A new modified Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference

Abstract

This paper proposes a new modification of the traditional Bayesian method for measurement uncertainty analysis.  The new modified Bayesian method is derived from the law of aggregation of information (LAI) and the rule of transformation between the frequentist view and Bayesian view.  It can also be derived from the original Bayes Theorem in continuous form.  We focus on a problem that is often encountered in measurement science: a measurement gives a series of observations.  We consider two cases: (1) there is no genuine prior information about the measurand, so the uncertainty evaluation is purely Type A, and (2) prior information is available and is represented by a normal distribution.  The traditional Bayesian method (also known as the reformulated Bayes Theorem) fails to provide a valid estimate of standard uncertainty in either case.  The new modified Bayesian method provides the same solutions to these two cases as its frequentist counterparts.  The differences between the new modified Bayesian method and the traditional Bayesian method are discussed.  This paper reveals that the traditional Bayesian method is not a self-consistent operation, so it may lead to incorrect inferences in some cases, such as the two cases considered.  In the light of the frequentist-Bayesian transformation rule and the law of aggregation of information (LAI), the frequentist and Bayesian inference are virtually equivalent, so they can be unified, at least in measurement uncertainty analysis.  The unification is of considerable interest because it may resolve the long-standing debate between frequentists and Bayesians.  The unification may also lead to an indisputable, uniform revision of the GUM (Evaluation of measurement data - Guide to the expression of uncertainty in measurement (JCGM 2008)).