Bayesian False Discovery Rate Wavelet Shrinkage: Theory and Applications

Statistical inference in the wavelet domain remains vibrant area of contemporary statistical research because desirable properties of wavelet representations and the need of scientific community to process, explore, and summarize massive data sets. Prime examples are biomedical, geophysical, and internet related data. In this paper we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. The shrinkage/thresholding approach by implicit or explicit simultaneous testing of many hypotheses 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/thresholding. (i) In the spirit of Efron and Tibshirani’s recent 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 also 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 techniques.