Empirical Bayes approaches to mixture problems and wavelet regression

We consider model selection in a hierarchical Bayes formulation of the sparse normal linear model in which individual variables have, independently, an unknown prior probability of being included in the model. The focus is on orthogonal designs, which are of particular importance in nonparametric regression via wavelet shrinkage. Empirical Bayes estimates of hyperparameters are easily obtained via the EM algorithm, and this approach is contrasted with a recent conditional likelihood proposal. Our model selection approach yields a straightforward method for data dependent threshold selection in wavelet regression. Performance on standard test sets and data examples is encouraging, especially if a translation invariant form of the estimator is used. Since the method produces separate threshold estimates on each wavelet resolution level, it also comfortably handles stationary correlated error structures.