The normal kernel coupler: an adaptive Markov Chain Monte Carlo method for efficiently sampling from multi-modal distributions

Abstract

The Normal Kernel Coupler (NKC) is an adaptive Markov Chain Monte Carlo (MCMC) method which maintains a set of current state vectors. At each iteration one state vector is updated using a density estimate formed by applying a normal kernel to the full set of states.We give proofs showing that this sampler is ergodic (irreducible, Harris recurrent and aperiodic) for any continuous distribution on d-dimensional Euclidean space. We also show that the NKC outperforms standard MCMC methods on a variety of uni-modal and bimodal problems in low to moderate dimensions.Further, we address practical issues in using the NKC by giving direction for the selection of various parameters and by providing a run-length diagnostic. Using these we give a systematic method for initializing the NKC, selecting the kernel variance, and determining the number of MCMC iterations.We demonstrate the utility of the NKC on a problem of current interest in cancer genetics which has two distinct and dissimilar modes and show that the results are consistent with current scientific understanding.Finally, we introduce Hydra, a software library for MCMC. We show how to use Hydra to implement both a variable-at-a-time Metropolis sampler and the NKC for our example problem.