Non-Euclidean spaces routinely arise in modern statistical applications such as in medical imaging, robotics, and computer vision, to name a few. While traditional Bayesian approaches are applicable to such settings by considering an ambient Euclidean space as the parameter space, we demonstrate the benefits of integrating manifold structure into the Bayesian framework, both theoretically and computationally. Moreover, existing Bayesian approaches which are designed specifically for manifold-valued parameters are primarily model-based, which are typically subject to inaccurate uncertainty quantification under model misspecification. In this article, we propose a robust model-free Bayesian inference for parameters defined on a Riemannian submanifold, which is shown to provide valid uncertainty quantification from a frequentist perspective. Computationally, we propose a Markov chain Monte Carlo to sample from the posterior on the Riemannian submanifold, where the mixing time, in the large sample regime, is shown to depend only on the intrinsic dimension of the parameter space instead of the potentially much larger ambient dimension. Our numerical results demonstrate the effectiveness of our approach on a variety of problems, such as reduced-rank multiple quantile regression, principal component analysis, and Fr\'{e}chet mean estimation.

翻译：在现代统计应用中，非欧几里得空间经常出现，例如医学成像、机器人学和计算机视觉等领域。尽管传统的贝叶斯方法可以通过将环境欧几里得空间视为参数空间来适用于这些场景，但我们从理论和计算两方面证明了将流形结构整合到贝叶斯框架中的优势。此外，现有的专为流形值参数设计的贝叶斯方法主要基于模型，在模型误设情况下通常会导致不准确的不确定性量化。在本文中，我们提出了一种针对定义在黎曼子流形上的参数的鲁棒无模型贝叶斯推断方法，该方法从频率学派角度提供了有效的不确定性量化。在计算方面，我们提出了一种马尔可夫链蒙特卡洛方法，用于在黎曼子流形上从后验分布采样，在大样本情况下，其混合时间仅取决于参数空间的内在维度，而非可能大得多的环境维度。我们的数值结果在多种问题上验证了该方法的有效性，例如降秩多重分位数回归、主成分分析和弗雷歇均值估计。