On the empirical Bayes approach to adaptive filtering in the Gaussian model

We discuss empirical Bayes approach to the problem of adaptive estimation of a linear functional of an infinite dimensional normal mean vector. In the continuous setting, this problem corresponds, via Fourier duality, to the pointwise recovery of a signal observed in the white Gaussian noise. The minimax approach to this problem is essentially equivalent to the Bayes filtering of a stationary Gaussian process corrupted by a white Gaussian noise. The proposed method of adaptation essentially combines two classical approaches: the Wiener filter and the empirical Bayes approach. Our main purpose is to demonstrate how this approach works, in a prototypical nonparametric problem. We will also briefly discuss an interesting phenomenon of (Bayesian) underand oversmoothing.