Adaptive Bayesian inference in the Gaussian sequence model using exponential-variance priors

Abstract

We revisit the problem of estimating the mean of an infinite dimensional normal distribution in a Bayesian paradigm. Of particular interest is obtaining adaptive estimation procedures so that the posterior distribution attains optimal rate of convergence without the knowledge of the true smoothness of the underlying parameter of interest. Belitser & Ghosal (2003) studied a class of power-variance priors and obtained adaptive posterior convergence rates assuming that the underlying smoothness lies inside a countable set on which the prior is specified. In this article, we propose a different class of exponential-variance priors, which leads to optimal rate of posterior convergence (up to a logarithmic factor) adaptively over all the smoothness levels in the positive real line. Our proposal draws a close parallel with signal estimation in a white noise model using rescaled Gaussian process prior with squared exponential covariance kernel.