Bayesian inference and prediction analysis of the power law process based on a natural conjugate prior

Under a natural conjugate prior with four hyperparameters, the importance sampling (IS) technique is applied to the Bayesian analysis of the power law process (PLP). Samples of the parameters of the PLP are obtained from IS. Based on these samples, not only the posterior analysis of parameters and some parameter functions in the PLP are performed conveniently, but also single-sample and two-sample prediction procedures are constructed easily. Furthermore, the sensitivity of the posterior mean of the parameter functions in the PLP is studied with respect to the hyperparameters of the natural conjugate prior and it can guide the selections of the hyperparameters directly. Coupled this sensitivity with the relations between the prior moments and the hyperparameters in the natural conjugate prior, it is possible to give directions about the selections of the prior moments to a certain degree. After some numerical experiments illustrate the rationality and feasibility of the proposed methods, an engineering example demonstrates its application.