# Bayesian optimization of generalized data

## Abstract

Direct application of Bayes' theorem to generalized data yields a posterior probability distribution function (PDF) that is a product of a prior PDF of generalized data and a likelihood function, where generalized data consists of model parameters, measured data, and model defect data. The prior PDF of generalized data is defined by prior expectation values and a prior covariance matrix of generalized data that naturally includes covariance between any two components of generalized data. A set of constraints imposed on the posterior expectation values and covariances of generalized data via a given model is formally solved by the method of Lagrange multipliers. Posterior expectation values of the constraints and their covariance matrix are conventionally set to zero, leading to a likelihood function that is a Dirac delta function of the constraining equation. It is shown that setting constraints to values other than zero is analogous to introducing a model defect. Since posterior expectation values of any function of generalized data are integrals of that function over all generalized data weighted by the posterior PDF, all elements of generalized data may be viewed as nuisance parameters marginalized by this integration. One simple form of posterior PDF is obtained when the priormore »

- Authors:

- ORNL
- Rensselaer Polytechnic Institute (RPI)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1482461

- DOE Contract Number:
- AC05-00OR22725

- Resource Type:
- Conference

- Resource Relation:
- Journal Volume: 4; Conference: 4th International Workshop on Nuclear Data Covariances - Aix en Provence, , France - 10/2/2017 4:00:00 AM-10/6/2017 4:00:00 AM

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Arbanas, Goran, Feng, Jinghua, Clifton, Zia J., Holcomb, Andrew M., Pigni, Marco T., Wiarda, Dorothea, Chapman, Christopher W., Sobes, Vladimir, Liu, Emily, and Danon, Y.
```*Bayesian optimization of generalized data*. United States: N. p., 2018.
Web. doi:10.1051/epjn/2018038.

```
Arbanas, Goran, Feng, Jinghua, Clifton, Zia J., Holcomb, Andrew M., Pigni, Marco T., Wiarda, Dorothea, Chapman, Christopher W., Sobes, Vladimir, Liu, Emily, & Danon, Y.
```*Bayesian optimization of generalized data*. United States. doi:10.1051/epjn/2018038.

```
Arbanas, Goran, Feng, Jinghua, Clifton, Zia J., Holcomb, Andrew M., Pigni, Marco T., Wiarda, Dorothea, Chapman, Christopher W., Sobes, Vladimir, Liu, Emily, and Danon, Y. Thu .
"Bayesian optimization of generalized data". United States. doi:10.1051/epjn/2018038. https://www.osti.gov/servlets/purl/1482461.
```

```
@article{osti_1482461,
```

title = {Bayesian optimization of generalized data},

author = {Arbanas, Goran and Feng, Jinghua and Clifton, Zia J. and Holcomb, Andrew M. and Pigni, Marco T. and Wiarda, Dorothea and Chapman, Christopher W. and Sobes, Vladimir and Liu, Emily and Danon, Y.},

abstractNote = {Direct application of Bayes' theorem to generalized data yields a posterior probability distribution function (PDF) that is a product of a prior PDF of generalized data and a likelihood function, where generalized data consists of model parameters, measured data, and model defect data. The prior PDF of generalized data is defined by prior expectation values and a prior covariance matrix of generalized data that naturally includes covariance between any two components of generalized data. A set of constraints imposed on the posterior expectation values and covariances of generalized data via a given model is formally solved by the method of Lagrange multipliers. Posterior expectation values of the constraints and their covariance matrix are conventionally set to zero, leading to a likelihood function that is a Dirac delta function of the constraining equation. It is shown that setting constraints to values other than zero is analogous to introducing a model defect. Since posterior expectation values of any function of generalized data are integrals of that function over all generalized data weighted by the posterior PDF, all elements of generalized data may be viewed as nuisance parameters marginalized by this integration. One simple form of posterior PDF is obtained when the prior PDF and the likelihood function are normal PDFs. For linear models without a defect this PDF becomes equivalent to constrained least squares (CLS) method, that is, the χ2 minimization method.},

doi = {10.1051/epjn/2018038},

journal = {},

issn = {2491--9292},

number = ,

volume = 4,

place = {United States},

year = {2018},

month = {11}

}