Stereographic Markov Chain Monte Carlo

Abstract

High dimensional distributions, especially those with heavy tails, are notoriously difficult for off the shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves, results in empirically observed "stickiness" and poor theoretical mixing properties -lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the "blessings of dimensionality" that the mixing time decreases with dimension.