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Credible occurrence probabilities for extreme geophysical events: Earthquakes, volcanic eruptions, magnetic storms
Author(s) -
Love Jeffrey J.
Publication year - 2012
Publication title -
geophysical research letters
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.007
H-Index - 273
eISSN - 1944-8007
pISSN - 0094-8276
DOI - 10.1029/2012gl051431
Subject(s) - frequentist inference , confidence interval , volcano , credible interval , bayesian probability , geology , poisson distribution , event (particle physics) , bayes' theorem , bayesian inference , statistics , seismology , geophysics , mathematics , physics , quantum mechanics
Statistical analysis is made of rare, extreme geophysical events recorded in historical data – counting the number of events k with sizes that exceed chosen thresholds during specific durations of time τ. Under transformations that stabilize data and model‐parameter variances, the most likely Poisson‐event occurrence rate, k / τ , applies for frequentist inference and, also, for Bayesian inference with a Jeffreys prior that ensures posterior invariance under changes of variables. Frequentist confidence intervals and Bayesian (Jeffreys) credibility intervals are approximately the same and easy to calculate: ( 1 / τ )( k − z / 2 ) 2 , ( k + z / 2 ) 2, where z is a parameter that specifies the width, z = 1 ( z = 2) corresponding to 1 σ , 68.3% (2 σ , 95.4%). If only a few events have been observed, as is usually the case for extreme events, then these “error‐bar” intervals might be considered to be relatively wide. From historical records, we estimate most likely long‐term occurrence rates, 10‐yr occurrence probabilities, and intervals of frequentist confidence and Bayesian credibility for large earthquakes, explosive volcanic eruptions, and magnetic storms.