On the Equivalence of Causal Models: A Category-Theoretic Approach

• URL: http://arxiv.org/abs/2201.06981v1
• Date: Tue, 18 Jan 2022 13:43:06 GMT
• Title: On the Equivalence of Causal Models: A Category-Theoretic Approach
• Authors: Jun Otsuka, Hayato Saigo
• Abstract summary: We develop a criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors. It is shown that when one model is a $Phi$-abstraction of another, the intervention of the former can be consistently translated into that of the latter.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We develop a category-theoretic criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. Following Jacobs et al. (2019), we define a causal model as a probabilistic interpretation of a causal string diagram, i.e., a functor from the ``syntactic'' category $\textsf{Syn}_G$ of graph $G$ to the category $\textsf{Stoch}$ of finite sets and stochastic matrices. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors, which we call a $\Phi$-abstraction and $\Phi$-equivalence, respectively. It is shown that when one model is a $\Phi$-abstraction of another, the intervention calculus of the former can be consistently translated into that of the latter. We also identify the condition under which a model accommodates a $\Phi$-abstraction, when transformations are deterministic.

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