Adaptive Bayesian estimation in indirect Gaussian sequence space models

TL;DR

This paper develops an adaptive Bayesian approach for indirect Gaussian sequence space models, providing optimal and minimax concentration rates for the posterior distribution, and constructing a hierarchical prior that adapts to unknown parameters or classes.

Contribution

It introduces an adaptive hierarchical prior that achieves oracle and minimax rates without prior knowledge of the true parameter or class.

Findings

Derived bounds for posterior concentration rates.

Constructed an adaptive hierarchical prior.

Proved convergence of the Bayesian estimator at optimal rates.

Abstract

In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value $\theta^\circ$ that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both $\theta^\circ$ and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each $\theta^\circ$ separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive. This means that, given a parameter $\theta^\circ$ or a class of parameters, respectively, the posterior distribution contracts at the oracle rate or at the minimax rate over the class. Notably, the hierarchical prior does not depend neither on $\theta^\circ$ nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.