There has recently been a great research interest in thresholding methods f
or nonlinear wavelet regression over spaces of smooth functions. Near-minim
ax convergence rates were, in particular, established for simple hard and s
oft thresholding rules over Besov and Triebel bodies. In this paper, we pro
pose a Bayesian approach where the functional properties of the underlying
signal in noise are directly modeled using Besov norm priors on its wavelet
decomposition coefficients. In the context of maximum a posteriori estimat
ion, we first prove that general thresholding rules are obtained in (genera
lized) dual spaces. In this Tikhonov-type regularization framework, we show
that nonstandard soft thresholding estimators are in particular obtained i
n possibly non-Gaussian noise situations. In the case of the minimum mean s
quare error criterion, a Gibbs sampler is finally presented to estimate the
model parameters and the posterior mean estimate of the underlying signal
of interest. (C) 2001 Elsevier Science B.V. All rights reserved.