On optimality of Bayesian wavelet estimators

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 $B^{s}_{p,q}$ for $p \geq 2$. For $1 \leq p < 2$, the Bayes Factor is still optimal for $(2s+2)/(2s+1) \leq p < 2$ and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case.