Multiple shrinkage and subset selection in wavelets

This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data compression, via dimension reduction, and for denoising, via shrinkage of the coefficients. In both areas, there is need for systematic guidance in the choice of subsets, and for assessment and incorporation of the uncertainty associated with the selection process. This motivates looking at a formal probabilistic treatment of uncertainty in the selection of subsets of basis elements. We do this using a Bayesian hierarchical model, assigning a probability distribution to the space of subsets of basis elements. This defines a multiple shrinkage estimator for the wavelet coefficients that is based on subset mixing. We show how the induced shrinkage rule can mimic many standard thresholding policies. In addition, the subset mixing shrinkage paradigm gives an efficient way of incorporating prior information about the problem. The degree of shrinkage can be calibrated based on a number of prior hyperparamters, whose specification is discussed in detail. One drawback is the higher computational burden, compared to many competing estimators. We discuss fast computational implementations. In particular, we consider easy-to-compute analytic approximations as well as importance sampling and MCMC methods. We illustrate our approach with an application. Some key words: Wavelet shrinkage, Variable Selection, Importance sampling, Gibbs sampling. 1. Introduction This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are a family of basis or basis-like functions that can be used to approximate other functions. They combine powerful properties such as orthonormality, compact support, varying degrees of smoothness, localiz...