Adaptive Bernstein-von Mises theorems in Gaussian white noise

TY - JOUR

T1 - Adaptive Bernstein-von Mises theorems in Gaussian white noise

AU - Ray, Kolyan

PY - 2017/12

Y1 - 2017/12

N2 - 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^∞, 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.

AB - 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^∞, 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.

KW - Bayesian inference

KW - posterior asymptotics

KW - adaptation

KW - credible set

KW - confidence set

U2 - 10.1214/16-AOS1533

DO - 10.1214/16-AOS1533

M3 - Article

SN - 0090-5364

VL - 45

SP - 2511

EP - 2536

JO - ANNALS OF STATISTICS

JF - ANNALS OF STATISTICS

IS - 6

ER -