Bayesian probability theory in conjunction with the model of extremes is used to develop a Bayesian distribution of extreme earthquake occurrences by assuming that earthquakes represent a Poisson process with exponential distribution of magnitudes. The Bayesian distribution represents the probability that Mmax, the largest earthquake expected to occur within a period of t years, will exceed some specified magnitude m, and may be computed from the relationship,
where n″ and t″ represent updated (posterior) Bayesian estimates of the number of earthquakes and the time period of observation, respectively, and F∽(m) is the Bayesian distribution of magnitudes, each updated from prior estimates of seismicity using historical observations of earthquake occurrences. Through the application of Bayesian probability theory, the above distribution contains two features that make it more powerful and reliable than the conventional extreme-value distribution. First, it incorporates statistical uncertainty in the estimation of seismicity in addition to the inherent uncertainty associated with the random nature of earthquake occurrences. Second, it provides a rigorous means of combining prior information on seismicity, such as geologica data, with available historical observations of earthquake occurrences. The last feature represents a major step toward the ability of earthquake engineers to systematically use information from geologists and seismologists, regarding the seismic potential of faults and small regions, in the probabilistic assessment of seismic hazards and provides a means of periodically updating estimates of earthquake occurrence probabilities as new seismicity data become available.