Stereographic Markov chain Monte Carlo
Jun Yang, Krzysztof Łatuszyński, Gareth O. Roberts
The Annals of Statistics · 2024 · Volume 52 · Issue 6
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 the 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 convergence is faster in higher dimensions.
Keywords
60J20, 60J25, 65C05, blessings of dimensionality, heavy tailed distributions, Piecewise deterministic Markov processes, Random walk Metropolis, stereographic projection, uniform ergodicity
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