Bayesian wavelet denoising: Besov priors and non-Gaussian noises

There has recently been a great research interest in thresholding methods for nonlinear wavelet regression over spaces of smooth functions. Near-minimax convergence rates were, in particular, established for simple hard and soft thresholding rules over Besov and Triebel bodies. In this paper, we propose 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 estimation, we first prove that general thresholding rules are obtained in (generalized) dual spaces. In this Tikhonov-type regularization framework, we show that nonstandard soft thresholding estimators are in particular obtained in possibly non-Gaussian noise situations. In the case of the minimum mean square 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.