Many models for point process data are defined through a thinning procedure where locations of a base process (often Poisson) are either kept (observed) or discarded (thinned). In this paper, we go back to the fundamentals of the distribution theory for point processes to establish a link between the base thinning mechanism and the joint density of thinned and observed locations in any of such models. In practice, the marginal model of observed points is often intractable, but thinned locations can be instantiated from their conditional distribution and typical data augmentation schemes can be employed to circumvent this problem. Such approaches have been employed in the recent literature, but some inconsistencies have been introduced across the different publications. We concentrate on an example: the so-called sigmoidal Gaussian Cox process. We apply our approach to resolve contradicting viewpoints in the data augmentation step of the inference procedures therein. We also provide a multitype extension to this process and conduct Bayesian inference on data consisting of positions of two different species of trees in Lansing Woods, Michigan. The emphasis is put on intertype dependence modeling with Bayesian uncertainty quantification.

翻译：许多点过程数据模型通过稀释过程定义，其中基础过程（通常为泊松过程）的位置点被保留（观测）或舍弃（稀释）。本文回归点过程分布理论的基本原理，建立了此类模型中基础稀释机制与稀释/观测位置联合密度之间的理论联系。在实际应用中，观测点的边缘模型往往难以处理，但可通过从条件分布中实例化稀释位置，并采用典型的数据增强方案来规避此问题。现有文献已应用此类方法，但不同出版物间存在某些不一致之处。我们以Sigmoidal Gaussian Cox过程为例，应用本方法解决了其推断过程中数据增强步骤的矛盾观点。同时提出了该过程的多类型扩展，并对密歇根州兰辛森林中两种树木的位置数据进行了贝叶斯推断，重点在于通过贝叶斯不确定性量化实现类型间依赖关系建模。