Bayesian false discovery rate wavelete shrinkage: theory and applications

The interest in inference in the wavelet domain remains vibrant area of statistical research because of needs of scientific community to process and explore massive data sets. Prime examples are geophysical, biomedical, and internet related data. In this paper we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. This approach had been considered by many researchers and goes back to the early 1990's. Even the early proposal, the universal thresholding, could be interpreted as a test of multiple hypotheses in the wavelet domain. We propose two new approaches to wavelet shrinkage. (i) In the spirit of Efron's work on local false discovery rate, we propose the theoretical counterpart Bayesian Local False Discovery Rate, BLFDR, where the underlying model assumes unknown variances. This approach to wavelet shrinkage can be connected with shrinkage based on Bayes factors. (ii) The second proposal to wavelet shrinkage explored in this paper is Bayesian False Discovery Rate, BaFDR. This proposal is based on ordering of posterior probabilities of hypotheses in Bayesian testing of multiple hypotheses. We demonstrate that both approaches result in a competitive shrinkage methods by contrasting them to some popular shrinkage technique