On the Robustness of the Bayes and Wiener Estimators under Model Uncertainty

TL;DR

This paper introduces a divergence-based minimax approach for robust estimation, demonstrating that the Bayes estimator is optimal without dynamics and the noncausal Wiener filter is optimal with dynamics under model uncertainty.

Contribution

It develops a new divergence family-based minimax framework for robust estimation, extending classical estimators to uncertain statistical models.

Findings

Bayes estimator is optimal for static signals.

Noncausal Wiener filter is optimal for dynamic signals.

The approach accounts for model uncertainty using Tau-divergence.

Abstract

This paper deals with the robust estimation problem of a signal given noisy observations. We assume that the actual statistics of the signal and observations belong to a ball about the nominal statistics. This ball is formed by placing a bound on the Tau-divergence family between the actual and the nominal statistics. Then, the robust estimator is obtained by minimizing the mean square error according to the least favorable statistics in that ball. Therefore, we obtain a divergence family-based minimax approach to robust estimation. We show in the case that the signal and observations have no dynamics, the Bayes estimator is the optimal solution. Moreover, in the dynamic case, the optimal offline estimator is the noncausal Wiener filter.