On the Equivalence of Causal Models: A Category-Theoretic Approach
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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 _G of graph G to the category 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 Φ-abstraction and Φ-equivalence, respectively. It is shown that when one model is a Φ-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 Φ-abstraction, when transformations are deterministic.
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