# A Bayesian approach for parameter estimation and prediction using a computationally intensive model

## Abstract

Bayesian methods have been successful in quantifying uncertainty in physics-based problems in parameter estimation and prediction. In these cases, physical measurements y are modeled as the best fit of a physics-based model $$\eta (\theta )$$, where θ denotes the uncertain, best input setting. Hence the statistical model is of the form $$y=\eta (\theta )+\epsilon ,$$ where $$\epsilon $$ accounts for measurement, and possibly other, error sources. When nonlinearity is present in $$\eta (\cdot )$$, the resulting posterior distribution for the unknown parameters in the Bayesian formulation is typically complex and nonstandard, requiring computationally demanding computational approaches such as Markov chain Monte Carlo (MCMC) to produce multivariate draws from the posterior. Although generally applicable, MCMC requires thousands (or even millions) of evaluations of the physics model $$\eta (\cdot )$$. This requirement is problematic if the model takes hours or days to evaluate. To overcome this computational bottleneck, we present an approach adapted from Bayesian model calibration. This approach combines output from an ensemble of computational model runs with physical measurements, within a statistical formulation, to carry out inference. A key component of this approach is a statistical response surface, or emulator, estimated from the ensemble of model runs. We demonstrate this approach with a case study in estimating parameters for a density functional theory model, using experimental mass/binding energy measurements from a collection of atomic nuclei. Lastly, we also demonstrate how this approach produces uncertainties in predictions for recent mass measurements obtained at Argonne National Laboratory.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Physics Division
- Argonne National Lab. (ANL), Argonne, IL (United States). Mathematics and Computer Science Division

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1378523

- Alternate Identifier(s):
- OSTI ID: 1407869

- Report Number(s):
- LLNL-JRNL-737147; LA-UR-14-26925

Journal ID: ISSN 0954-3899

- Grant/Contract Number:
- AC52-07NA27344; AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Physics. G, Nuclear and Particle Physics

- Additional Journal Information:
- Journal Volume: 42; Journal Issue: 3; Journal ID: ISSN 0954-3899

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

### Citation Formats

```
Higdon, Dave, McDonnell, Jordan D., Schunck, Nicolas, Sarich, Jason, and Wild, Stefan M.
```*A Bayesian approach for parameter estimation and prediction using a computationally intensive model*. United States: N. p., 2015.
Web. doi:10.1088/0954-3899/42/3/034009.

```
Higdon, Dave, McDonnell, Jordan D., Schunck, Nicolas, Sarich, Jason, & Wild, Stefan M.
```*A Bayesian approach for parameter estimation and prediction using a computationally intensive model*. United States. doi:10.1088/0954-3899/42/3/034009.

```
Higdon, Dave, McDonnell, Jordan D., Schunck, Nicolas, Sarich, Jason, and Wild, Stefan M. Thu .
"A Bayesian approach for parameter estimation and prediction using a computationally intensive model". United States.
doi:10.1088/0954-3899/42/3/034009. https://www.osti.gov/servlets/purl/1378523.
```

```
@article{osti_1378523,
```

title = {A Bayesian approach for parameter estimation and prediction using a computationally intensive model},

author = {Higdon, Dave and McDonnell, Jordan D. and Schunck, Nicolas and Sarich, Jason and Wild, Stefan M.},

abstractNote = {Bayesian methods have been successful in quantifying uncertainty in physics-based problems in parameter estimation and prediction. In these cases, physical measurements y are modeled as the best fit of a physics-based model $\eta (\theta )$, where θ denotes the uncertain, best input setting. Hence the statistical model is of the form $y=\eta (\theta )+\epsilon ,$ where $\epsilon $ accounts for measurement, and possibly other, error sources. When nonlinearity is present in $\eta (\cdot )$, the resulting posterior distribution for the unknown parameters in the Bayesian formulation is typically complex and nonstandard, requiring computationally demanding computational approaches such as Markov chain Monte Carlo (MCMC) to produce multivariate draws from the posterior. Although generally applicable, MCMC requires thousands (or even millions) of evaluations of the physics model $\eta (\cdot )$. This requirement is problematic if the model takes hours or days to evaluate. To overcome this computational bottleneck, we present an approach adapted from Bayesian model calibration. This approach combines output from an ensemble of computational model runs with physical measurements, within a statistical formulation, to carry out inference. A key component of this approach is a statistical response surface, or emulator, estimated from the ensemble of model runs. We demonstrate this approach with a case study in estimating parameters for a density functional theory model, using experimental mass/binding energy measurements from a collection of atomic nuclei. Lastly, we also demonstrate how this approach produces uncertainties in predictions for recent mass measurements obtained at Argonne National Laboratory.},

doi = {10.1088/0954-3899/42/3/034009},

journal = {Journal of Physics. G, Nuclear and Particle Physics},

number = 3,

volume = 42,

place = {United States},

year = {Thu Feb 05 00:00:00 EST 2015},

month = {Thu Feb 05 00:00:00 EST 2015}

}

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