Bayesian uncertainty analysis compared with the application of the GUM and its supplements

The Guide to the Expression of Uncertainty in Measurement (GUM) has proven to be a major step towards the harmonization of uncertainty evaluation in metrology. Its procedures contain elements from both classical and Bayesian statistics. The recent supplements 1 and 2 to the GUM appear to move the guidelines towards the Bayesian point of view, and they produce a probability distribution that shall encode one's state of knowledge about the measurand. In contrast to a Bayesian uncertainty analysis, however, Bayes' theorem is not applied explicitly. Instead, a distribution is assigned for the input quantities which is then 'propagated' through a model that relates the input quantities to the measurand. The resulting distribution for the measurand may coincide with a distribution obtained by the application of Bayes' theorem, but this is not true in general. The relation between a Bayesian uncertainty analysis and the application of the GUM and its supplements is investigated. In terms of a simple example, similarities and differences in the approaches are illustrated. Then a general class of models is considered and conditions are specified for which the distribution obtained by supplement 1 to the GUM is equivalent to a posterior distribution resulting from the application of Bayes' theorem. The corresponding prior distribution is identified and assessed. Finally, we briefly compare the GUM approach with a Bayesian uncertainty analysis in the context of regression problems.