We present a flexible hierarchical Bayesian model and develop a comprehensive Bayesian decision theoretic framework for point process theory. We closely investigate the commonly used point process model for independent events, the Poisson process, using a finite mixture of exponential family components to model the intensity function. We employ a Bayesian hierarchical framework for parameter estimation and illustrate the Bayesian computations involved. We demonstrate the effectiveness of the Bayes rule under the Kullback–Leibler and Hellinger loss functions and compare them with the usual estimator, the posterior mean under squared error loss. The methodology is exemplified through simulations and a motivating application involving estimation of the intensity surface of homicide incidents in Chicago during 2015.

我们提出了一种灵活的分层贝叶斯模型，并为点过程理论开发了一个综合的贝叶斯决策理论框架。我们使用指数族成分的有限混合对强度函数进行建模，仔细研究独立事件的常用点过程模型（泊松过程）。我们采用贝叶斯分层框架进行参数估计，并说明所涉及的贝叶斯计算。我们证明了在Kullback-Leibler和Hellinger损失函数下贝叶斯规则的有效性，并将它们与通常的估计量（平方误差损失下的后均值）进行比较。该方法通过模拟和激励性应用进行举例说明，该应用涉及估计2015年芝加哥凶杀事件的强度表面。