# Bayesian estimation of design loads

J.M. van Noortwijk^{1,2}, H.J. Kalk^{1}
and E.H. Chbab^{3
1} HKV Consultants, Lelystad, The Netherlands.

^{2} Faculty of Electrical Engineering, Mathematics and
Computer Science,

Delft University of Technology, Delft, The Netherlands.

^{3} Institute for Inland Water Management and Waste Water
Treatment,

Ministry of Transport, Public Works, and Water Management, Lelystad,

The Netherlands.

Probabilistic design of structures is usually based on estimates of design loads with a large average return period. Design loads are often estimated using classical statistical methods. A shortcoming of this approach is that statistical uncertainties are not taken into account. In this paper, a method based on Bayesian statistics is presented. Using Bayes’ theorem, the prior distribution representing information about the uncertainty of the statistical parameters can be updated to the posterior distribution as soon as data becomes available. Nine predictive probability distributions are considered for determining extreme quantiles of loads: the exponential, Rayleigh, normal, lognormal, gamma, Weibull, Gumbel, generalised gamma and generalised extreme-value. The Bayesian method has been successfully applied to estimate the discharge of the rivers Rhine and Meuse with an average return period of 1,250 years while taking account of the statistical uncertainties involved. In order that the observations ‘speak for themselves’, the non-informative Jeffreys priors were chosen as priors. The Bayes estimates are compared to the classical maximum-likelihood estimates. Furthermore, so-called Bayes factors are used to determine weights corresponding to how well a probability distribution fits the observed data; that is, the better the fit, the higher the weighting.

**Key words**: Bayesian analysis,
non-informative Jeffreys prior, Bayes weights, river discharges, maximum
likelihood.