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.

[1]  X. Zeng,et al.  Evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods , 2016 .

[2]  Q. Kang,et al.  Optimization and uncertainty assessment of strongly nonlinear groundwater models with high parameter dimensionality , 2010 .

[3]  M. Newton.  Approximate Bayesian-inference With the Weighted Likelihood Bootstrap , 1994 .

[4]  Jasper A. Vrugt,et al.  High‐dimensional posterior exploration of hydrologic models using multiple‐try DREAM(ZS) and high‐performance computing , 2012 .

[5]  Elena Volpi,et al.  Sworn testimony of the model evidence: Gaussian Mixture Importance (GAME) sampling , 2017 .

[6]  Nial Friel,et al.  Estimating the evidence – a review , 2011, 1111.1957.

[7]  Alain Dassargues,et al.  Conceptual model uncertainty in groundwater modeling: Combining generalized likelihood uncertainty estimation and Bayesian model averaging , 2008 .

[8]  Indranil Pan,et al.  Marginal Likelihood Based Model Comparison in Fuzzy Bayesian Learning , 2020, IEEE Transactions on Emerging Topics in Computational Intelligence.

[9]  J. Vrugt,et al.  Bayesian model selection in hydrogeophysics: Application to conceptual subsurface models of the South Oyster Bacterial Transport Site, Virginia, USA , 2017 .

[10]  D. Parkinson,et al.  A Nested Sampling Algorithm for Cosmological Model Selection , 2005, astro-ph/0508461.

[11]  H. Philippe,et al.  Computing Bayes factors using thermodynamic integration. , 2006, Systematic biology.

[12]  W. Nowak,et al.  Model selection on solid ground: Rigorous comparison of nine ways to evaluate Bayesian model evidence , 2014, Water resources research.

[13]  Jasper A. Vrugt,et al.  Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation , 2016, Environ. Model. Softw..

[14]  James D. Hamilton.  Time Series Analysis , 1994 .

[15]  Mary F. Wheeler,et al.  Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates , 2014 .

[16]  Jens Christian Refsgaard,et al.  Review of strategies for handling geological uncertainty in groundwater flow and transport modeling , 2012 .

[17]  F. Feroz,et al.  Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses , 2007, 0704.3704.

[18]  S. P. Neuman,et al.  On model selection criteria in multimodel analysis , 2007 .

[19]  G. Niu,et al.  Assessing five evolving microbial enzyme models against field measurements from a semiarid savannah—What are the mechanisms of soil respiration pulses? , 2014 .

[20]  F. Feroz,et al.  MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics , 2008, 0809.3437.

[21]  M. P. Hobson,et al.  polychord: nested sampling for cosmology , 2015, Monthly Notices of the Royal Astronomical Society: Letters.

[22]  A. Gelfand,et al.  Inequalities between expected marginal log‐likelihoods, with implications for likelihood‐based model complexity and comparison measures , 2003 .

[23]  David Anderson,et al.  Multimodel Ranking and Inference in Ground Water Modeling , 2004, Ground water.

[24]  Michael Andrew Christie,et al.  Calibration of channelized subsurface flow models using nested sampling and soft probabilities , 2015 .

[25]  Ming Ye,et al.  Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators , 2018 .

[26]  Eric Laloy,et al.  Mass conservative three‐dimensional water tracer distribution from Markov chain Monte Carlo inversion of time‐lapse ground‐penetrating radar data , 2012 .

[27]  Jichun Wu,et al.  Effects of grain size and structural heterogeneity on the transport and retention of nano-TiO2 in saturated porous media. , 2016, The Science of the total environment.

[28]  L. Shawn Matott,et al.  Evaluating uncertainty in integrated environmental models: A review of concepts and tools , 2009 .

[29]  Jeroen P. van der Sluijs,et al.  A framework for dealing with uncertainty due to model structure error , 2004 .

[30]  Mary F. Wheeler,et al.  Nested sampling algorithm for subsurface flow model selection, uncertainty quantification, and nonlinear calibration , 2013 .

[31]  Ming Ye,et al.  Maximum likelihood Bayesian averaging of spatial variability models in unsaturated fractured tuff , 2003 .

[32]  J. Hidalgo,et al.  Theoretical and observational constraints on Tachyon Inflation , 2017, 1711.06693.

[33]  Dan Lu,et al.  Maximum likelihood Bayesian model averaging and its predictive analysis for groundwater reactive transport models , 2015 .

[34]  D. Higdon,et al.  Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling , 2009 .

[35]  J. A. Vrugt,et al.  Distributed Soil Moisture from Crosshole Ground‐Penetrating Radar Travel Times using Stochastic Inversion , 2013, 1701.01634.

[36]  S. P. Neuman,et al.  Maximum likelihood Bayesian averaging of uncertain model predictions , 2003 .

[37]  Mary F. Wheeler,et al.  Hybrid nested sampling algorithm for Bayesian model selection applied to inverse subsurface flow problems , 2014, J. Comput. Phys..

[38]  J. Skilling.  Nested sampling for general Bayesian computation , 2006 .