Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Causal Inference

Ed. by Imai, Kosuke / Pearl, Judea / Petersen, Maya Liv / Sekhon, Jasjeet / van der Laan, Mark J.

See all formats and pricing
More options …

The Inflation Technique for Causal Inference with Latent Variables

Elie WolfeORCID iD: https://orcid.org/0000-0002-6960-3796 / Robert W. Spekkens / Tobias Fritz
Published Online: 2019-07-16 | DOI: https://doi.org/10.1515/jci-2017-0020


The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the inflation technique for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution’s incompatibility with the causal structure (of which Bell inequalities and Pearl’s instrumental inequality are prominent examples). We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum (and post-quantum) generalizations of the notion of a causal model.

Keywords: causal inference with latent variables; inflation technique; causal compatibility inequalities; marginal problem; Bell inequalities; Hardy paradox; graph symmetries; quantum causal models; GPT causal models, triangle scenario


  • 1.

    Pearl J. Causality: Models, reasoning, and inference. Cambridge University Press; 2009.Google Scholar

  • 2.

    Spirtes P, Glymour C, Scheines R. Causation, prediction, and search. Lecture Notes in Statistics. New York: Springer; 2011.Google Scholar

  • 3.

    Studený M. Probabilistic conditional independence structures. Information Science and Statistics. London: Springer; 2005.Google Scholar

  • 4.

    Koller D. Probabilistic graphical models: Principles and techniques. MIT Press; 2009.Google Scholar

  • 5.

    Pearl J. Theoretical impediments to machine learning with seven sparks from the causal revolution. arXiv:1801.04016 (2018).Google Scholar

  • 6.

    Rosset D, Gisin N, Wolfe E. Universal bound on the cardinality of local hidden variables in networks. Quantum Inf Comput 2018;18.Google Scholar

  • 7.

    Geiger D, Meek C. Graphical models and exponential families. In: Proc 14th conf uncert artif intell. AUAI; 1998. p. 156–65.Google Scholar

  • 8.

    Lee CM, Spekkens RW. Causal inference via algebraic geometry: Feasibility tests for functional causal structures with two binary observed variables. J Causal Inference. 2017;5:20160013.Google Scholar

  • 9.

    Chaves R. Polynomial Bell inequalities. Phys Rev Lett. 2016;116:010402.Google Scholar

  • 10.

    Geiger D, Meek C. Quantifier elimination for statistical problems. In: Proc 15th conf uncert artif intell. AUAI; 1999. p. 226–35.Google Scholar

  • 11.

    Garcia LD, Stillman M, Sturmfels B. Algebraic geometry of bayesian networks. J Symb Comput. 2005;39:331.CrossrefGoogle Scholar

  • 12.

    Garcia LD. Algebraic statistics in model selection. In: Proc 20th conf uncert artif intell. AUAI; 2004. p. 177–84.Google Scholar

  • 13.

    Garcia-Puente LD, Spielvogel S, Sullivant S. Identifying causal effects with computer algebra. In: Proc 26th conf uncert artif intell. AUAI; 2010. p. 193–200.Google Scholar

  • 14.

    Tian J, Pearl J. On the testable implications of causal models with hidden variables. In: Proc 18th conf uncert artif intell. AUAI; 2002. p. 519–27.Google Scholar

  • 15.

    Kang C, Tian J. Inequality constraints in causal models with hidden variables. In: Proc 22nd conf uncert artif intell. AUAI; 2006. p. 233–40.Google Scholar

  • 16.

    Kang C, Tian J. Polynomial constraints in causal bayesian networks. In: Proc 23rd conf uncert artif intell. AUAI; 2007. p. 200–8.Google Scholar

  • 17.

    Bell JS. On the Einstein-Podolsky-Rosen paradox. Physics. 1964;1:195.CrossrefGoogle Scholar

  • 18.

    Clauser JF, Horne MA, Shimony A, Holt RA. Proposed experiment to test local hidden-variable theories. Phys Rev Lett. 1969;23:880.CrossrefGoogle Scholar

  • 19.

    Wood CJ, Spekkens RW. The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning. New J Phys. 2015;17:033002.Google Scholar

  • 20.

    Brunner N, Cavalcanti D, Pironio S, Scarani V, Wehner S. Bell nonlocality. Rev Mod Phys. 2014;86:419.CrossrefGoogle Scholar

  • 21.

    Fritz T. Beyond Bell’s theorem: correlation scenarios. New J Phys. 2012;14:103001.Google Scholar

  • 22.

    Henson J, Lal R, Pusey MF. Theory-independent limits on correlations from generalized Bayesian networks. New J Phys. 2014;16:113043.Google Scholar

  • 23.

