On Bayesian procedure for maximum earthquake magnitude estimation

Abstract

This work is focused on the Bayesian procedure for the estimation of the regional maximum possible earthquake magnitude <em>m</em><sub>max</sub>. The paper briefly discusses the currently used Bayesian procedure for m<sub>max</sub>, as developed by Cornell, and a statistically justifiable alternative approach is suggested. The fundamental problem in the application of the current Bayesian formalism for <em>m</em><sub>max</sub> estimation is that one of the components of the posterior distribution is the sample likelihood function, for which the range of observations (earthquake magnitudes) depends on the unknown parameter <em>m</em><sub>max</sub>. This dependence violates the property of regularity of the maximum likelihood function. The resulting likelihood function, therefore, reaches its maximum at the maximum observed earthquake magnitude <em>m</em><sup>obs</sup><sub>max</sub> and not at the required maximum <em>possible</em> magnitude <em>m</em><sub>max</sub>. Since the sample likelihood function is a key component of the posterior distribution, the posterior estimate of <em>m^</em><sub>max</sub> is biased. The degree of the bias and its sign depend on the applied Bayesian estimator, the quantity of information provided by the prior distribution, and the sample likelihood function. It has been shown that if the maximum posterior estimate is used, the bias is negative and the resulting underestimation of <em>m</em><sub>max</sub> can be as big as 0.5 units of magnitude. This study explores only the maximum posterior estimate of <em>m</em><sub>max</sub>, which is conceptionally close to the classic maximum likelihood estimation. However, conclusions regarding the shortfall of the current Bayesian procedure are applicable to all Bayesian estimators, <em>e.g.</em> posterior mean and posterior median. A simple, <em>ad hoc</em> solution of this non-regular maximum likelihood problem is also presented.