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. 1 Introduction Consider the standard non-parametric regression problem: y i = g(t i ) + ffl i ; i = 1; : : : ; n; (1.1) and suppose we wish to recover the unknown function g from additive noise ffl i given noisy data y i at discrete point...