Regular Article

Bayesian optimization of generalized data

¹ Nuclear Data and Criticality Safety Group, Reactor and Nuclear Systems Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6171, USA
² Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
³ Department of Physics and Astronomy, University of Alabama, Huntsville, AL 35899, USA
⁴ Nuclear & Radiological Engineering & Medical Physics, Georgia Institute of Technology, Atlanta, GA 30332-0745, USA

^* e-mail: arbanasg@ornl.gov

Received: 31 October 2017
Received in final form: 11 April 2018
Accepted: 28 May 2018
Published online: 14 November 2018

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 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.