Sub-Cauchy Sampling: Escaping the Dark Side of the Moon

TL;DR

This paper presents Sub-Cauchy Sampling, a new MCMC algorithm that effectively samples from heavy-tailed distributions by leveraging a geometric transformation, improving Bayesian inference in challenging high-dimensional problems.

Contribution

The paper introduces a novel Sub-Cauchy Projection-based MCMC method that is proven to be uniformly ergodic for heavy-tailed targets, expanding the toolkit for Bayesian modeling with such distributions.

Findings

Proves uniform ergodicity for sub-Cauchy targets.

Demonstrates empirical effectiveness in high-dimensional problems.

Offers a simple, broadly applicable approach for heavy-tailed Bayesian inference.

Abstract

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.