Adaptive Bernstein-von Mises theorems in Gaussian white noise

TL;DR

This paper establishes Bernstein-von Mises theorems for adaptive Bayesian methods in Gaussian white noise models, providing theoretical support for credible sets and confidence intervals in nonparametric settings.

Contribution

It introduces a general framework for adaptive Bernstein-von Mises theorems in Gaussian white noise, justifying plug-in procedures and constructing optimal confidence sets.

Findings

Theoretical Bernstein-von Mises results for adaptive procedures.

Construction of optimal frequentist confidence sets.

Simulations illustrating the geometric aspects of the approach.

Abstract

We investigate Bernstein-von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved.