Bayesian inference of a negative quantity from positive measurement results

In this paper Bayesian analysis is applied to a Cesium fountain frequency standard to estimate the density shift correction and its uncertainty. Cesium 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 so called atomic density shift of the reference atomic transition. The Bayes theorem allows the a priori knowledge of the sign of the collisional correction that is definitively negative to be rigourously embedded into the analysis. In the H-maser frequency measurements using INRIM Cesium fountain as reference, we used the Bayesian analysis and we report a reduction on the final combined uncertainty by more than 25% that demonstrates the Bayesian analysis as a relevant tool in primary frequency-metrology.