Manifold-valued parameters 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 muchlarger ambient dimension. Our numerical results demonstrate the effectiveness of our approach on a variety of problems, such as multiple quantile regression, reduced-rank regression, and Fréchet mean estimation.

翻译：流形值参数在现代统计应用中频繁出现，例如医学成像、机器人学和计算机视觉等领域。虽然传统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此类场景，但我们从理论和计算两方面论证了将流形结构整合到贝叶斯框架中的优势。此外，现有专门针对流形值参数设计的贝叶斯方法主要是基于模型的，在模型设定错误的情况下通常存在不确定性量化不准确的问题。本文提出了一种针对黎曼子流形上定义的参数的鲁棒无模型贝叶斯推断方法，该方法从频率学派的视角被证明能够提供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，用于从黎曼子流形上的后验分布中采样，其中在大样本体系下，混合时间被证明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。我们的数值结果展示了该方法在多种问题上的有效性，例如多重分位数回归、降秩回归和弗雷歇均值估计。