, Volume 121, Issue 1–2, pp 93–149 | Cite as

Probabilities Of Causation: Three Counterfactual Interpretations And Their Identification

  • Judea Pearl


According to common judicial standard, judgment in favor ofplaintiff should be made if and only if it is “more probable than not” thatthe defendant's action was the cause for the plaintiff's damage (or death). This paper provides formal semantics, based on structural models ofcounterfactuals, for the probability that event x was a necessary orsufficient cause (or both) of another event y. The paper then explicates conditions under which the probability of necessary (or sufficient)causation can be learned from statistical data, and shows how data fromboth experimental and nonexperimental studies can be combined to yieldinformation that neither study alone can provide. Finally, we show thatnecessity and sufficiency are two independent aspects of causation, andthat both should be invoked in the construction of causal explanations for specific scenarios.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aldrich, J.: 1993, 'Cowles' Exogeneity and Core Exogeneity', Technical Report Discussion Paper 9308, Department of Economics, University of Southampton, Southampton.Google Scholar
  2. Angrist, J. D., G. W. Imbens, and D. B. Rubin: 1996, 'Identification of Causal Effects Using Instrumental Variables (with Comments)', Journal of the American Statistical Association 91, 444–472.Google Scholar
  3. Balke, A. and J. Pearl: 1994a, 'Counterfactual Probabilities: Computational Methods, Bounds and Applications', in R. Lopez de Mantaras and D. Poole (eds.), Uncertainty in Artificial Intelligence 10, Morgan Kaufmann, San Mateo, CA, pp. 46–54.Google Scholar
  4. Balke, A. and J. Pearl: 1994b, 'Probabilistic Evaluation of Counterfactual Queries', in Proceedings of the Twelfth National Conference on Artificial Intelligence, Volume I, MIT Press, Cambridge, MA, pp. 230–237.Google Scholar
  5. Balke, A. and J. Pearl: 1995, 'Counterfactuals and Policy Analysis in Structural Models', in P. Besnard and S. Hanks (eds.), Uncertainty in Artificial Intelligence 11, Morgan Kaufmann, San Francisco, CA, pp. 11–18.Google Scholar
  6. Balke, A. and J. Pearl: 1997, 'Nonparametric Bounds on Causal Effects from Partial Compliance Data', Journal of the American Statistical Association 92, 1–6.Google Scholar
  7. Breslow, N. E. and N. E. Day: 1980, Statistical Methods in Cancer Research, Vol. 1; The Analysis of Case-Control Studies, IARC, Lyon.Google Scholar
  8. Cartwright, N.: 1989, Nature's Capacities and Their Measurement, Clarendon, Oxford.Google Scholar
  9. Cheng, P. W.: 1997, 'From Covariation to Causation: A Causal Power Theory', Psychological Review 104, 367–405.Google Scholar
  10. Cole, P.: 1997, 'Causality in Epidemiology, Health Policy, and Law', Journal of Marketing Research 27, 10279–10285.Google Scholar
  11. Dawid, A. P.: 1997, 'Causal Inference without Counterfactuals', Technical Report, Department of Statistical Science, University College London. Forthcoming (with discussion), Journal of the American Statistical Association.Google Scholar
  12. Dhrymes, P. J.: 1970, Econometrics, Springer-Verlag, New York.Google Scholar
  13. Eells, E.: 1991, Probabilistic Causality, Cambridge University Press, Cambridge.Google Scholar
  14. Engle, R. F., Hendry, D. F., and Richard, J. F.: 1983, 'Exogeneity', Econometrica 51, 277–304.Google Scholar
  15. Fine, K.: 1975, 'Review of Lewis' Counterfactuals', Mind 84, 451–458.Google Scholar
  16. Fine, K.: 1985, Reasoning with Arbitrary Objects, B. Blackwell, New York.Google Scholar
  17. Finkelstein, M. O. and Levin, B.: 1990, Statistics for Lawyers, Springer-Verlag, New York.Google Scholar
  18. Fisher, F. M.: 1970, 'A Correspondence Principle for Simultaneous Equations Models', Econometrica 38, 73–92.Google Scholar
