Wavelet thresholding via a Bayesian approach

F. Abramovich, T. Sapatinas*, B. W. Silverman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

Original languageEnglish
Pages (from-to)725-749
Number of pages25
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Volume60
Issue number4
DOIs
StatePublished - 1998

Keywords

  • Adaptive estimation
  • Anaesthetics
  • Bayes model
  • Besov spaces
  • Nonparametric regression
  • Thresholding
  • Wavelet transform

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