Probability measures on the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Due to their widespread use, but also as an algorithmic building block, efficient sampling of distributions on the sphere is highly desirable. We propose a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere. In particular, the shrinkage based algorithm works in any dimension, is straight-forward to implement and has no tuning parameters. We verify reversibility and show that under weak regularity conditions geodesic slice sampling is uniformly ergodic. Numerical experiments show that the proposed slice samplers achieve excellent mixing on challenging targets including the Bingham distribution and mixtures of von Mises-Fisher distributions. In these settings our approach outperforms standard samplers such as random-walk Metropolis Hastings and Hamiltonian Monte Carlo.

翻译：球面上的概率测度构成一类重要的统计模型，例如用于方向数据或形状建模。由于其广泛应用以及作为算法构建模块的需求，高效采样球面分布具有极高价值。本文提出了基于收缩与理想化的测地线切片采样马尔可夫链，旨在从球面分布生成近似样本。特别地，基于收缩的算法适用于任意维度、实现简单且无需调节参数。我们验证了算法的可逆性，并证明在弱正则条件下测地线切片采样具有均匀遍历性。数值实验表明，所提切片采样器在包含Bingham分布与von Mises-Fisher混合分布的挑战性目标上取得了优异混合性能。在这些场景中，我们的方法优于随机游走Metropolis Hastings与哈密顿蒙特卡洛等标准采样器。