    Fritz T. Beyond Bell’s theorem II: Scenarios with arbitrary causal structure. Commun Math Phys. 2015;341:391.Google Scholar

  • 24.

    Chaves R, Budroni C. Entropic nonsignaling correlations. Phys Rev Lett. 2016;116:240501.Google Scholar

  • 25.

    Chaves R, Luft L, Maciel TO, Gross D, Janzing D, Schölkopf B. Inferring latent structures via information inequalities. In: Proc 30th conf uncert artif intell. AUAI; 2014. p. 112–21.Google Scholar

  • 26.

    Weilenmann M, Colbeck R. Non-Shannon inequalities in the entropy vector approach to causal structures. Quantum. 2018;2:57.CrossrefGoogle Scholar

  • 27.

    Kela A, von Prillwitz K, Åberg J, Chaves R, Gross D. Semidefinite tests for latent causal structures. arXiv:1701.00652 (2017).Google Scholar

  • 28.

    Tavakoli A, Skrzypczyk P, Cavalcanti D, Acín A. Nonlocal correlations in the star-network configuration. Phys Rev A. 2014;90:062109.Google Scholar

  • 29.

    Rosset D, Branciard C, Barnea TJ, Pütz G, Brunner N, Gisin N. Nonlinear Bell inequalities tailored for quantum networks. Phys Rev Lett. 2016;116:010403.Google Scholar

  • 30.

    Tavakoli A. Bell-type inequalities for arbitrary noncyclic networks. Phys Rev A. 2016;93:030101.Google Scholar

  • 31.

    Pearl J. On the Testability of Causal Models with Latent and Instrumental Variables. In: Proc 11th conf uncert artif intell. AUAI; 1995. p. 435–43.Google Scholar

  • 32.

    Steudel B, Ay N. Information-theoretic inference of common ancestors. Entropy. 2015;17:2304.CrossrefGoogle Scholar

  • 33.

    Chaves R, Luft L, Gross D. Causal structures from entropic information: geometry and novel scenarios. New J Phys. 2014;16:043001.Google Scholar

  • 34.

    Evans RJ. Graphical methods for inequality constraints in marginalized DAGs. In: Proc 2012 IEEE intern work MLSP. IEEE; 2012. p. 1–6.Google Scholar

  • 35.

    Fritz T, Chaves R. Entropic inequalities and marginal problems. IEEE Trans Inf Theory. 2013;59:803.CrossrefGoogle Scholar

  • 36.

    Pienaar J. Which causal structures might support a quantum–classical gap? New J Phys. 2017;19:043021.Google Scholar

  • 37.

    Braunstein SL, Caves CM. Information-theoretic Bell inequalities. Phys Rev Lett. 1988;61:662.CrossrefGoogle Scholar

  • 38.

    Schumacher BW. Information and quantum nonseparability. Phys Rev A. 1991;44:7047.CrossrefGoogle Scholar

  • 39.

    Leifer MS, Spekkens RW. Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys Rev A. 2013;88:052130.Google Scholar

  • 40.

    Chaves R, Majenz C, Gross D. Information–theoretic implications of quantum causal structures. Nat Commun. 2015;6:5766.CrossrefGoogle Scholar

  • 41.

    Ried K, Agnew M, Vermeyden L, Janzing D, Spekkens RW, Resch KJ. A quantum advantage for inferring causal structure. Nat Phys. 2015;11:414.CrossrefGoogle Scholar

  • 42.

    Costa F, Shrapnel S. Quantum causal modelling. New J Phys. 2016;18:063032.Google Scholar

  • 43.

    Allen J-MA, Barrett J, Horsman DC, Lee CM, Spekkens RW. Quantum common causes and quantum causal models. Phys Rev X. 2017;7:031021.Google Scholar

  • 44.

    Hardy L. Quantum theory from five reasonable axioms, quant-ph/0101012 (2001).Google Scholar

  • 45.

    Barrett J. Information processing in generalized probabilistic theories. Phys Rev A. 2007;75:032304.Google Scholar

  • 46.

    Boldi P, Vigna S. Fibrations of graphs. Discrete Math. 2002;243:21.CrossrefGoogle Scholar

  • 47.

    Branciard C, Rosset D, Gisin N, Pironio S. Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Phys Rev A. 2012;85:032119.Google Scholar

  • 48.

    Dür W, Vidal G, Cirac JI. Three qubits can be entangled in two inequivalent ways. Phys Rev A. 2000;62:062314.Google Scholar

  • 49.

