Hierarchical Bayes Analysis of Rare Events Using High-Dispersion Poisson Mixtures

Modeling the occurrence of rare events such as multiyear ice or iceberg encounters, ship collisions, and several types of accidental events is often challenging because considerable dispersion is found to be associated with discrete count data. This may be due to fluctuations in the processes generating the events, or they may arise because of a complicated mixture of causal events or there may be other unexplained discontinuities. In such cases, the traditional use of the Poisson distribution is inadequate, especially when the event frequency is subsequently used to formulate design criteria based on extreme values. In this paper, the use of discrete Poisson mixtures is suggested as opposed to the simple Poisson process and continuous Poisson mixtures. One objective is to ensure that the uncertainty regarding event occurrence is well represented in both the central and tail parts of count data. The analysis of discrete Poisson mixtures involves the estimation of the number k of mixture components, the k Poisson occurrence rates, and the k weights of the mixture. Until recently such an analysis was considered daunting at best. However, the analysis can be re-cast as an equivalent Hierarchical Bayes (HB) net using an auxiliary variable vector Z of variable dimension. A Markov Chain Monte Carlo analysis can then be used to obtain the posterior distributions of the dimensionality of the mixture, the mixture weights and the occurrence rates themselves. Also, posterior distributions can be found for iceberg collision risks and iceberg scour rates. The approach is illustrated for an iceberg risk estimation.