This paper outlines methodological aspects of the statistical evaluation of earthquake forecast hypotheses. The recent debates concerning predictability of earthquakes clearly show how this problem is centred on the difficulty of systematically testing the numerous methodologies that in the years have been proposed and sustained by the supporters of prediction. This difficulty starts, sometimes, from the lack of a quantitative and rigorous definition of the concerned precursor, and other times from the lack of continuous observations, upon which statistical analyses could be based. The application of rigorous statistical methods, with the aim of verifying any prediction method, requires a univocal definition of the hypothesis, or the model, characterising the concerned anomaly or precursor, so as it can be recognised objectively in any circumstance and by any observer. A simple definition of an earthquake forecasting hypothesis could consist of the identification of particular sub-volumes of the total time-space volume (usually named alarm volumes) within which the probability of occurrence of strong earthquakes is higher than the average. The test of a similar model needs the observation of a sufficient number of past cases upon which to carry out a statistical analysis aimed to determine the rate at which the precursor has been followed (success rate) or not followed (false alarm, rate) by the target seismic event, or the rate at which a target event has been preceded (alarm rate) or not preceded (failure rate) by the precursor. A valid forecast hypothesis is expected to maximise success and minimise false alarms, with regard also to the maximisation of the probability gain. Some geophysicists prefer a statistical approach such as the B ayes criterion, based on the computation of the likelihood of an observed realisation of seismic events, and on the comparison of the likelihood obtained under different hypotheses. This method can be extended, as it has recently been the case for models of earthquake clustering, to algorithms that allow the computation of the density distribution of the conditional probability of earthquake occurrence in space, time and magnitude. Another approach to assess the validity of an earthquake forecast hypothesis could be that of estimating the total cost that the community has to pay in relation to earthquakes, and choosing the model that minimise such cost. Whatever method is chosen for building up a new hypothesis, the final assessment of its validity should be carried out by a test on a new and independent set of observations. The implementation of this test could, however, be problematic for seismicity characterised by long-term recurrence. (C) 2001 Elsevier Science B.V. All rights reserved.
