An adaptive approximate Bayesian computation MCMC with Global-Local proposals

Abstract

In this paper, we address the challenge of Markov Chain Monte Carlo (MCMC) algorithms within the approximate Bayesian Computation (ABC) framework, which often get trapped in local optima due to their inherent local exploration mechanism. We propose a novel Global-Local ABC-MCMC algorithm that combines the ``exploration" capabilities of global proposals with the ``exploitation" finesse of local proposals. We integrate iterated importance resampling into the likelihood-free framework to establish an effective global proposal distribution. For high-dimensional parameter spaces, we optimize the efficiency of the local sampler by leveraging Langevin dynamics and common random numbers. Furthermore, we introduce two adaptive schemes to enhance the algorithmic performance. The first scheme divides the update target of the importance proposal into a sequence of intermediate target distributions that progressively approximate the ABC posterior, thereby gradually updating the importance proposal distribution during the iterations. The second adaptive scheme automatically selects the optimal mixture of global and local moves through sequential optimization, based on a relative version of the expected squared jumping distance (ESJD). We theoretically and numerically demonstrate that our method is able to improve sampling efficiency and achieve more reliable convergence for complex posteriors. We develop a software package that is available at https://github.com/caofff/GL-ABC-MCMC.