Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T03:14:02.250Z Has data issue: false hasContentIssue false

Causal inference via string diagram surgery

A diagrammatic approach to interventions and counterfactuals

Published online by Cambridge University Press:  16 November 2021

Bart Jacobs*
Affiliation:
Department of Computing and Information Science, Radboud Universiteit Faculteit der Natuurwetenschappen Wiskunde en Informatica, Nijmegen, The Netherlands
Aleks Kissinger
Affiliation:
Department of Computer Science, Oxford University, Oxford, UK
Fabio Zanasi
Affiliation:
Department of Computer Science, University College London, London, UK
*
*Corresponding author. Email: [email protected]

Abstract

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.

Type
Paper
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M. and Spekkens, R. W. (2017). Quantum common causes and quantum causal models. Physical Review X 7 031021.CrossRefGoogle Scholar
Balke, A. and Pearl, J. (1994). Probabilistic evaluation of counterfactual queries. In: Hayes-Roth, B. and Korf, R. (eds.) Proceedings of the 12th National Conference on Artificial Intelligence. AAAI Press/The MIT Press, 230237.Google Scholar
Bonchi, F., Sobociński, P. and Zanasi, F. (2018). Deconstructing Lawvere with distributive laws. Journal of Logical Algebra Methods Programming 95 128146.CrossRefGoogle Scholar
Cabrera, B., Heindel, T., Heckel, R. and König, B. (2018). Updating probabilistic knowledge on condition/event nets using Bayesian networks. In: 29th International Conference on Concurrency Theory, CONCUR 2018, September 4–7, 2018, Beijing, China, 27:1–27:17.Google Scholar
Chiribella, G., D’Ariano, G. M. and Perinotti, P. (2008). Quantum circuit architecture. Physical Review Letters 101 060401.CrossRefGoogle ScholarPubMed
Cho, K. and Jacobs, B. (2019). Disintegration and Bayesian inversion via string diagrams. Mathematical Structures in Computer Science 29 (7) 938971.CrossRefGoogle Scholar
Clerc, F., Danos, V., Dahlqvist, F. and Garnier, I. (2017). Pointless learning. In: Esparza, J. and Murawski, A. (eds.) Foundations of Software Science and Computation Structures, vol. 10203. Lecture Notes Computer Science. Berlin: Springer, 355369.CrossRefGoogle Scholar
Coecke, B. and Heunen, C. (2016). Pictures of complete positivity in arbitrary dimension. Information and Computation 250 5058.CrossRefGoogle Scholar
Coecke, B. and Spekkens, R. W. (2012). Picturing classical and quantum Bayesian inference. Synthese 186 (3) 651696.CrossRefGoogle Scholar
Corradini, A. and Gadducci, F. (1999). An algebraic presentation of term graphs, via GS-monoidal categories. Applied Categorical Structures 7 (4) 299331.CrossRefGoogle Scholar
Costa, F. and Shrapnel, S. (2016). Quantum causal modelling. New Journal of Physics 18 (6) 063032.CrossRefGoogle Scholar
Fong, B. (2012). Causal theories: A categorical perspective on Bayesian networks. Master’s thesis, Univ. of Oxford. see arxiv.org/abs/1301.6201.Google Scholar
Fritz, T. (2020). A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statistics. Advances in Mathematics 370 107239.CrossRefGoogle Scholar
Gutoski, G. and Watrous, J. (2007). Toward a general theory of quantum games. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing. ACM, 565574.CrossRefGoogle Scholar
Henson, J., Lal, R. and Pusey, M. F. (2014). Theory-independent limits on correlations from generalized Bayesian networks. New Journal of Physics 16 (11), 113043.CrossRefGoogle Scholar
Huang, Y. and Valtorta, M. (2008). On the completeness of an identifiability algorithm for semi-Markovian models. Annals of Mathematics and Artificial Intelligence 54 (4) 363408.CrossRefGoogle Scholar
