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