    Hardy L. Nonlocality for two particles without inequalities for almost all entangled states. Phys Rev Lett. 1993;71:1665.CrossrefGoogle Scholar

  • 50.

    Mansfield S, Fritz T. Hardy’s non-locality paradox and possibilistic conditions for non-locality. Found Phys. 2012;42:709.CrossrefGoogle Scholar

  • 51.

    Bell JS. On the problem of hidden variables in quantum mechanics. Rev Mod Phys. 1966;38:447.CrossrefGoogle Scholar

  • 52.

    Donohue JM, Wolfe E. Identifying nonconvexity in the sets of limited-dimension quantum correlations. Phys Rev A. 2015;92:062120.Google Scholar

  • 53.

    Steeg GV, Galstyan A. A sequence of relaxations constraining hidden variable models. In: Proc 27th conf uncert artif intell. AUAI; 2011. p. 717–26.Google Scholar

  • 54.

    Cirel’son BS. Quantum generalizations of Bell’s inequality. Lett Math Phys. 1980;4:93. Available at http://www.tau.ac.il/~tsirel/download/qbell80.html.CrossrefGoogle Scholar

  • 55.

    Popescu S, Rohrlich D. Quantum nonlocality as an axiom. Found Phys. 1994;24:379.CrossrefGoogle Scholar

  • 56.

    Barrett J, Pironio S. Popescu-Rohrlich correlations as a unit of nonlocality. Phys Rev Lett. 2005;95:140401.Google Scholar

  • 57.

    Liang Y-C, Spekkens RW, Wiseman HM. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Phys Rep. 2011;506:1.CrossrefGoogle Scholar

  • 58.

    Roberts D. Aspects of quantum non-locality. PhD thesis. University of Bristol; 2004.Google Scholar

  • 59.

    Pitowsky I. George Boole’s ‘Conditions of possible experience’ and the quantum puzzle. Br J Philos Sci. 1994;45:95.CrossrefGoogle Scholar

  • 60.

    Pitowsky I. Quantum probability – Quantum logic. Lecture Notes in Physics. vol. 321. Springer-Verlag; 1989.Google Scholar

  • 61.

    Kellerer HG. Verteilungsfunktionen mit gegebenen Marginalverteilungen. Z Wahrscheinlichkeitstheorie. 1964;3:247.CrossrefGoogle Scholar

  • 62.

    Leggett AJ, Garg A. Quantum mechanics versus macroscopic realism: is the flux there when nobody looks? Phys Rev Lett. 1985;54:857.CrossrefGoogle Scholar

  • 63.

    Araújo M, Túlio Quintino M, Budroni C, Terra Cunha M, Cabello A. All noncontextuality inequalities for the n-cycle scenario. Phys Rev A. 2013;88:022118.Google Scholar

  • 64.

    Horodecki R, Horodecki P, Horodecki M, Horodecki K. Quantum entanglement. Rev Mod Phys. 2009;81:865.CrossrefGoogle Scholar

  • 65.

    Abramsky S, Brandenburger A. The sheaf-theoretic structure of non-locality and contextuality. New J Phys. 2011;13:113036.Google Scholar

  • 66.

    Vorob’ev NN. Consistent families of measures and their extensions. Theory Probab Appl. 1960;7:147.Google Scholar

  • 67.

    Budroni C, Miklin N, Chaves R. Indistinguishability of causal relations from limited marginals. Phys Rev A. 2016;94:042127.Google Scholar

  • 68.

    Kahle T. Neighborliness of marginal polytopes. Beitr Algebra Geom. 2010;51:45.Google Scholar

  • 69.

    Andersen ED. Certificates of primal or dual infeasibility in linear programming. Comput Optim Appl. 2001;20:171.CrossrefGoogle Scholar

  • 70.

    Garuccio A. Hardy’s approach, Eberhard’s inequality, and supplementary assumptions. Phys Rev A. 1995;52:2535.CrossrefGoogle Scholar

  • 71.

    Cabello A. Bell’s theorem with and without inequalities for the three-qubit Greenberger-Horne-Zeilinger and W states. Phys Rev A. 2002;65:032108.Google Scholar

  • 72.

    Braun D, Choi M-S. Hardy’s test versus the Clauser-Horne-Shimony-Holt test of quantum nonlocality: Fundamental and practical aspects. Phys Rev A. 2008;78:032114.Google Scholar

  • 73.

    Mančinska L, Wehner S. A unified view on Hardy’s paradox and the Clauser–Horne–Shimony–Holt inequality. J Phys A. 2014;47:424027.Google Scholar

  • 74.

    Ghirardi G, Marinatto L. Proofs of nonlocality without inequalities revisited. Phys Lett A. 2008;372:1982.CrossrefGoogle Scholar

  • 75.

