Adaptive Bernstein–von Mises theorems in Gaussian white noise
研究了高斯白噪声模型中自适应非参数贝叶斯过程的伯恩斯坦-冯·米塞斯定理，为基于后验分布构建最优频率学派置信集提供了理论依据，并通过模拟展示了几何特性。
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.