Bayesian Analysis of a change-point Poisson Process

For a Poisson process with a change-point, a uniform prior is commonly used for the change-point, but it is more realistic to put a unimodal prior on it, which outlines an important feature of prior beliefs. We consider a couple of unimodal priors on the change-point first and use ML-II approach to obtain the empirical Bayes estimators in this paper. The Bayes factor is used for the selection of a suitable prior. The procedure is applied to the British coal-mining disaster data. Finally, a comparison among these empirical Bayes estimators is made by Monte Carlo integration. It turns out that the ML-II Beta prior fit the data most, which corresponds to the prior belief of unimodality.