Geodesic slice sampling on the sphere

Probability measures that are constrained to the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Therefore, and as building block methodology, efficient sampling of distributions on the sphere is highly appreciated. 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 algorithmic parameters such that no tuning is necessary. Apart from the verification of reversibility we show under weak regularity conditions on the target distribution that geodesic slice sampling is uniformly geometrically ergodic, i.e., uniform exponential convergence to the target is proven.