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

Jun Otsuka, Hayato Saigo
Proceedings of the First Conference on Causal Learning and Reasoning, PMLR 177:634-646, 2022.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v177-otsuka22a, title = {On the Equivalence of Causal Models: A Category-Theoretic Approach}, author = {Otsuka, Jun and Saigo, Hayato}, booktitle = {Proceedings of the First Conference on Causal Learning and Reasoning}, pages = {634--646}, year = {2022}, editor = {Schölkopf, Bernhard and Uhler, Caroline and Zhang, Kun}, volume = {177}, series = {Proceedings of Machine Learning Research}, month = {11--13 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v177/otsuka22a/otsuka22a.pdf}, url = {https://proceedings.mlr.press/v177/otsuka22a.html}, 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.} }
Endnote
%0 Conference Paper %T On the Equivalence of Causal Models: A Category-Theoretic Approach %A Jun Otsuka %A Hayato Saigo %B Proceedings of the First Conference on Causal Learning and Reasoning %C Proceedings of Machine Learning Research %D 2022 %E Bernhard Schölkopf %E Caroline Uhler %E Kun Zhang %F pmlr-v177-otsuka22a %I PMLR %P 634--646 %U https://proceedings.mlr.press/v177/otsuka22a.html %V 177 %X 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.
APA
Otsuka, J. & Saigo, H.. (2022). On the Equivalence of Causal Models: A Category-Theoretic Approach. Proceedings of the First Conference on Causal Learning and Reasoning, in Proceedings of Machine Learning Research 177:634-646 Available from https://proceedings.mlr.press/v177/otsuka22a.html.

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