Bayesian computation of design discharges

Probabilistic design of river dikes is usually based on estimates of a design discharge. Dutch design discharges are currently estimated using classical statistical methods. A shortcoming of this approach is that statistical uncertainties are not taken into account and that probability distributions are given equal weight. In the paper, a method based on Bayesian statistics is presented. Seven probability distributions for annual maxima are investigated for determining extreme quantiles of discharges: the exponential, Rayleigh, normal, lognormal, gamma, Weibull, and Gumbel. Bayes factors are used to determine weights corresponding to how well a probability distribution fits the observed data. Predictive exceedance probabilities are obtained using two different Bayesian computation methods: numerical integration and Markov Chain Monte Carlo (MCMC). MCMC methods can be used to draw samples from the posterior density. The pros and cons of numerical integration and MCMC are given and illustrated by estimating the discharge of the river Rhine with an average return period of 1,250 years. 1,250 years, while taking account of the statistical uncertainties involved. In this paper, a Bayesian method for estimating design discharges is presented. Section 2 considers Bayesian estimation of both parameters and quantiles associated with large average return periods. Section 3 and 4 are devoted to determining noninformative prior distributions and Bayes weights, respectively. Section 5 presents two well-known computational methods for calculating posterior distributions and predictive exceedance probabilities: numerical integration and Markov Chain Monte Carlo. In Section 6, the two Bayesian computational methods are compared by performing a Bayesian statistical analyis of the annual maximum discharges of the river Rhine. Section 7 ends with conclusions. 2 BAYESIAN ESTIMATION According to (amongst others) Slijkhuis et al. (1999) and Siu & Kelly (1998), uncertainties in risk analysis can primarily be divided into two categories: inherent uncertainties and epistemic uncertainties. Inherent uncertainties represent randomness or variability in nature. For example, even in the event of sufficient data, one cannot predict the maximum discharge that will occur next year. In this paper, we study inherent uncertainty in time (e.g., fluctuation of the discharge in time). Epistemic uncertainties represent the lack of knowledge about a physical system. In this paper, we study statistical uncertainty (due to lack of sufficient data); it includes parameter uncertainty (when the parameters of the distribution are unknown) and distribution-type uncertainty (when the type of distribution is unknown). Statistical uncertainty can be reduced as more data becomes available. The only statistical theory which combines modelling inherent uncertainty and statistical uncertainty is Bayesian statistics. The theorem of Bayes (1763) provides a solution to the problem of how to learn from data. In the framework of estimating the parameters θ = ′ ( ,..., ) θ θ 1 d of a probability distribution ( | ) x θ , Bayes’ theorem can be written as π π π π π ( | ) ( | ) ( )