Improving the Asymptotic Performance of Markov Chain Monte-Carlo by Inserting Vortices

TL;DR

This paper introduces a novel method to convert reversible Markov chains into non-reversible ones by inserting vortices, which guarantees reduced asymptotic variance and improved MCMC performance.

Contribution

The paper proposes a general graphical method for transforming reversible chains into non-reversible chains with theoretical variance reduction guarantees.

Findings

Non-reversible chains outperform reversible ones in asymptotic variance.

The method applies to chains with non-tree state connectivity.

The approach offers a new direction for enhancing MCMC efficiency.

Abstract

We present a new way of converting a reversible finite Markov chain into a non-reversible one, with a theoretical guarantee that the asymptotic variance of the MCMC estimator based on the non-reversible chain is reduced. The method is applicable to any reversible chain whose states are not connected through a tree, and can be interpreted graphically as inserting vortices into the state transition graph. Our result confirms that non-reversible chains are fundamentally better than reversible ones in terms of asymptotic performance, and suggests interesting directions for further improving MCMC.