Haar-Weave-Metropolis kernel

TL;DR

This paper introduces the Haar-Weave-Metropolis kernel, a novel Markov chain Monte Carlo method combining deterministic transforms with Haar measure to improve robustness and convergence, especially for heavy-tailed distributions.

Contribution

It proposes a new Markov kernel that integrates deterministic transforms preserving structure with Haar measure, enhancing global and local information utilization.

Findings

Outperforms existing methods in effective sample size

Achieves higher mean square jump distance per second

Demonstrates robustness for heavy-tailed target distributions

Abstract

Recently, many Markov chain Monte Carlo methods have been developed with deterministic reversible transform proposals inspired by the Hamiltonian Monte Carlo method. The deterministic transform is relatively easy to reconcile with the local information (gradient etc.) of the target distribution. However, as the ergodic theory suggests, these deterministic proposal methods seem to be incompatible with robustness and lead to poor convergence, especially in the case of target distributions with heavy tails. On the other hand, the Markov kernel using the Haar measure is relatively robust since it learns global information about the target distribution introducing global parameters. However, it requires a density preserving condition, and many deterministic proposals break this condition. In this paper, we carefully select deterministic transforms that preserve the structure and create a Markov kernel, the Weave-Metropolis kernel, using the deterministic transforms. By combining with the Haar measure, we also introduce the Haar-Weave-Metropolis kernel. In this way, the Markov kernel can employ the local information of the target distribution using the deterministic proposal, and thanks to the Haar measure, it can employ the global information of the target distribution. Finally, we show through numerical experiments that the performance of the proposed method is superior to other methods in terms of effective sample size and mean square jump distance per second.