We propose a theoretically justified and practically applicable slice sampling based Markov chain Monte Carlo (MCMC) method for approximate sampling from probability measures on Riemannian manifolds. The latter naturally arise as posterior distributions in Bayesian inference of matrix-valued parameters, for example belonging to either the Stiefel or the Grassmann manifold. Our method, called geodesic slice sampling, is reversible with respect to the distribution of interest, and generalizes Hit-and-run slice sampling on $\mathbb{R}^{d}$ to Riemannian manifolds by using geodesics instead of straight lines. We demonstrate the robustness of our sampler's performance compared to other MCMC methods dealing with manifold valued distributions through extensive numerical experiments, on both synthetic and real data. In particular, we illustrate its remarkable ability to cope with anisotropic target densities, without using gradient information and preconditioning.

翻译：我们提出了一种理论上合理且实际可用的基于切片采样的马尔可夫链蒙特卡洛（MCMC）方法，用于近似采样黎曼流形上的概率测度。后者自然地出现在矩阵参数贝叶斯推断中的后验分布中，例如属于Stiefel流形或Grassmann流形的参数。我们的方法称为测地盘切片采样，对目标分布具有可逆性，并通过使用测地线代替直线，将$\mathbb{R}^{d}$上的Hit-and-run切片采样推广到黎曼流形上。通过在合成数据和真实数据上进行的大量数值实验，我们证明了我们的采样器在处理流形值分布方面与其他MCMC方法相比的稳健性能。特别地，我们展示了其在无需梯度信息和预处理的情况下应对各向异性目标密度的卓越能力。