Gibbs posteriors are proportional to a prior distribution multiplied by an exponentiated loss function, with a key tuning parameter weighting information in the loss relative to the prior and providing a control of posterior uncertainty. Gibbs posteriors provide a principled framework for likelihood-free Bayesian inference, but in many situations, including a single tuning parameter inevitably leads to poor uncertainty quantification. In particular, regardless of the value of the parameter, credible regions have far from the nominal frequentist coverage even in large samples. We propose a sequential extension to Gibbs posteriors to address this problem. We prove the proposed sequential posterior exhibits concentration and a Bernstein-von Mises theorem, which holds under easy to verify conditions in Euclidean space and on manifolds. As a byproduct, we obtain the first Bernstein-von Mises theorem for traditional likelihood-based Bayesian posteriors on manifolds. All methods are illustrated with an application to principal component analysis.

翻译：吉布斯后验正比于先验分布乘以指数化损失函数，其中关键调优参数权衡损失相对于先验的信息权重，并控制后验不确定性。吉布斯后验为无似然贝叶斯推断提供了理论框架，但在许多场景中，单一调优参数不可避免地导致不确定性量化效果不佳。具体而言，无论参数取值如何，即使在较大样本量下，可信区域的频率覆盖度也远未达到名义水平。为解决此问题，我们提出了吉布斯后验的序贯扩展。我们证明所提出的序贯后验具有集中性并满足伯恩斯坦-冯·米塞斯定理，该定理在欧氏空间及流形上易于验证的条件下成立。作为副产品，我们首次获得了流形上传统基于似然的贝叶斯后验的伯恩斯坦-冯·米塞斯定理。所有方法均通过主成分分析的应用进行说明。