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.
This paper proposes a new approach to sparsity, called the horseshoe estimator, which arises from a prior based on multivariate-normal scale mixtures. We describe the estimator's advantages over existing approaches, including its robustness, adaptivity to different sparsity patterns and analytical tractability. We prove two theorems: one that characterizes the horseshoe estimator's tail robustness and the other that demonstrates a super-efficient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using both real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers obtained by Bayesian model averaging under a point-mass mixture prior. Copyright 2010, Oxford University Press. Abstract This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate choice of prior distribution from this family is affected by the data. It is assumed that the predictors represent distinct observables; the corresponding regression coefficients are assigned independent prior distributions. For each regression coefficient subject to deletion from the model, the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere, that is, a “spike and slab” distribution. The random error component is assigned a normal distribution with mean 0 and standard deviation ... This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the 'relevance vector machine' (RVM), a model of identical functional form to the popular and state-of-the-art 'support vector machine' (SVM). We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages. These include the benefits of probabilistic predictions, automatic estimation of 'nuisance' parameters, and the facility to utilise arbitrary basis functions (e.g. non-'Mercer' kernels). We detail the Bayesian framework and associated learning algorithm for the RVM, and give some illustrative examples of its application along with some comparative benchmarks. We offer some explanation for the exceptional degree of sparsity obtained, and discuss and demonstrate some of the advantageous features, and potential extensions, of Bayesian relevance learning.