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

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 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.

Abstraction between Structural Causal Models: A Review of Definitions and Properties

“…Finally, Otsuka and Saigo [2022] offers an alternative category-theoretical definition, by requiring an abstraction α to be defined by first finding a graph homomorphism from G M m to G M M , expressing the micromodel M m and the macromodel M M as functors, and finally by seeing the abstraction as a natural transformation between the two functors. Interventions, once again used to enforce interventional consistency, take the form of endofuctors.…”

“…This separation of concerns is already implicitly present in Rischel [2020], Rischel and Weichwald [2021] where an abstraction is defined on two layers, as a mapping a between variables and a collection of mappings α X ′ between outcomes. This distinction is given stronger emphasis in Otsuka and Saigo [2022] where an explicit mapping between graphs (via a graph homomorphism) and a mapping between outcomes (via a natural transformation) are required; this setup follows from the category-theoretical approach of representing a model (e.g., a casual model in Jacobs et al [2019] or a database in Spivak [2014]) at a syntactic level capturing the underlying structure (in a free category generated from a graph) and at a semantic level (as a functor to a category that instantiates specific values).…”

“…Let us first consider a map G M m → G M M . First of all, notice that between two graphs we can typically establish a function on nodes X m → X M , or a stricter structure-preserving functor C M m → C M M , where C M m , C M M are the free categories generated from the DAGs of the micromodel and the macromodel [Otsuka and Saigo, 2022]. Other solutions, such as a function on edges, are not considered in the literature.…”

“…This is a property expressing the requirement that each node in the micromodel is abstracted, and no node can be ignored. This translates the idea of abstraction as a strict coarsening of the variables in a micromodel; structural functionality is implied by functoriality in Otsuka and Saigo [2022] .…”

“…A study of the formal properties of abstraction in the context of structural causal models (SCM) has been proposed in a few recent papers [Rubenstein et al, 2017, Beckers and Halpern, 2019, Beckers et al, 2020, Rischel, 2020, Rischel and Weichwald, 2021, Otsuka and Saigo, 2022. All these works have in common the presentation of abstraction as a map between SCMs that would be consistent (or approximately consistent) with respect to interventions; that is, a map that would commute (or approximately commute) with respect to interventions, thus enforcing the intuition that working first at the microlevel and then abstracting to the macrolevel would produce the same result as immediately switching to the macrolevel and then working there.…”