Comparing occurrence rates of events of interest in science, business, and medicine is an important topic. Because count data are often under-reported, we desire to account for this error in the response when constructing interval estimators. In this article, we derive a Bayesian interval for the difference of two Poisson rates when counts are potentially under-reported. The under-reporting causes a lack of identifiability. Here, we use informative priors to construct a credible interval for the difference of two Poisson rate parameters with under-reported data. We demonstrate the efficacy of our new interval estimates using a real data example. We also investigate the performance of our newly derived Bayesian approach via simulation and examine the impact of various informative priors on the new interval.