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.

翻译：空间和时间中点模式的随机模型广泛用于发布预测或评估风险，且常影响涉及社会的重要决策。我们将一致评分函数和适当评分规则的概念——这些是用于比较评估预测性能的统计原理工具——适配到点过程框架中，并将新方法与现有方法纳入此框架。参照地震似然模型检验，我们证明现有技术可应用于比先前预期更广泛的背景。特别地，泊松对数似然可用于基于单元期望的理论性比较预测评估。我们通过模拟研究以及意大利业务地震预测的比较评估来展示该方法。