On Optimality of Bayesian Wavelet Estimators

Abstract

We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space for "p" ≥ 2. For 1 ≤ "p" > 2, the Bayes Factor is still optimal for (2"s"+2)/(2"s"+1) ≤ "p" > 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..

On posterior distribution of Bayesian wavelet thresholding

“…Remark: The above rate of convergence is the same as in Abramovich et al (2004) for posterior mean and posterior median, except an extra log factor in our case, which we think might be an artifact of our proofs.…”

“…This paper intends to fill this gap. Using the same prior as in Abramovich et al (1998Abramovich et al ( , 2004, we show that the posterior distribution has the same convergence rate as the point estimators proposed in those papers.…”

“…This prior is proposed in Abramovich et al (1998), and Abramovich et al (2004) investigated the optimality of some Bayesian estimators with this prior. The choice of the hyperparameters must satisfy some conditions for the prior to put positive mass on B s p,q (Abramovich et al, 1998, Theorem 1), although this is not our focus here.…”

“…In particular, they study the rate of convergence of different point estimators including the posterior mean and posterior median as well as other estimators derived from the posterior distribution. The theoretical results in Abramovich et al (2004) show that some Bayesian estimators can achieve the better-than-linear rates if an appropriate prior is chosen that implicitly implements shrinkage or thresholding rule similar to the frequentist approach.…”

“…Bayesian approach to function estimation in Besov spaces has been investigated in Abramovich et al (1998Abramovich et al ( , 2004. In these approaches, after specifying an appropriate prior, the Bayesian estimator is obtained from the posterior and investigated from the frequentist point of view.…”