There are many potential pitfalls in assigning statistical significance to ''successful'' earthquake predictions, including sensitivity to the stochastic model for earthquake occurrence, dependence among the parameters used to specify an earthquake, and the stochastic alternative model for the predictions. There is no simple resolution to these difficulties, but it is clear that assuming earthquake location and size variables are independent can lead to the erroneous impression that predictions are statistically significant when they are not, and that assuming that main shocks follow a Poisson process in time (and space) can make chance predictions appear statistically significant, and can make sound predictions appear statistically insignificant. It is possible to avoid assuming any probability distribution for the occurrence of earthquakes, treating the seismicity history as fixed (conditioning on the observed seismicity) and testing the performance of the prediction algorithm against various random predictors; this ''conditional'' approach seems preferable. ¿ American Geophysical Union 1996