    Eiter T, Makino K, Gottlob G. Computational aspects of monotone dualization: A brief survey. Discrete Appl Math. 2008;156:2035.CrossrefGoogle Scholar

  • 76.

    Barrett C, Fontaine P, Tinelli C. The satisfiability modulo theories library. 2016. www.SMT-LIB.org.Google Scholar

  • 77.

    Fordan A. Projection in constraint logic programming. Ios Press; 1999.Google Scholar

  • 78.

    Dantzig GB, Eaves BC. Fourier-Motzkin elimination and its dual. J Comb Theory, Ser A. 1973;14:288.CrossrefGoogle Scholar

  • 79.

    Bastrakov SI, Zolotykh NY. Fast method for verifying Chernikov rules in Fourier-Motzkin elimination. Comput Math Math Phys. 2015;55:160.CrossrefGoogle Scholar

  • 80.

    Balas E. Projection with a minimal system of inequalities. Comput Optim Appl. 1998;10:189.CrossrefGoogle Scholar

  • 81.

    Jones CN, Kerrigan EC, Maciejowski JM. On polyhedral projection and parametric programming. J Optim Theory Appl. 2008;138:207.CrossrefGoogle Scholar

  • 82.

    Jones C. Polyhedral tools for control. PhD thesis. University of Cambridge; 2005.Google Scholar

  • 83.

    Jones C, Kerrigan EC, Maciejowski J. Equality set projection: A new algorithm for the projection of polytopes in halfspace representation. Tech Rep. Cambridge University Engineering Dept; 2004.Google Scholar

  • 84.

    Spekkens RW. The paradigm of kinematics and dynamics must yield to causal structure. In: Questioning the foundations of physics: which of our fundamental assumptions are wrong? Springer International; 2015. p. 5–16.Google Scholar

  • 85.

    Henson J. Causality, Bell’s theorem, and ontic definiteness. arXiv:1102.2855 (2011).Google Scholar

  • 86.

    Barrett J, Linden N, Massar S, Pironio S, Popescu S, Roberts D. Nonlocal correlations as an information-theoretic resource. Phys Rev A. 2005;71:022101.Google Scholar

  • 87.

    Scarani V. The device-independent outlook on quantum physics. Acta Phys Slovaca. 2012;62:347.Google Scholar

  • 88.

    Bancal J-D. On the device-independent approach to quantum physics. Springer International; 2014.Google Scholar

  • 89.

    Chaves R, Fritz T. Entropic approach to local realism and noncontextuality. Phys Rev A. 2012;85:032113.Google Scholar

  • 90.

    Barnum H, Wilce A. Post-classical probability theory. arXiv:1205.3833 (2012).Google Scholar

  • 91.

    Janotta P, Hinrichsen H. Generalized probability theories: what determines the structure of quantum theory? J Phys A. 2014;47:323001.Google Scholar

  • 92.

    Fraser TC, Wolfe E. Causal compatibility inequalities admitting quantum violations in the triangle structure. Phys Rev A. 2018;98:022113.Google Scholar

  • 93.

    Barnum H, Caves CM, Fuchs CA, Jozsa R, Schumacher B. Noncommuting mixed states cannot be broadcast. Phys Rev Lett. 1996;76:2818.CrossrefGoogle Scholar

  • 94.

    Barnum H, Barrett J, Leifer M, Wilce A. Cloning and broadcasting in generic probabilistic theories. quant-ph/0611295 (2006).Google Scholar

  • 95.

    Popescu S. Nonlocality beyond quantum mechanics. Nat Phys. 2014;10:264.CrossrefGoogle Scholar

  • 96.

    Yang TH, Navascués M, Sheridan L, Scarani V. Quantum Bell inequalities from macroscopic locality. Phys Rev A. 2011;83:022105.Google Scholar

  • 97.

    Rohrlich D. PR-box correlations have no classical limit. In: Quantum theory: A two-time success story. Milan: Springer; 2014. p. 205–11.Google Scholar

  • 98.

    Pawlowski M, Scarani V. Information causality. arXiv:1112.1142 (2011).Google Scholar

  • 99.

    Fritz T, Sainz AB, Augusiak R, Brask JB, Chaves R, Leverrier A, Acin A. Local orthogonality as a multipartite principle for quantum correlations. Nat Commun. 2013;4:2263.CrossrefGoogle Scholar

  • 100.

    Sainz AB, Fritz T, Augusiak R, Brask JB, Chaves R, Leverrier A, Acín A. Exploring the local orthogonality principle. Phys Rev A. 2014;89:032117.Google Scholar

  • 101.

