A Repelling–Attracting Metropolis Algorithm for Multimodality

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.

Irreversible samplers from jump and continuous Markov processes

“…We also tested our method against a recently considered multimodal setting [66]. In the first setting considered, the target distribution is highly multimodal in 2D with unevenly distributed modes.…”

“…5 and 6. In [66], a repulsing-attracting Metropolis (RAM) sampler was proposed with a structure specifically designed to efficiently handle these types of multimodal distributions. We use this as a gold-standard comparison, since this method was already shown to outperform parallel tempering and alternatives [38] in this setting.…”

“…One designs a proposal distribution to generate samples and uses an accept-reject procedure to ensure that the target distribution is maintained. A focus has been on developing clever proposals [66,33,44] to specify a Markov process with good mixing rates, but traditional methods are often strongly coupled to a specific challenge setting, like multimodal targets [66] or heavy tailed distributions [33]. In practice, one often does not know the structure of the target distribution, which might additionally exhibit a combination of these factors.…”

A framework for adaptive MCMC targeting multimodal distributions

“…Therefore, constructing an MCMC algorithm which enables fast exploration of the state space for complicated target functions is of great interest, especially as multimodal distributions are common in applications. The examples include, but are not limited to, problems in genetics [15,31], astrophysics [20,21,49] and sensor network localisation [28].…”

“…Other common approaches include Metropolis-Hastings algorithms with a special de-sign of the proposal distribution accounting for the necessity of moving between the modes [51,49] and MultiNest algorithms based on nested sampling [20,21].…”