Bayesian prior elicitation and selection for extreme values

Abstract: A major issue of extreme value analysis is the determination of the shape parameter $\xi$ common to Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions, which drives the tail behavior, and is of major impact on the estimation of return levels and periods. Many practitioners make the choice of a Bayesian framework to conduct this assessment for accounting of parametric uncertainties, which are typically high in such analyses characterized by a low number of observations. Nonetheless, such approaches can provide large credibility domains for $\xi$, including negative and positive values, which does not allow to conclude on the nature of the tail. Considering the block maxima framework, a generic approach of the determination of the value and sign of $\xi$ arises from model selection between the Fr\'echet, Gumbel and Weibull possible domains of attraction conditionally to observations. Opposite to the common choice of the GEV as an appropriate model for {\it sampling} extreme values, this model selection must be conducted with great care. The elicitation of proper, informative and easy-to use priors is conducted based on the following principle: for all parameter dimensions they act as posteriors of noninformative priors and virtual samples. Statistics of these virtual samples can be assessed from prior predictive information, and a compatibility rule can be carried out to complete the calibration, even though they are only semi-conjugated. Besides, the model selection is conducted using a mixture encompassing framework, which allows to tackle the computation of Bayes factors. Motivating by a real case-study involving the elicitation of expert knowledge on meteorological magnitudes, the overall methodology is illustrated by toy examples too.