Divide and Conquer: A Mixture-Based Approach to Regional Adaptation for MCMC

The efficiency of Markov chain Monte Carlo (MCMC) algorithms can vary dramatically with the choice of simulation parameters. Adaptive MCMC (AMCMC) algorithms allow the automatic tuning of the parameters while the simulation is in progress. A multimodal target distribution may call for regional adaptation of Metropolis-Hastings samplers so that the proposal distribution varies across regions in the sample space. Establishing such a partition is not straightforward and, in many instances, the learning required for its specification takes place gradually, as the simulation proceeds. In the case in which the target distribution is approximated by a mixture of Gaussians, we propose an adaptation process for the partition. It involves fitting the mixture using the available samples via an online EM algorithm and, based on the current mixture parameters, constructing the regional adaptive algorithm with online recursion (RAPTOR). The method is compared with other regional AMCMC samplers and is tested on simulated as well as real data examples. Relevant theoretical proofs, code and datasets are posted as an online supplement.