Estimating Earthquake Probabilities by Jaynes’s Method of Maximum Entropy

Abstract

ABSTRACT An important midcentury statistical method, already applied in several other scientific disciplines, enables epistemic ignorance to be incorporated more fully in multidecadal earthquake forecasts. Codified by the physicist E. T. Jaynes in the 1950s, the method of maximum entropy (MaxEnt) is suited for settings of great epistemic uncertainty. Shortly after its initial formulation it was being applied by industrial engineers to rates of failure and expected lifetimes—engineering problems analogous to fault ruptures and their recurrence intervals. Here, I show how MaxEnt can improve upon previous estimates of time-dependent conditional probabilities in Cascadia, California, and New Zealand. The method formalizes probabilistic inferences from long geology-based earthquake histories as truncated-Gaussian curves of time-dependent hazard that match the observed mean and mean square of the historical recurrence data. For each example, rupture forecasts by Brownian passage time, lognormal, Weibull, or Poisson methods have been published previously. In the Cascadia example, the MaxEnt estimate for a 50 yr exposure, 8.3%, compares with previous estimates of up to 14% for passage time or lognormal models. For other datasets drawn from the third Uniform California Earthquake Rupture Forecast study of California earthquakes and from the southern Alpine fault of New Zealand, MaxEnt recurrence probabilities similarly differ by up to a factor of nearly 2.0 from estimates published earlier. The lognormal approach is a variant of the MaxEnt model in which the estimate’s constraints concern the logarithms of the observed recurrence intervals; there is a continuum of methods in between. The discrepancies between MaxEnt and these other probability claims are not trivial; they arise from assumptions built into other methods that do not correspond to the actual information content of the data at hand. Discrepancies like these are directly relevant to the communication of these aspects of earthquake hazards to stakeholders concerned with the consequent risks.