A Repelling-Attracting Metropolis Algorithm for Multimodality

TL;DR

The paper introduces the RAM algorithm, a simple yet effective Metropolis-Hastings variant that enhances exploration of multimodal distributions by combining downhill and uphill moves, reducing the need for tuning compared to tempering methods.

Contribution

The RAM algorithm innovatively combines reciprocal and standard Metropolis moves to improve multimodal sampling without complex tuning or intractable computations.

Findings

RAM explores multimodal distributions more efficiently than standard Metropolis.

The algorithm requires less tuning than tempering-based methods.

Demonstrated effectiveness across several examples.

Abstract

Although the Metropolis algorithm is simple to implement, it often has difficulties exploring multimodal distributions. We propose the repelling-attracting Metropolis (RAM) algorithm that maintains the simple-to-implement nature of the Metropolis algorithm, but is more likely to jump between modes. The RAM algorithm is a Metropolis-Hastings algorithm with a proposal that consists of a downhill move in density that aims to make local modes repelling, followed by an uphill move in density that aims to make local modes attracting. The downhill move is achieved via a reciprocal Metropolis ratio so that the algorithm prefers downward movement. The uphill move does the opposite using the standard Metropolis ratio which prefers upward movement. This down-up movement in density increases the probability of a proposed move to a different mode. Because the acceptance probability of the proposal involves a ratio of intractable integrals, we introduce an auxiliary variable which creates a term in the acceptance probability that cancels with the intractable ratio. Using several examples, we demonstrate the potential for the RAM algorithm to explore a multimodal distribution more efficiently than a Metropolis algorithm and with less tuning than is commonly required by tempering-based methods.