  19. Fleiss, J. L.: 1981, Statistical Methods for Rates and Proportions (second edition), John Wiley and Sons, New York.Google Scholar
  20. Galles, D. and Pearl, J. 1995, 'Testing Identifiability of Causal Effects', in P. Besnard and S. Hanks (eds.), Uncertainty in Artificial Intelligence 11, Morgan Kaufmann, San Francisco, CA, pp. 185–195. See also Pearl (2000), Chapter 4.Google Scholar
  21. Galles, D. and Pearl, J.: 1997, 'Axioms of Causal Relevance', Artificial Intelligence 97, 9–43.Google Scholar
  22. Galles, D. and J. Pearl: 1998, 'An Axiomatic Characterization of Causal Counterfactuals', Foundations of Science 3, 151–182.Google Scholar
  23. Glymour, C.: 1998, 'Psychological and Normative Theories of Causal Power and Probabilities of Causes', in G. F. Cooper and S. Moral (eds.), Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA, pp. 166–172.Google Scholar
  24. Goldszmidt, M. and J. Pearl: 1992, 'Rank-Based Systems: A Simple Approach to Belief Revision, Belief Update, and Reasoning about Evidence and Actions', in B. Nebel, C. Rich and W. Swartout (eds.), Proceedings of the Third International Conference on Knowledge Representation and Reasoning, Morgan Kaufmann, San Francisco, CA, pp. 661–672.Google Scholar
  25. Good, I. J.: 1961, 'A Causal Calculus, I', British Journal for the Philosophy of Science 11, 305–318.Google Scholar
  26. Good, I. J.: 1993, 'A Tentative Measure of Probabilistic Causation Relevant to the Philosophy of the Law', J. Statist. Comput. and Simulation 47, 99–105.Google Scholar
  27. Greenland, S. and J. Robins: 1988, 'Conceptual Problems in the Definition and Interpretation of Attributable Fractions', American Journal of Epidemiology 128, 1185–1197.]Google Scholar
  28. Hall, N.: 1998, Two Concepts of Causation (in press).Google Scholar
  29. Halpern, J. Y.: 1998, 'Axiomatizing Causal Reasoning', in G. F. Cooper and S. Moral (eds.), Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA, pp. 202–210.Google Scholar
  30. Heckermann, D. and R. Schachter: 1995a, 'Decision-Theoretic Foundations for Causal Reasoning', Journal of Artificial Intelligence Research 3, 405–430.Google Scholar
  31. Heckermann, D. and R. Schachter: 1995b, 'A Definition and Graphical Representation for Causality', in Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence, San Mateo, CA, pp. 262–273.Google Scholar
  32. Hendry, D.: 1995, Dynamic Econometrics, Oxford University Press, New York.Google Scholar
  33. Hennekens, C. H. and J. E. Buring, Epidemiology in Medicine, Brown, Little, Boston.Google Scholar
  34. Hume, D.: 1948, An Enquiry Concerning Human Understanding, Open Court Press, LaSalle.Google Scholar
  35. Imbens, G. W.: 1997, 'Book Reviews', Journal of Applied Econometrics 12.Google Scholar
  36. Kelsey, J. L., A. S. Whittemore, A. S. Evans, and W. D. Thompson: Methods in Observational Epidemiology, Oxford University Press, New York.Google Scholar
  37. Khoury, M. J., W. D. Flanders, S. Greenland, and M. J. Adams: 1989, 'On the Measurement of Susceptibility in Epidemiology Studies', American Journal of Epidemiology 129, 183–190.Google Scholar
  38. Kim, J.: 'Causes and Events: Mackie on Causation', Journal of Philosophy 68, 426–471.Google Scholar
  39. Lewis, D.: 1979, 'Counterfactual Dependence and Time's Arrow', Nous 13, 418–446.Google Scholar
  40. Lewis, D.: 1986, Philosophical Papers, Oxford University Press, New York.Google Scholar
  41. Mackie, J. L.: 1965, 'Causes and Conditions', American Philosophical Quarterly 2, 261–264. Reprinted in E. Sosa and M. Tooley (eds.), Causation, Oxford University Press.Google Scholar
  42. Marschak, J.: 1950, 'Statistical Inference in Economics', in T. Koopmans (ed.), Statistical Inference in Dynamic Economic Models, John Wiley and Sons, New York, pp. 1–50.Google Scholar
