🤖 AI Summary
This work addresses the challenges of inefficient sampling and excessive variance in Monte Carlo estimation arising from scale disparities among modes and low-density regions in multimodal distributions. To overcome these issues, the authors propose a modular Markov chain Monte Carlo (MCMC) method that constructs parallel Markov chains confined to subsets of the target space and combines their estimates via weights derived from inter-subset transition probabilities. By innovatively integrating parallel constrained sampling with a principled weighted fusion mechanism, the approach substantially enhances sampling efficiency and reduces estimator variance, yielding more robust multimodal expectation estimates within simulated annealing frameworks. Theoretical justification is provided through central limit theorem–type results, and numerical experiments—including Bayesian sparse regression with spike-and-slab priors—demonstrate the method’s superior performance.
📝 Abstract
We develop a modular approach to Markov chain Monte Carlo (MCMC) sampling for unnormalized target densities. In this approach, Markov chains are constructed in parallel, each constrained to a subset of the target space. The Monte Carlo estimates from the constrained chains are then combined with appropriate weights, calculated from the transition probabilities between subsets. In addition to the computational advantages arising from its parallelized structure, this modular MCMC approach enables variance reduction for Monte Carlo estimation in settings where sampling from low-density regions is required. We develop a central limit theorem-type result for the resulting Monte Carlo estimates and propose a method for estimating their standard errors. Furthermore, by applying this modular sampling technique to simulated tempering, we propose a method for Monte Carlo estimation of expectations with respect to multimodal target distributions. This approach effectively addresses a well-known challenge of tempering-based methods: sampling efficiency can be greatly reduced when separated modes of the target distribution have different scales. We demonstrate the efficiency of the proposed methods through numerical examples, including one arising from Bayesian sparse regression with a spike-and-slab prior.