On Regret of Parametric Mismatch in Minimum Mean Square Error Estimation

TL;DR

This paper analyzes how parametric mismatch affects MMSE estimation in Gaussian channels, deriving bounds on the additional error and revealing a trade-off between Fisher information and estimation regret.

Contribution

It introduces bounds on regret due to parametric mismatch in MMSE estimation and establishes a fundamental trade-off involving Fisher information and estimation accuracy.

Findings

Derived upper bounds on absolute and relative regret.

Established a trade-off between Fisher information and relative regret.

Showed that higher Fisher information reduces estimation regret.

Abstract

This paper studies the effect of parametric mismatch 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.