    Cabello A. Simple explanation of the quantum limits of genuine n-body nonlocality. Phys Rev Lett. 2015;114:220402.Google Scholar

  • 102.

    Barnum H, Müller MP, Ududec C. Higher-order interference and single-system postulates characterizing quantum theory. New J Phys. 2014;16:123029.Google Scholar

  • 103.

    Navascués M, Guryanova Y, Hoban MJ, Acín A. Almost quantum correlations. Nat Commun. 2015;6:6288.CrossrefGoogle Scholar

  • 104.

    Kang C, Tian J. Polynomial constraints in causal bayesian networks. In: Proc 23rd conf uncert artif intell. AUAI; 2007. p. 200–8.Google Scholar

  • 105.

    Navascués M, Pironio S, Acín A. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J Phys. 2008;10:073013.Google Scholar

  • 106.

    Pál KF, Vértesi T. Quantum bounds on Bell inequalities. Phys Rev A. 2009;79:022120.Google Scholar

  • 107.

    Wolfe E, et al. Quantum inflation: A general approach to quantum causal compatibility (in preparation).

  • 108.

    Navascués M, Wolfe E. The inflation technique completely solves the causal compatibility problem. arXiv:1707.06476 (2017).Google Scholar

  • 109.

    Avis D, Bremner D, Seidel R. How good are convex hull algorithms? Comput Geom. 1997;7:265.CrossrefGoogle Scholar

  • 110.

    Fukuda K, Prodon A. Double description method revisited. In: Combin & Comp Sci. Springer-Verlag; 1996. p. 91–111.Google Scholar

  • 111.

    Shapot DV, Lukatskii AM. Solution building for arbitrary system of linear inequalities in an explicit form. Am J Comput Math. 2012;02:1.CrossrefGoogle Scholar

  • 112.

    Avis D. A revised implementation of the reverse search vertex enumeration algorithm. In: Polytopes — Combinatorics and computation. DMV Seminar. vol. 29. Basel: Birkhäuser; 2000. p. 177–98.Google Scholar

  • 113.

    Gläßle T, Gross D, Chaves R. Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling. J Phys A. 2018;51:484002.Google Scholar

  • 114.

    Bremner D, Sikiric MD, Schürmann A. Polyhedral representation conversion up to symmetries. In: Polyhedral computation. CRM proc lecture notes, vol. 48. Amer Math Soc; 2009. p. 45–71.Google Scholar

  • 115.

    Schürmann A. Exploiting symmetries in polyhedral computations. Disc Geom Optim. 2013;265.Google Scholar

  • 116.

    Kaibel V, Liberti L, Schürmann A, Sotirov R. Mini-workshop: Exploiting symmetry in optimization. Oberwolfach Rep. 2010;7:2245.Google Scholar

  • 117.

    Rehn T, Schürmann A. C++ tools for exploiting polyhedral symmetries. In: Proc 3rd int congr conf math soft, ICMS’10. Springer-Verlag; 2010. p. 295–8.Google Scholar

  • 118.

    Lörwald S, Reinelt G. PANDA: a software for polyhedral transformations. EURO J Comp Optim. 2015;3:297.CrossrefGoogle Scholar

  • 119.

    Yeung RW. Beyond Shannon-type inequalities. In: Information theory and network coding. Springer US; 2008. p. 361–86.Google Scholar

  • 120.

    Kaced T. Equivalence of two proof techniques for non-Shannon-type inequalities. In: Information theory proceedings (ISIT). IEEE; 2013. p. 236–40.Google Scholar

  • 121.

    Dougherty R, Freiling CF, Zeger K. Non-Shannon information inequalities in four random variables. arXiv:1104.3602 (2011).Google Scholar

  • 122.

    Fine A. Hidden variables, joint probability, and the Bell inequalities. Phys Rev Lett. 1982;48:291.CrossrefGoogle Scholar

  • 123.

    Namioka I, Phelps R. Tensor products of compact convex sets. Pac J Math. 1969;31:469.CrossrefGoogle Scholar

  • 124.

    Bogart T, Contois M, Gubeladze J. Hom-polytopes. Math Z. 2013;273:1267.CrossrefGoogle Scholar

About the article

Received: 2017-08-16

Revised: 2018-08-31

Accepted: 2019-06-10

Published Online: 2019-07-16

Funding Source: John Templeton Foundation

Award identifier / Grant number: 69609

This project/publication was made possible in part through the support of a grant #69609 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.

Citation Information: Journal of Causal Inference, 20170020, ISSN (Online) 2193-3685, DOI: https://doi.org/10.1515/jci-2017-0020.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in