Bayesian inference of a negative quantity from positive measurement results

Caesium-fountain frequency standards realize the second in the International System of Units with a relative uncertainty approaching 10(-16). Among the main contributions to the accuracy budget, cold collisions play an important role because of the atomic density shift of the reference atomic transition. This paper describes an application of the Bayesian analysis of the clock frequency to estimate the density shift and describes how the Bayes theorem allows the a priori knowledge of the sign of the collisional coefficient to be rigorously embedded into the analysis. As an application, data from the INRIM caesium fountain are used and the Bayesian and orthodox analyses are compared. The Bayes theorem allows the orthodox uncertainty to be reduced by 28% and demonstrates that it is an important tool in primary frequency metrology.