 
Summary: On Optimality of Bayesian Wavelet
Estimators
FELIX ABRAMOVICH
Tel Aviv University
UMBERTO AMATO and CLAUDIA ANGELINI
Istituto per le Applicazioni del Calcolo
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 Bs
p;q for p 2. For 1 £ p < 2, the Bayes Factor is still optimal for (2s+2)/(2s+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.
Key words: Bayes Factor, Bayes model, Besov spaces, minimax estimation, nonlinear
estimation, nonparametric regression, posterior mean, posterior median, wavelets
1. Introduction
Consider the standard nonparametric regression model:
yi ¼ f
