On the application of Supplement 1 to the GUM to non-linear problems

Supplement 1 to the GUM (GUM-S1) produces an arbitrarily large sample from a probability distribution for the measurand which is used for the calculation of an estimate and its associated uncertainty. In the presence of Gaussian observations on one or several input quantities this distribution is equivalent to the Bayesian posterior obtained for a particular choice of a non-informative prior. Recently, a reference prior under partial information was proposed as an alternative non-informative prior in this context. Since for non-linear problems different results are obtained with this prior than by application of GUM-S1, the question arises whether GUM-S1 should actually be recommended for non-linear problems. We address this question by comparing the properties of the GUM-S1 distribution and the posterior distribution obtained by the proposed alternative prior. The comparison is supplemented by also considering a hybrid prior which assigns a constant prior for the measurand. We specify the conditions when the same results are reached. While the GUM-S1 distribution is always proper, we show that the proposed reference prior under partial information and the hybrid prior can fail to yield a proper posterior. On the basis of this (most important) criterion we can already recommend application of GUM-S1. Finally, we show that the prior underlying GUM-S1 can be derived as a (conditional) data-translated likelihood prior that exploits the symmetry and invariance of the considered likelihood function.