BAYES THEOREM AND THE PROBABILISTIC PREDICTION OF INTER-ARRIVAL TIMES FOR STRONG EARTHQUAKES FELT IN MEXICO CITY

Abstract

The probabilistic estimation or prediction of future inter-arrival times (T) of strong earthquake magnitudes (M≥6.5, Richter scale), felt in Mexico City from 1908 to 1979, is considered in terms of Bayes theorem. P(T_r/M)={P(T_r)P(M/T_r)}/Σ(T_j)P(M/T_j). We require the posterior probability P(T_r/M), i.e., the conditional probability of the event T_r, given that M has occurred. T₀ determine the prior probability P(T_j) and the likelihood function P(M/T_j), we use two fundamental assumptions: (i) Random occurrence with the associated equal prior probabilities, (ii) Binomial likelihood function. However, the discrete Bayesian analysis of our earthquake sample is evaluated on the basis of the proportion of successes (π), rather than in the number of successes. Thus, the Bayesian posterior distribution is the distribution of (π) conditional on observing the sample with X successes. The posterior distribution can be written in the form, where r stands for the specific midpoint earthquake magnitude class, s stand for the specific time-interval class. The discrete Bayesian analysis of strong earthquakes felt in Mexico City, suggests that the probabilistic estimation or prediction of strong earthquakes (magnitudes) felt in Mexico City as follows:
Observed Predicted
(Occurrence time)(Occurrence time)
October 25, 1981 (M_s=7.3) June, 1980 (M=7.3)
June 7, 1982 (M_s=7.0) January, 1982 (M=7.3)
September, 1985 (M=7.8) April, 1986 (M=7.8)
It should be noted that very strong agreement exists between the Observed occurrence time and the Predicted occurrence time.