The d-Separation Criterion in Categorical Probability
Tobias Fritz, Andreas Klingler; 24(46):1−49, 2023.
Abstract
The d-separation criterion is used to determine the compatibility of a joint probability distribution with a directed acyclic graph by examining certain conditional independences. In this study, we investigate this problem within the context of categorical probability theory. We introduce a categorical definition of causal models, a categorical concept of d-separation, and establish an abstract version of the d-separation criterion. This approach offers two main advantages. Firstly, categorical d-separation is an intuitive criterion based on topological connectedness. Secondly, our findings are applicable to various domains, including measure-theoretic probability (with standard Borel spaces) and beyond probability theory, such as deterministic and possibilistic networks. Consequently, it provides a clear 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.
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