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：我们提出了一种具有理论依据且实际可行的基于切片采样的马尔可夫链蒙特卡洛方法，用于近似采样黎曼流形上的概率测度。该类测度天然地出现在矩阵值参数（例如属于Stiefel流形或Grassmann流形）的贝叶斯推断后验分布中。所提方法——称为测地线切片采样——对目标分布具有可逆性，并通过采用测地线替代直线，将$\mathbb{R}^{d}$上的Hit-and-run切片采样推广至黎曼流形。通过大量合成数据与真实数据的数值实验，我们证明了该方法在处理流形值分布时相较于其他MCMC方法的稳健性能。特别地，我们展示了其在无需梯度信息和预条件处理的情况下，对异向目标密度分布具有卓越的适应能力。