  We propose using a Bayes procedure with uniform improper prior to determine
credible belts for the mean of a Poisson distribution in the presence of
background and for the continuous problem of measuring a non-negative quantity
$\theta$ with a normally distributed measurement error. Within the Bayesian
framework, these belts are optimal. The credible limits are then examined from
a frequentist point of view and found to have good frequentist and conditional
frequentist properties.
