Bayesian prediction in doubly stochastic Poisson process

A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process , where is the time of occurrence of the th event and the mark is its characteristic (size), is supposed to be a non-homogeneous Poisson process on with intensity measure , where is known, whereas is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.