On posterior distribution of Bayesian wavelet thresholding

TL;DR

This paper studies the convergence rate of the posterior distribution in Bayesian wavelet thresholding within Besov spaces, providing insights into its asymptotic behavior without focusing on specific estimators.

Contribution

It offers a nonparametric Bayesian analysis of the posterior distribution's convergence rate, extending previous results on Bayesian estimators to the distribution itself.

Findings

Posterior distribution converges at the same rate as Bayesian estimators.

Analysis applies to general Besov spaces.

Provides theoretical foundation for Bayesian wavelet methods.

Abstract

We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators.