Log-likelihood of earthquake models: evaluation of models and forecasts

There has been debate in the Collaboratory for the Study of Earthquake Predictability project about the most appropriate form of the likelihood function to use to evaluate earthquake forecasts in specified discrete space-time intervals, and also to evaluate the validity of the model itself. The debate includes whether the likelihood function should be discrete in nature, given that the forecasts are in discrete space-time bins, or continuous. If discrete, can different bins be assumed to be statistically independent, and is it satisfactory to assume that the forecasted count in each bin will have a Poisson distribution? In order to discuss these questions, we start with the most simple models (homogeneous Poisson), and progressively develop the model complexity to include self exciting point process models. For each, we compare the discrete and continuous time likelihoods. Examples are given where it is proven that the counts in discrete space-time bins are not Poisson. We argue that the form of the likelihood function is intrinsic to the given model, and the required forecast for some specified space-time region simply determines where the likelihood function should be evaluated. We show that continuous time point process models where the likelihood function is also defined in continuous space and time can easily produce forecasts over discrete space-time intervals.