Bayesian Approach to Wavelet Decomposition and Shrinkage

We consider Bayesian approach to wavelet decomposition. We show how prior knowledge about a function's regularity can be incorporated into a prior model for its wavelet coefficients by establishing a relationship between the hyperparameters of the proposed model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation may be seen as giving insight into the meaning of the Besov space parameters themselves. Furthermore, we consider Bayesian wavelet-based function estimation that gives rise to different types of wavelet shrinkage in non-parametric regression. Finally, we discuss an extension of the proposed Bayesian model by considering random functions generated by an overcomplete wavelet dictionary.