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 $$(T_{i},Y_{i})_{i\ge 1}$$ ( T i , Y i ) i ≥ 1 , where $$T_{i}$$ T i is the time of occurrence of the $$i$$ i th event and the mark $$Y_{i}$$ Y i is its characteristic (size), is supposed to be a non-homogeneous Poisson process on $$\mathbb {R}_{+}^{2}$$ R + 2 with intensity measure $$P\times \varTheta $$ P × Θ , where $$P$$ P is known, whereas $$\varTheta $$ Θ 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 $$\varTheta $$ Θ 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. Copyright The Author(s) 2014