The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probability (with standard Borel spaces) and beyond probability theory, including to deterministic and possibilistic networks. It therefore provides a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed random variables as well as deterministic and possibilistic variables.

翻译：d-分离标准通过某些有条件独立,检测出联合概率分布与定向单曲图的兼容性。在这项工作中,我们从绝对概率理论的角度研究这一问题,方法是引入因果模型的绝对定义、d-分离的绝对概念,并证明d-分离标准的抽象版本。这个方法有两个主要好处。首先,绝对的d-分离是基于地形关联性的一个非常直观的标准。第二,我们的结果既适用于测量-理论概率(标准波雷尔空间),也不适用于概率理论,包括确定性和可捕捉性网络。因此,它提供了当地和全球Markov特性与连续和混合随机变量以及确定性和可捕性变量的因果关系的清晰证据。