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

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 textsfSynG of graph G to the category textsfStoch 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.