Adaptive Bernstein–von Mises theorems in Gaussian white noise
The Annals of Statistics · 2017 · Volume 45 · Issue 6
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.
Keywords
62G08, 62G15, 62G20, Adaptation, Bayesian inference, confidence set, Credible set, posterior asymptotics
Enable JavaScript for citation tools and additional paper interactions.