On the empirical Bayes approach to adaptive filtering

We consider an empirical Bayes approach to adaptive estimation in a sequence model corresponding, via Faurier transform, to the pointwise recovery of a signal in the continuous Gaussian white noise model. The vell-knovn minimax approach to this problem is closely related to the Bayes filtering of stationary Gaussian processes corrupted by a Gaussian vhite noise. The proposed method of adaptive filtering combines two well-known techniques: the Wiener filter and empirical Bayes approach. Our main purpose is to demonstrate how this method works, in a prototypical nonparametric problem. We also discuss an interesting phenomenon of (Bayesian) under- and oversmoothing.