Bayesian and frequentist prediction limits for the Poisson distribution

Prediction limits for Poisson distribution are useful in real life when predicting the occurrences of some phenomena, for example, the number of infections from a disease per year among school children, or the number of hospitalizations per year among patients with cardiovascular disease. In order to allocate the right resources and to estimate the associated cost, one would want to know the worst (i.e., an upper limit) and the best (i.e., the lower limit) scenarios. Under the Poisson distribution, we construct the optimal frequentist and Bayesian prediction limits, and assess frequentist properties of the Bayesian prediction limits. We show that Bayesian upper prediction limit derived from uniform prior distribution and Bayesian lower prediction limit derived from modified Jeffreys non informative prior coincide with their respective frequentist limits. This is not the case for the Bayesian lower prediction limit derived from a uniform prior and the Bayesian upper prediction limit derived from a modified Jeffreys prior distribution. Furthermore, it is shown that not all Bayesian prediction limits derived from a proper prior can be interpreted in a frequentist context. Using a counterexample, we state a sufficient condition and show that Bayesian prediction limits derived from proper priors satisfying our condition cannot be interpreted in a frequentist context. Analysis of simulated data and data on Atlantic tropical storm occurrences are presented.