Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling

Abstract

Markov chain Monte Carlo (MCMC) methods have found widespread use in many fields of study to estimate the average properties of complex systems, and for posterior inference in a Bayesian framework. Existing theory and experiments prove convergence of well-constructed MCMC schemes to the appropriate limiting distribution under a variety of different conditions. In practice, however this convergence is often observed to be disturbingly slow. This is frequently caused by an inappropriate selection of the proposal distribution used to generate trial moves in the Markov Chain. Here we show that significant improvements to the efficiency of MCMC simulation can be made by using a self-adaptive Differential Evolution learning strategy within a population-based evolutionary framework. This scheme, entitled Differential Evolution Adaptive Metropolis or DREAM, runs multiple different chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution in randomized subspaces during the search. Ergodicity of the algorithm is proved, and various examples involving nonlinearity, highdimensionality, and multimodality show that DREAM is generally superior to other adaptive MCMC sampling approaches. The DREAM scheme significantly enhances the applicability of MCMC simulation to complex, multi-modal search problems.

Refined System Parameters and TTV Study of Transiting Exoplanetary System Hat-P-20

“…In order to improve the efficiency of MCMC simulation, Vrugt et al (2009) proposed a self-adaptive randomized subspace sampling algorithm DREAM based on the DE-MC algorithm of ter Braak (2006). The applicability of DREAM to complex, multi-modal search problems is generally superior to most MCMC sampling approaches because this scheme runs multiple different chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution in randomized subspace sampling (Vrugt et al 2009).…”

“…The applicability of DREAM to complex, multi-modal search problems is generally superior to most MCMC sampling approaches because this scheme runs multiple different chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution in randomized subspace sampling (Vrugt et al 2009). …”

“…Therefore, accounting for the effects of stellar activity on transit light curves becomes essential for acquiring accurate physical parameters and appropriately characterizing transiting exoplanetary systems. Based on previous work of Tregloan-Reed et al (2013, 2015 and Montalto et al (2014), we develop a highly efficient tool by employing the DREAM algorithm  (Vrugt et al 2009) to simultaneously model multiple transiting light curves deformed by spots in the stellar photosphere. Utilizing our new tool, we comprehensively analyze the published and new transit light curves of HAT-P-20.…”

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

“…As an exact analytical solution of p hjd ð Þis not available in most practical cases, we resort to MCMC simulation to generate samples from the posterior pdf [see e.g., Robert and Casella, 2004]. The state-of-the-art DREAM ZS ð Þ [ter Braak and Vrugt, 2008;Vrugt et al, 2009; algorithm is used to approximate the posterior distribution. A detailed description of this sampling scheme including a proof of ergodicity and detailed balance can be found in the cited references.…”

“…Our method uses Gaussian process generation via circulant embedding [Dietrich and Newsam, 1997] to decouple the variogram from grid cell specific values, and implements dimensionality reduction by interpolation to enable MCMC simulation with the DREAM ZS ð Þ algorithm [Vrugt et al, 2009;. We use the Matern function to infer the conductivity values jointly with the field smoothness and other geostatistical parameters (mean, sill, integral scales, anisotropy direction(s) and anisotropy ratio(s)).…”