GNGTS 2013 - Atti del 32° Convegno Nazionale

This formalization enlightens a basic aspect, i.e. the fundamental role played by the likelihood P ( E | H i ) to perform relative evaluations of the competing models. It also puts in evidence that ex-ante evaluations P*(H i ) could play an important role but they are not sufficient to judge the relative feasibility of the model under study. A possible straightforward way to determine the likelihood function P ( E | H i ) is the one considered by Albarello and D’Amico (2008) and many others. Given the PSHA computational model H i and the set of sites E Δ t * where ground shaking has been monitored during the control interval Δ t , the model’s Likelihood L i can be estimated from the control sample E Δt* . If the events e s are mutually independent (in the PSHA computational model considered) and if, over the duration of the control period, a total of N out of S sites have experienced ground shaking > g 0 , then we have (4) Of course one should account that several possible combinations sites/events may exist that result in the same configuration of the available evidence: all sites characterized in H i by the same exceedance probability are equivalent. By taking this into account, when the probabilities P(e S |H i )=P are equal, the number of such equivalent configurations will be the binomial coefficient C= N*!/(N*-S)!S! . Then the Likelihood L i will correspond to the Binomial probability density with parameters N*, S and P . The binomial coefficient makes possible the comparison of likelihoods relative to different amounts of data considered as Evidence. It is worth to note that likelihood could be also considered to evaluate the “absolute” heuristic potential of the H i model (testing): when its value is lower than any minimum threshold (0.05 to say) the relevant model could be considered as “unreliable” and the relevant contribution can be discarded in computing the comprehensive hazard curve. Other possibilities exist for testing any PSHA procedure against the evidence E (e.g., Schorlemmer et al. , 2007). Counting is one of these procedures. In this case, a binary variable e s ( g 0 ) is defined which assumes the value of 1 in the case that during the time interval Δ t * (which has the same extension as the hazard exposure time Δ t ) at least one earthquake occurred producing a ground motion in excess of g 0 at the s-th site; otherwise e s ( g 0 )=0. We define the “control sample” E Δ t * as the set of S realizations of the variable e s (g 0 ) at S sites. The i-th considered PSHA computational model H i provides a probability p si for the event e s ( g 0 )=1 given by (5) where the dependence on g 0 is omitted to simplify the notation. Expectation µ s and standard deviation σ s relative to the Bernoulli variable e s results to be (6) and (7) respectively. The number N* of sites out of the S sites considered for testing that experienced at least one earthquake during Δt* with ground-shaking >g 0 is (8) In terms of probabilistic forecasts provided by the H i PSHA computational model, is a random variate with expectation (9) 11 GNGTS 2013 S essione 2.1