Integrating MT-DREAMzs and nested sampling algorithms to estimate marginal likelihood and comparison with several other methods

Abstract

Marginal likelihood (or Bayesian evidence) estimation is the basis for model evaluation and model selection, which is the key research content of the uncertainty quantification of groundwater model structure. Recently, the nested sampling estimator (NSE) has been widely used as an efficient method to estimate the marginal likelihood. The Metropolis-Hasting (M-H) algorithm is usually used to explore the model likelihood space for NSE. However, M-H is inefficient in searching complex likelihood functions, especially for the multimodal distribution functions, because of the sampling structure of M-H algorithm. The multi-try differential evolution adaptive Metropolis (MT-DREAMzs) algorithm is effective and robust in searching complex probability space, and it is incorporated into NSE to improve the performance of marginal likelihood estimation in this study. In addition, the developed MT-DREAMzs-based NSE (NSE_MT) is evaluated by comparing with several other marginal likelihood estimators including DREAMzs-based and M-H-based NSEs (NSE_DR, NSE_MH), Gaussian mixture importance (GAME) sampling and a new nested sampling algorithm (PolyChord). Based on three analytical functions (up to 100 dimensions) and one groundwater solute transport case studies, the results showed that NSE_MT outperforms NSE_MH in the convergence, efficiency and accuracy of marginal estimation, and NSE_MT is slightly more efficient and accurate than NSE_DR. The accuracy of NSE_MH, NSE_DR and NSE_MT deteriorates with increasing dimensionality of target distributions/model parameters. In addition, GAME sampling and PolyChord sampling outperform other three NSEs in the accuracy and stability of estimation for high dimensional (up to 100-variate) distributions, and PolyChord is slightly less accurate and stable than GAME. Compared with GAME and PolyChord, NSE_MT and NSE_DR are competent for less than 10-dimensional distributions with higher computational efficiency.