On regret of parametric mismatch in minimum mean square error estimation

This paper studies the effect of parametric mis-match in minimum mean square error (MMSE) estimation. In particular, we consider the problem of estimating the input signal from the output of an additive white Gaussian channel whose gain is fixed, but unknown. The input distribution is known, and the estimation process consists of two algorithms. First, a channel estimator blindly estimates the channel gain using past observations. Second, a mismatched MMSE estimator, optimized for the estimated channel gain, estimates the input signal. We analyze the regret, i.e., the additional mean square error, that is raised in this process. We derive upper-bounds on both absolute and relative regrets. Bounds are expressed in terms of the Fisher information. We also study regret for unbiased, efficient channel estimators, and derive a simple trade-off between Fisher information and relative regret. This trade-off shows that the product of a certain function of relative regret and Fisher information equals the signal-to-noise ratio, independent of the input distribution. The trade-off relation implies that higher Fisher information results to smaller expected relative regret.