Wavelet Thresholding via a Bayesian Approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coe cients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior speci ed, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any speci c Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coe cients. However, prior knowledge about a function's regularity properties might be hard to elicit; with this in mind, we propose a standard choise of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set collected in an anaesthesiological study.