ABSTRACT
The present paper deals with Bayesian frameworks of analysis for mixtures of Poisson processes. Numerous applications exist in the specific fields of risk and reliability analysis and in the more general field of probabilistic modeling, especially in cases where data are sparse and infrequent and when there are strong suspicions that failures/undesirable events arise from more than one distinct (but unknown) causes. Example areas include: systems with non-identical but unknown failure behavior, common cause failures, inspection of deteriorating infrastructure, spatial non-homogeneity, and signal detection. A Bayesian analysis of mixed Poisson-type data involves 3 layers of variables: observable (discrete) failure data X , non-observable failure rates θ associated with individual Poisson processes X | θ , and hyperparameters α which characterize the prior/posterior mixing distributions of θ . A full Bayesian analysis of mixtures is quite involved which leads us to investigate simplifications related to (1) discretization of the parameters θ (this gives rise to mixtures of “scenarios”) and (2) discretization of the parameter space associated with the hyper-parameters. Another avenue is explored which is based on Lindley’s second-order approximation of the mixture likelihood function. This approach is seen as quite an attractive alternative to 2-level discretization.
