  Stochastic models of point patterns in space and time are widely used to
issue forecasts or assess risk, and often they affect societally relevant
decisions. We adapt the concept of consistent scoring functions and proper
scoring rules, which are statistically principled tools for the comparative
evaluation of predictive performance, to the point process setting, and place
both new and existing methodology in this framework. With reference to
earthquake likelihood model testing, we demonstrate that extant techniques
apply in much broader contexts than previously thought. In particular, the
Poisson log-likelihood can be used for theoretically principled comparative
forecast evaluation in terms of cell expectations. We illustrate the approach
in a simulation study and in a comparative evaluation of operational earthquake
forecasts for Italy.
