Adaptive rates of contraction of posterior distributions in Bayesian wavelet regression

• Posterior rate of contraction is nearly optimal when parameters of the prior depend on level.
• Adaptive estimation with nearly optimal rates is achieved by a hyperprior.
• Both normal and double exponential priors achieve nearly optimal rates.

In the last decade, many authors studied asymptotic optimality of Bayesian wavelet estimators such as the posterior median and the posterior mean. In this paper, we consider contraction rates of the posterior distribution in Bayesian wavelet regression in L2/l2L2/l2 neighborhood of the true parameter, which lies in some Besov space. Using the common spike-and-slab-type of prior with a point mass at zero mixed with a Gaussian distribution, we show that near-optimal rates (that is optimal up to extra logarithmic terms) can be obtained. However, to achieve this, we require that the ratio between the log-variance of the Gaussian prior component and the resolution level is not constant over different resolution levels. Furthermore, we show that by putting a hyperprior on this ratio, the approach is adaptive in that knowledge of the value of the smoothness parameter is no longer necessary. We also discuss possible extensions to other priors such as the sieve prior.