WAVELET THRESHOLDING VIA A BAYESIAN-APPROACH

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response functi on, designed to capture the sparseness of wavelet expansion that is co mmon to most applications. For the prior specified, the posterior medi an yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Be sov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insig ht into the meaning of the Besov space parameters. Moreover, the relat ionship established makes it possible in principle to incorporate prio r knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a f unction's regularity properties might be difficult to elicit; with thi s in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to ill ustrate our method, and comparisons are made with other thresholding m ethods. We also present an application to a data set that was collecte d in an anaesthesiological study.