Stereographic Multi-Try Metropolis Algorithms for Heavy-tailed Sampling

Abstract: Markov chain Monte Carlo (MCMC) methods for sampling from heavy-tailed distributions present unique challenges, particularly in high dimensions. Multi-proposal MCMC algorithms have recently gained attention for their potential to improve performance, especially through parallel implementation on modern hardware. This paper introduces a novel family of gradient-free MCMC algorithms that combine the multi-try Metropolis (MTM) with stereographic MCMC framework, specifically designed for efficient sampling from heavy-tailed targets. The proposed stereographic multi-try Metropolis (SMTM) algorithm not only outperforms traditional Euclidean MTM and existing stereographic random-walk Metropolis methods, but also avoids the pathological convergence behavior often observed in MTM and demonstrates strong robustness to tuning. These properties are supported by scaling analysis and extensive simulation studies.