Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

TL;DR

This paper introduces a new reversible Markov chain Monte Carlo algorithm optimized for high-dimensional, heavy-tailed distributions, demonstrating superior convergence properties through theoretical analysis and simulations.

Contribution

The paper proposes a novel MCMC method with a heavy-tailed invariant distribution, showing improved high-dimensional convergence over existing algorithms.

Findings

Outperforms pCN and RWM algorithms in high dimensions for heavy-tailed targets.

At least as effective as pCN for light-tailed distributions.

Demonstrates superior convergence rates through asymptotic theory and simulations.

Abstract

The purpose of this paper is to introduce a new Markov chain Monte Carlo method and exhibit its efficiency by simulation and high-dimensional asymptotic theory. Key fact is that our algorithm has a reversible proposal transition kernel, which is designed to have a heavy-tailed invariant probability distribution. The high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distribution. As the number of dimension of the state space goes to infinity, we will show that our algorithm has a much better convergence rate than that of the preconditioned Crank Nicolson (pCN) algorithm and the random-walk Metropolis (RWM) algorithm. We also show that our algorithm is at least as good as the pCN algorithm and better than the RWM algorithm for light-tailed target probability distribution.