We introduce a Markov chain Monte Carlo algorithm based on Sub-Cauchy Projection, a geometric transformation that generalizes stereographic projection by mapping Euclidean space into a spherical cap of a hyper-sphere, referred to as the complement of the dark side of the moon. We prove that our proposed method is uniformly ergodic for sub-Cauchy targets, namely targets whose tails are at most as heavy as a multidimensional Cauchy distribution, and show empirically its performance for challenging high-dimensional problems. The simplicity and broad applicability of our approach open new opportunities for Bayesian modeling and computation with heavy-tailed distributions in settings where most existing methods are unreliable.

翻译：我们提出一种基于亚柯西投影的马尔可夫链蒙特卡洛算法，该几何变换通过将欧几里得空间映射至超球面的球冠区域（即月球暗面的补集）来推广球极投影。我们证明所提方法对亚柯西目标具有一致遍历性，此类目标的尾部至多与多维柯西分布同等厚重，并通过实验验证了该方法在处理高维复杂问题时的性能。本方法兼具简洁性与广泛适用性，为现有方法大多失效场景下的重尾分布贝叶斯建模与计算开辟了新途径。