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-分离准则通过特定的条件独立性来检测联合概率分布与有向无环图的兼容性。本文通过引入因果模型的范畴化定义、d-分离的范畴化概念，并证明d-分离准则的抽象版本，在范畴概率论框架下研究该问题。这一方法具有两个主要优势：首先，范畴化d-分离是基于拓扑连通性的非常直观的准则；其次，我们的结论既适用于测度论概率（标准波莱尔空间），也适用于概率论之外的领域，包括确定性和可能性网络。因此，它为连续随机变量、混合随机变量以及确定性和可能性变量，提供了局部马尔可夫性质、全局马尔可夫性质与因果兼容性之间等价性的简洁证明。