Reducing reliability bias in assessments of extreme weather risk using calibrating priors

Abstract. A number of recent climate studies have used univariate parametric statistical models to estimate return periods of extreme weather events based on the method of maximum likelihood. Using simulations over multiple training datasets, we find that using maximum likelihood gives predictions of extreme return levels that are exceeded more often than expected. For instance, when using the generalised extreme value distribution (GEVD) with 50 annual data values, fitted using maximum likelihood, we find that 200-year return levels are exceeded more than twice as often as expected; i.e. they are exceeded in more than 1 in 100 simulated years. This bias, which we refer to as a predictive coverage probability (PCP) bias, would be expected to lead to unreliable predictions. We review the theory related to Bayesian prediction using right Haar priors which gives an objective way to incorporate parameter uncertainty into predictions for some statistical models and which eliminates the bias. We consider a number of commonly used parametric statistical models and give the right Haar priors in each case. Where possible, we give analytical solutions for the resulting predictions. Where analytical solutions are not possible, we apply either an asymptotic approximation for the Bayesian prediction integral or ratio of uniforms sampling. For the fully parameterised GEVD and the generalised Pareto distribution with a known location parameter, neither of which have a right Haar prior, we test a number of methods and find one that gives big reductions in the PCP bias relative to maximum likelihood predictions. Finally, we revisit the De Bilt extreme temperature example considered in a number of previous studies and generate revised, and shorter, estimates for the return period of the 2018 heatwave. Software for fitting predictive distributions with parameter uncertainty has been developed by the first author and will be available as an R package.