Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators

Abstract

Estimating marginal likelihood is of central importance to Bayesian model selection and/or model averaging. The nested sampling method has been recently used together with the Metropolis-Hasting (M-H) sampling algorithm for estimating marginal likelihood. This study develops a new implementation of nested sampling by using the DiffeRential Evolution Adaptive Metropolis (DREAMzs) sampling algorithm. The two implementations of nested sampling are evaluated for four models of a synthetic groundwater flow modeling. The DREAMzs-based nested sampling outperforms the M-H-based nested sampling in terms of their accuracy, convergence, efficiency, and stability, which is attributed to the fact that DREAMzs is more robust than M-H for parameter sampling. The nested sampling method is also compared with four other methods (arithmetic mean, harmonic mean, stabilized harmonic mean, and thermodynamic integration) commonly used for estimating marginal likelihood and posterior probability of the four groundwater models. The comparative study requires hundreds of millions of model executions, which would not be possible without using surrogate models to replace the original models. Using the arithmetic mean estimator as the reference, the comparison reveals that thermodynamic integration outperforms nested sampling in terms of accuracy and stability, whereas nested sampling is more computationally efficient to reach to convergence. The harmonic mean and stabilized harmonic mean methods give the worst marginal likelihood estimation. Accurate marginal likelihood estimation is important for accurate estimation of posterior model probability and better predictive performance of Bayesian model averaging.