On use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent

The International Organization for Standardization (ISO) Guide to the Expression of Uncertainty in Measurement is being increasingly recognized as a de facto international standard. The ISO Guide recommends a standardized way of expressing uncertainty in all kinds of measurements and provides a comprehensive approach for combining information to evaluate that uncertainty. The ISO Guide supports uncertainties evaluated from statistical methods, Type A, and uncertainties determined by other means, Type B. The ISO Guide recommends classical (frequentist) statistics for evaluating the Type A components of uncertainty; but it interprets the combined uncertainty from a Bayesian viewpoint. This is inconsistent. In order to overcome this inconsistency, we suggest that all Type A uncertainties should be evaluated through a Bayesian approach. It turns out that the estimates from a classical statistical analysis are either equal or approximately equal to the corresponding estimates from a Bayesian analysis with non-informative prior probability distributions. So the classical (frequentist) estimates may be used provided they are interpreted from the Bayesian viewpoint. The procedure of the ISO Guide for evaluating the combined uncertainty is to propagate the uncertainties associated with the input quantities. This procedure does not yield a complete specification of the distribution represented by the result of measurement and its associated combined standard uncertainty. So the correct coverage factor for a desired coverage probability of an expanded uncertainty interval cannot always be determined. Nonetheless, the ISO Guide suggests that the coverage factor may be computed by assuming that the distribution represented by the result of measurement and its associated standard uncertainty is a normal distribution or a scaled-and-shifted t-distribution with degrees of freedom determined from the Welch-Satterthwaite formula. This assumption may be unjustified and the coverage factor so determined may be incorrect. A popular convention is to set the coverage factor as 2. When the distribution represented by the result of measurement and its associated standard uncertainty is not completely determined, the 2-standard-uncertainty interval may be interpreted in terms of its minimum coverage probability for an applicable class of probability distributions.