## 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 inte-gration outperforms nested sampling in terms of accuracy and stability, whereas nested sampling is more computationally efﬁcient to reach to convergence. The harmonic mean andmore »

- Authors:

- Key Laboratory of Surficial Geochemistry, Ministry of Education, School of Earth Sciences and Engineering, Nanjing University, Nanjing China
- Department of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee FL USA

- Publication Date:

- Research Org.:
- Florida State Univ., Tallahassee, FL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1537334

- Alternate Identifier(s):
- OSTI ID: 1419574

- Grant/Contract Number:
- SC0008272

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Water Resources Research

- Additional Journal Information:
- Journal Volume: 54; Journal Issue: 2; Journal ID: ISSN 0043-1397

- Publisher:
- American Geophysical Union (AGU)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Environmental Sciences & Ecology; Marine & Freshwater Biology; Water Resources

### Citation Formats

```
Zeng, Xiankui, Ye, Ming, Wu, Jichun, Wang, Dong, and Zhu, Xiaobin. Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators. United States: N. p., 2018.
Web. doi:10.1002/2017wr020782.
```

```
Zeng, Xiankui, Ye, Ming, Wu, Jichun, Wang, Dong, & Zhu, Xiaobin. Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators. United States. doi:10.1002/2017wr020782.
```

```
Zeng, Xiankui, Ye, Ming, Wu, Jichun, Wang, Dong, and Zhu, Xiaobin. Thu .
"Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators". United States. doi:10.1002/2017wr020782. https://www.osti.gov/servlets/purl/1537334.
```

```
@article{osti_1537334,
```

title = {Improved Nested Sampling and Surrogate-Enabled Comparison With Other Marginal Likelihood Estimators},

author = {Zeng, Xiankui and Ye, Ming and Wu, Jichun and Wang, Dong and Zhu, Xiaobin},

abstractNote = {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 inte-gration 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.},

doi = {10.1002/2017wr020782},

journal = {Water Resources Research},

number = 2,

volume = 54,

place = {United States},

year = {2018},

month = {2}

}

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