We establish a general Bernstein--von Mises theorem for approximately linear semiparametric functionals of fractional posterior distributions based on nonparametric priors. This is illustrated in a number of nonparametric settings and for different classes of prior distributions, including Gaussian process priors. We show that fractional posterior credible sets can provide reliable semiparametric uncertainty quantification, but have inflated size. To remedy this, we further propose a \textit{shifted-and-rescaled} fractional posterior set that is an efficient confidence set having optimal size under regularity conditions. As part of our proofs, we also refine existing contraction rate results for fractional posteriors by sharpening the dependence of the rate on the fractional exponent.

翻译：本文针对基于非参数先验的分数后验分布中近似线性半参函数，建立了广义的伯恩斯坦-冯·米塞斯定理。该定理在多种非参数设定下及不同先验分布类别（包括高斯过程先验）中得到了验证。研究表明，分数后验置信集虽能提供可靠的半参数不确定性量化，但其规模存在膨胀效应。为此，我们进一步提出一种"平移-重缩放"分数后验集，该集合在正则条件下可作为具有最优规模的置信集。作为证明过程的一部分，我们还通过优化分数指数对收敛速度的依赖关系，改进了现有分数后验收缩率结果。