Compound Auxiliary Metropolis: Incorporating Auxiliary Variables into Multi-Candidate MCMC

2026-07-16

AI Summary

This work addresses the challenge of inefficient global exploration in traditional Multiple-Try Metropolis (MTM) algorithms when sampling from complex target distributions—particularly those that are multimodal or severely non-Gaussian—due to poor local transition efficiency. To overcome this limitation, the authors propose the Composite Auxiliary Metropolis (CAM) method, which introduces flexibly designed auxiliary variables within the MTM framework. These auxiliary variables, combined with the current state, generate multiple candidate proposals and enable either state-independent exploration or reference-distribution-guided strategies. Empirical results demonstrate that CAM substantially outperforms standard MTM on multimodal and slowly mixing distributions, and in the most challenging scenarios, even surpasses the No-U-Turn Sampler, effectively mitigating the local trapping issues inherent in conventional approaches.

Abstract

Multiple-try Metropolis (MTM) is a Markov chain Monte Carlo (MCMC) algorithm that improves local transition efficiency by evaluating multiple candidate draws at each iteration. However, for complicated target distributions exhibiting severely non-Gaussian topography or multiple well-separated modes, locally optimal transitions may be insufficient for effective global exploration. In this work, we propose compound auxiliary Metropolis (CAM), a general multi-candidate MCMC method that incorporates both the local state of the chain and auxiliary information into the multi-candidate framework of MTM. Using an auxiliary generating distribution, CAM accommodates a flexible definition of auxiliary information. As examples, we consider three different auxiliary variables: one that promotes state-independent exploration and two that use a reference distribution to improve mixing. These auxiliaries are tested against distributions that present challenging targets for modern MCMC methods. In particular, we focus on the challenges presented by multiple well-separated modes and topography that requires long mixing for local MCMC moves. We find that CAM is able to sample effectively from these distributions, using MTM as a baseline to evaluate the benefit introduced by the auxiliary information. CAM also compares favourably with the No-U-Turn Sampler, showing similar performance for milder test distributions and better performance for the most difficult settings.

Problem

• Markov chain Monte Carlo
• multi-modal distributions
• non-Gaussian topography
• global exploration
• mixing efficiency

Innovation

• Compound Auxiliary Metropolis
• Multiple-try Metropolis
• auxiliary variables
• multi-modal sampling
• Markov chain Monte Carlo