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 and provide a colouring theorem that characterizes 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 recent publications, but conceptual flaws have been introduced in this literature. We concentrate on an example: the so-called sigmoidal Gaussian Cox process. We apply our general theory to resolve what are contradicting viewpoints in the data augmentation step of the inference procedures therein. Finally, we provide a multitype extension to this process and conduct Bayesian inference on data consisting of positions of 2 different species of trees in Lansing Woods, Illinois.

翻译：许多点过程数据模型通过稀疏化过程定义，其中基过程（通常为泊松过程）的位置被保留（观测）或丢弃（稀疏化）。本文回归点过程分布理论的基础，提出一个着色定理，该类模型中稀疏化位置与观测位置的联合密度由此定理刻画。实践中，观测点的边缘模型往往难以处理，但可以通过条件分布实例化稀疏化位置，并采用典型的数据增强方案绕过该问题。尽管近期文献已采用此类方法，但其中存在概念性缺陷。我们聚焦于一个示例：所谓的S形高斯Cox过程。应用一般理论，我们解决了该过程推断步骤中数据增强环节的矛盾观点。最后，我们为此过程提供多类型扩展，并对伊利诺伊州兰辛森林两种不同树种的位置数据进行贝叶斯推断。