-to-noise ratios (SNRs) of the excess mean-square estimation error incurred by the mismatched estimator is twice the relative entropy D(P ||Q) (in nats). This representation of relative entropy can be generalized to nonreal-valued random variables, and can be particularized to give new general representations of mutual information in terms of conditional means. Inspired by the new representation, we also propose a definition of free relative entropy which fills a gap in, and is consistent with, the literature on free probability.