On the robustness of the Bayes and Wiener estimators under model uncertainty

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 a suitable divergence (or distance) 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-based minimax approach to robust estimation. Choosing a set of divergences, called Tau divergence family, we show that the Bayes estimator based on the nominal statistics is the optimal solution. Moreover, in the dynamic case, the optimal offline estimator is the noncausal Wiener filter based on the nominal statistics.