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 version of the algorithm can be implemented such that it runs efficiently and has no tuning parameters. We verify reversibility and prove 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 distributions arising in rigid-registration problems 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重要性，高效采样球面上的分布具有显著意义。我们提出一种基于收缩的测地线切片采样马尔可夫链及其理想化版本，旨在生成球面分布的近似样本。特别地，该算法的收缩版本可实现高效运行且无需调参。我们验证了可逆性，并证明在弱正则性条件下测地线切片采样具有一致遍历性。数值实验表明，所提出的切片采样器在包括刚体配准问题产生的分布和冯·米塞斯-费希尔混合分布在内的挑战性目标上均能实现优异的混合效果。在这些场景中，我们的方法优于随机游走Metropolis-Hastings和哈密顿蒙特卡洛等标准采样器。