Setting confidence belts

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 a 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.