Upper Limits in the Case That Zero Events are Observed: An Intuitive Solution to the Background Dependence Puzzle

We compare the ``unified approach'' for the estimation of upper limits withan approach based on the Bayes theory, in the special case that no events areobserved. The ``unified approach'' predicts, in this case, an upper limit thatdecreases with the increase in the expected level of background. This seemsabsurd. On the other hand, the Bayesian approach leads to a result which isbackground independent. An explanation of the Bayesian result is presented,together with suggested reasons for the paradoxical result of the ``unifiedapproach''.