Abstract: The main purpose of this article is to study the wavelet shrinkage method from a Bayesian viewpoint. Nonparametric mixed-effects models are proposed and used for interpretation of the Bayesian structure. Bayes and empirical Bayes estimation are discussed. The latter is shown to have the Gauss-Markov type optimality (i.e., BLUP), to be equivalent to a method of regularization estimator (MORE), and to be minimax in a certain class. Characterization of prior and posterior regularity is discussed. The smoothness of posterior estimators is controlled via prior parameters. Computational issues including the use of generalized cross validation are discussed, and examples are presented.
Key words and phrases: Bayesian regression, Besov spaces, best linear unbiased prediction (BLUP), Gauss-Markov estimation, generalized cross validation, nonparametric regression, Sobolev regularization, wavelet shrinkage.