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

2026/01/16 by Sebastiano Grazzi, Sifan Liu, Gareth O. Roberts +1 · 1 voice · 1 citation
Mathematics · #stat.CO #stat.ME

paper · pdf

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.

Citations

• Central Limit Theorem for ergodic averages of Markov chains & the comparison of sampling algorithms for heavy-tailed distributions
• Stereographic Multi-Try Metropolis Algorithms for Heavy-tailed Sampling
• Sampling with time-changed Markov processes
• Robust Bayesian Model Averaging for Linear Regression Models With Heavy-Tailed Errors
• Methods and applications of PDMP samplers with boundary conditions
• Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality
• Mean-Square Analysis of Discretized Itô Diffusions for Heavy-tailed Sampling
• Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm
• Convergence properties of data augmentation algorithms for high-dimensional robit regression
• Speed Up Zig-Zag
• The Barker proposal: combining robustness and efficiency in gradient-based MCMC
• Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients
• Optimal Scaling of Random-Walk Metropolis Algorithms on General Target Distributions
• Complexity Results for MCMC derived from Quantitative Bounds
• On the Geometric Ergodicity of Hamiltonian Monte Carlo
• Variational Inference: A Review for Statisticians
• On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression
• Adam: A Method for Stochastic Optimization
• Black Box Variational Inference
• Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm
• Bayesian inference for logistic models using Polya-Gamma latent variables
• Bayesian Inference for Logistic Models Using Pólya–Gamma Latent Variables
• The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo
• A weakly informative default prior distribution for logistic and other regression models
• Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution
• Weak convergence and optimal scaling of random walk Metropolis algorithms
• Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms
• Bayesian Analysis of Binary and Polychotomous Response Data

Cited by

• Diffeomorphic Markov Chain Monte Carlo: fast mixing for heavy-tailed distributions

Discussions

• Sebastiano Grazzi, Sifan Liu, Gareth O. Roberts, Jun Yang: Sub-Cauchy Sampling: Escaping the Dark Side of the Moon https://arxiv.org/abs/2601.11066 https://arxiv.org/pdf/2601.11066 https://arxiv.org/h [bsky, 1 points, 0 comments]

Related

• Stereographic Markov Chain Monte Carlo
• Adaptive Stereographic MCMC
• Integral geometry for Markov chain Monte Carlo: overcoming the curse of search-subspace dimensionality
• Monte Carlo on manifolds in high dimensions
• Multiple projection MCMC algorithms on submanifolds