  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 rigourously
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 to be an important tool in primary frequency-metrology.
