Check here for a new preprint by Frank van der Meulen, Moritz Schauer, Peter Spreij and me on a Bayesian approach to wavelet de-noising with the caravan prior. Julia implementation is available here.
The basic idea is that wavelet coefficients of real signals typically exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). We present a Bayesian approach to wavelet de-noising, that exploits this dependence between neighbouring wavelet coefficients by a priori modelling them via a Markov chain-based prior. We call this prior the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method due to Johnstone and Silverman, see here and here. We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.