According to both domain expertise knowledge and empirical evidence, wavelet coefficients of real signals typically exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). A wavelet de-noising approach that takes into account such features of the signal may in practice outperform other, more vanilla methods, both in terms of the estimation error and visual appearance of the estimates. Motivated by this observation, we present a Bayesian approach to wavelet de-noising, where dependencies between neighbouring wavelet coefficients are a priori modelled via a Markov chain-based prior, that we term 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). We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.
Several Bayesian approaches to wavelet de-noising are discussed in @cite_17 , pp. 412--415 and 426--428. However, the method that gained the greatest acclaim in the wavelet de-noising context is the empirical Bayes method of @cite_15 , which we will refer to as EBayes. EBayes relies on the spike-and-slab prior, with its hyperparameters optimised by maximising the marginal likelihood, see @cite_31 . The simulation studies in @cite_15 demonstrate overall excellent performance of EBayes in wavelet de-noising, and Bayesian point estimates resulting from it possess a natural shrinkage property. In fact, the coefficients @math 's can even be estimated exactly as zero if the posterior median is used as a point estimate, and in that case the solution to the estimation problem is truly sparse. We thus consider EBayes as a benchmark in this article. This is in line with earlier works using EBayes as a benchmark in the sparse normal means model, see, e.g., @cite_11 and @cite_9 , who study the horseshoe prior.
@cite_31 @cite_9 @cite_17 @cite_15 @cite_11
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This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold. Details of the calculations needed for implementing the procedure are included. In practice, the estimates are quick to compute and there is software available. Simulations on the standard model functions show excellent performance, and applications to data drawn from various fields of application are used to explore the practical performance of the approach. By using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence, overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of Besov classes, covering also the case of f of bounded variation. The rates obtained are optimal for any value of the parameter p in (0, ∞], simultaneously for a wide range of loss functions, each dominating the Lq norm of the σth derivative, with σ ≥ 0 and 0 < q ≤ 2. Attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points, and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points. Results for all relevant combinations of these scenarios are obtained. In some cases a key feature of the theory is a particular boundary-corrected wavelet basis, details of which are discussed. Overall, the approach described seems so far unique in combining the properties of fast computation, good theoretical properties and good performance in simulations and in practice. A key feature appears to be that the estimate of sparsity adapts to three different zones of estimation, first where the signal is not sparse enough for thresholding to be of benefit, second where an appropriately chosen threshold results in substantially improved estimation, and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate. An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density y, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are nearly black sequences where only a proportion η is allowed to be nonzero, and sequences with normalized p norm bounded by η, for η > 0 and 0 1. Simulations show excellent performance. For appropriately chosen functions y, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. Received 28 May 2002; revised 18 September 2002; accepted 24 September 2002; published 14 February 2003. [1] When an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site, one constraint is that the unknown function is known to be everywhere nonnegative. In this work, we develop a statistically rigorous method for enforcing function nonnegativity in Bayesian inverse problems. The proposed method behaves similarly to a Gaussian process with a linear variogram (i.e., unrestricted Brownian motion) for parameter values significantly greater than zero. The method uses the method of images to define a prior probability density function based on reflected Brownian motion that implicitly enforces nonnegativity. This work focuses on problems where the unknown is a function of a single variable (e.g., time). A Markov chain Monte Carlo (MCMC) method, specifically, a highly efficient Gibbs sampler, is implemented to generate conditional realizations of the unknown function. The new method is applied to the estimation of the trichloroethylene (TCE) and perchloroethylene (PCE) contamination history in an aquifer at Dover Air Force Base, Delaware, based on concentration profiles obtained from an underlying aquitard. INDEX TERMS: 1831 Hydrology: Groundwater quality; 1869 Hydrology: Stochastic processes; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: stochastic inverse modeling, contaminant source identification, inference under constraints, Markov chain Monte Carlo (MCMC), Gibbs sampling, Bayesian inference