  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.
