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

Jun Otsuka, Hayato Saigo

Keywords: Causal Models, Abstraction, Category Theory, String Diagrams

TL;DR: The paper gives a category-theoretic criterion of equivalence of causal models

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 Syn_G of graph $G$ to the category 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.