On the use of Bayesian Monte-Carlo in evaluation of nuclear data

¹ CEA, DEN, Cadarache, 13108 Saint-Paul-les-Durance, France
² AREVA-TA, 13593 Aix-en-Provence, France

^a e-mail: cyrille.de-saint-jean@cea.fr

Published online: 13 September 2017

Abstract

As model parameters, necessary ingredients of theoretical models, are not always predicted by theory, a formal mathematical framework associated to the evaluation work is needed to obtain the best set of parameters (resonance parameters, optical models, fission barrier, average width, multigroup cross sections) with Bayesian statistical inference by comparing theory to experiment. The formal rule related to this methodology is to estimate the posterior density probability function of a set of parameters by solving an equation of the following type: pdf(posterior) ∼ pdf(prior) × a likelihood function. A fitting procedure can be seen as an estimation of the posterior density probability of a set of parameters (referred as x^→) knowing a prior information on these parameters and a likelihood which gives the probability density function of observing a data set knowing x^→. To solve this problem, two major paths could be taken: add approximations and hypothesis and obtain an equation to be solved numerically (minimum of a cost function or Generalized least Square method, referred as GLS) or use Monte-Carlo sampling of all prior distributions and estimate the final posterior distribution. Monte Carlo methods are natural solution for Bayesian inference problems. They avoid approximations (existing in traditional adjustment procedure based on chi-square minimization) and propose alternative in the choice of probability density distribution for priors and likelihoods. This paper will propose the use of what we are calling Bayesian Monte Carlo (referred as BMC in the rest of the manuscript) in the whole energy range from thermal, resonance and continuum range for all nuclear reaction models at these energies. Algorithms will be presented based on Monte-Carlo sampling and Markov chain. The objectives of BMC are to propose a reference calculation for validating the GLS calculations and approximations, to test probability density distributions effects and to provide the framework of finding global minimum if several local minimums exist. Application to resolved resonance, unresolved resonance and continuum evaluation as well as multigroup cross section data assimilation will be presented.