Huang, Y. and Valtorta, M. (2012). Pearl’s calculus of intervention is complete. CoRR, abs/1206.6831.Google Scholar
Jacobs, B. and Zanasi, F. (2021). The logical essentials of Bayesian reasoning. In: Barthe, G., Katoen, J.-P. and Silva, A. (eds.) Foundations of Probabilistic Programming. Cambridge Univ. Press, 295331.Google Scholar
Jacobs, B., Kissinger, A. and Zanasi, F. (2019). Causal inference by string diagram surgery. In: Foundations of Software Science and Computation Structures - 22nd International Conference, FOSSACS 2019, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019, Prague, Czech Republic, April 6–11, 2019, Proceedings, 313–329.CrossRefGoogle Scholar
Jacobs, B. and Zanasi, F. (2016). A predicate/state transformer semantics for Bayesian learning. Electr. Notes Theor. Comput. Sci. 325, 185200.CrossRefGoogle Scholar
Jacobs, B. and Zanasi, F. (2017). A formal semantics of influence in Bayesian reasoning. In: Larsen, K., Bodlaender, H. and Raskin, J.-F. (eds.) Mathematics Foundation of Computer Science, vol. 83. LIPIcs. Schloss Dagstuhl, 21:1–21:14.Google Scholar
Kissinger, A. and Uijlen, S. (2017). A categorical semantics for causal structure. In : 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20–23, 2017, 1–12.CrossRefGoogle Scholar
Lawvere, F. W. (1963). Functorial semantics of algebraic theories. Proceedings of the National Academy of Sciences of the United States of America 50 (5) 869.CrossRefGoogle ScholarPubMed
Lawvere, F. W. (1969). Ordinal sums and equational doctrines. In: Eckmann, B. (ed.) Seminar on Triples and Categorical Homology Theory, vol. 80. Lecture Notes in Mathematics. Springer-Verlag, 141–155.CrossRefGoogle Scholar
Leifer, M. S. and Spekkens, R. W. (2013). Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Physical Review A 88 052130.CrossRefGoogle Scholar
Mooij, J. M., Magliacane, S. and Claassen, T. (2020). Joint causal inference from multiple contexts. Journal of Machine Learning Research 21 (99) 1108.Google Scholar
Nielsen, M. (2018). If correlation doesn’t imply causation, then what does? Available at http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does, accessed: 2018-11-15.Google Scholar
Pearl, J. (2000). Causality: Models, Reasoning and Inference. Cambridge University Press.Google Scholar
Pearl, J. and Verma, T. S. (1991). A Theory of Inferred Causation. San Mateo, CA: Morgan Kaufmann.Google Scholar
Pienaar, J. and Brukner, Č. (2015). A graph-separation theorem for quantum causal models. New Journal of Physics 17 (7) 073020.CrossRefGoogle Scholar
Ried, K., Agnew, M., Vermeyden, L., Janzing, D., Spekkens, R. W. and Resch, K. J. (2015). A quantum advantage for inferring causal structure. Nature Physics 11 17452473.CrossRefGoogle Scholar
Selinger, P. (2011). A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, vol. 813. Lecture Notes in Physics. Springer, 289–355.Google Scholar
Shpitser, I. and Pearl, J. (2006). Identification of joint interventional distributions in recursive semi-Markovian causal models. In: Proceedings of the 21th National Conference on Artificial Intelligence, Menlo Park, CA; Cambridge, MA; London: AAAI Press; MIT Press; 1999, 1219–1226.Google Scholar
Tian, J. and Pearl, J. (2002). A general identification condition for causal effects. In: Proceedings of the Eighteenth National Conference on Artificial Intelligence and Fourteenth Conference on Innovative Applications of Artificial Intelligence, July 28–August 1, 2002, Edmonton, Alberta, Canada, 567–573.Google Scholar
Wootters, W. K. and Zurek, W. H. (1982). A single quantum cannot be cloned. Nature 299 (5886) 802803.CrossRefGoogle Scholar