Bayesian methods for type A evaluation of standard uncertainty

One of the aims of the revision of the Guide to the expression of Uncertainty in Measurement (GUM), is to make it self-consistent as the current edition would combine frequentist and Bayesian statistical methods. The methods for type A evaluation of standard uncertainty in the GUM are considered to be 'frequentist' and hence would have to be replaced by their Bayesian counterparts. The aim of this paper is to propose Bayesian methods for four mainstream type A evaluations of standard uncertainty, based on the same assumptions as those underlying the current methods for type A evaluation given in the GUM. Specific attention is given to the computational aspects of the Bayesian methods. An approach involving weakly informative prior distributions is proposed to ensure a proper posterior distribution, still allowing the data to dominate it. The applications for which Bayesian methods are proposed include the calculation of a mean of a series of observations, analysis of variance, ordinary least squares and errors-in-variables regression. Examples H.3 (ordinary least squares) and H.5 (analysis of variance) from the GUM are reworked accordingly. The second aim of this paper is to propose Bayesian methods for type A evaluation that take the information typically available in metrology into account in the form of weakly informative prior distributions. These methods can applied in a wide variety of situations. By comparing the results from the proposed Bayesian methods involving weakly informative priors with their classical counterparts, it is seen that in most instances these methods give very similar estimates and standard uncertainties as classical methods do under the same conditions and assumptions. Estimates are often identical up to the last meaningful digit, and the standard uncertainties are usually slightly yet persistently larger in comparison to its classical counterpart. In many cases these differences turn out to have limited or little practical meaning, save in the case of a small series of observations, where the differences in the computed standard uncertainty of the mean is larger.