  43. Michie, D.: 1997, 'Adapting Good's q Theory to the Causation of Individual Events', Technical report, University of Edinburgh, UK (submitted for publication in Machine Intelligence 15).Google Scholar
  44. Mill, J. S.: 1843, System of Logic, Volume 1, John W. Parker, London.Google Scholar
  45. Neyman, J.: 1923, 'On the Application of Probability Theory to Agricultural Experiments. Essays on Principles. Section 9', English Translation (1990), Statistical Science 5(4), 465–480.Google Scholar
  46. Pearl, J.: 1993, 'Comment: Graphical Models, Causality, and Interventions', Statistical Science 8, 266–269.Google Scholar
  47. Pearl, J.: 1994, 'A Probabilistic Calculus of Actions', in R. Lopez de Mantaras and D. Poole (eds.), Uncertainty in Artificial Intelligence 10, Morgan Kaufmann, San Mateo, CA, pp. 454–462.Google Scholar
  48. Pearl, J.: 1995, 'Causal Diagrams for Experimental Research', Biometrika 82, 669–710.Google Scholar
  49. Pearl, J.: 1996a, 'Causation, Action, and Counterfactuals', in Y. Shoham (ed.), Theoretical Aspects of Rationality and Knowledge, Proceedings of the Sixth Conference, Morgan Kaufmann, San Francisco, CA, pp. 51–73.Google Scholar
  50. Pearl, J.: 1996b, 'Structural and Probabilistic Causality', in D. R. Shanks, K. J. Holyoak and D. L. Medin (eds.), The Psychology and Learning and Motivations, Volume 34, Academic Press, San Diego, CA, pp. 393–435.Google Scholar
  51. Pearl, J.: 1998, 'On the Definition of Actual Cause', Technical Report R-259, Department of Computer Science, University of California, Los Angeles, CA. Also in Pearl (2000), Chapter 10.Google Scholar
  52. Pearl, J.: 2000, Causality, Cambridge University Press (forthcoming).Google Scholar
  53. Robertson, D. W.: 1997, 'The Common Sense of Cause in Fact', Texas Law Review 75, 1765–1800.Google Scholar
  54. Robins, J. M. and S. Greenland, 1989, 'The Probability of Causation under a Stochastic Model for Individual Risk', Biometrics 45, 1125–1138.Google Scholar
  55. Robins, J. M.: 1986, 'A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period-Applications to Control of the Healthy Workers Survivor Effect', Mathematical Modeling 7, 1393–1512.Google Scholar
  56. Robins, J. M.: 1987, 'A Graphical Approach to the Identification and Estimation of Causal Parameters in Mortality Studies with Sustained Exposure Period', Journal of Chronic Diseases 40, 139–161S.Google Scholar
  57. Rosenbaum, P. and D. Rubin: 1983, 'The Central Role of Propensity Score in Observational Studies for Causal Effects', Biometrica 70, 41–55.Google Scholar
  58. Rubin, D. B.: 1974, 'Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies', Journal of Educational Psychology 66, 688–701.Google Scholar
  59. Schlesselman, J. J.: 1982, Case-Control Studies: Design Conduct Analysis, Oxford University Press, New York.Google Scholar
  60. Shep, M. C.: 1958, 'Shall We Count the Living or the Dead?', New England Journal of Medicine 259, 1210–1214.Google Scholar
  61. Simon, H. A. and N. Rescher: 1966, 'Cause and Counterfactual', Philosophy and Science 33, 323–340.Google Scholar
  62. Simon, H. A.: 1953, 'Causal Ordering and Identifiability', in Wm. C. Hood and T. C. Koopmans (eds.), Studies in Econometric Methods, John Wiley and Sons, New York, pp. 49–74.Google Scholar
  63. Skyrms, B.: 1980, Causal Necessity, Yale University Press, New Haven, CT.Google Scholar
  64. Sobel, M. E.: 1990, 'Effect Analysis and Causation in Linear Structural Equation Models', Psychometrika 55, 495–515.Google Scholar
  65. Spirtes, P., C. Glymour, and R. Scheines: 1993, Causation, Prediction, and Search, Springer-Verlag, New York.Google Scholar
  66. Strotz, R. H. and H. O. A. Wold: 1960, 'Recursive versus Nonrecursive Systems: An Attempt at Synthesis', Econometrica 28, 417–427.Google Scholar
  67. Suppes, P.: 1970, A Probabilistic Theory of Causality, North-Holland Publishing Co., Amsterdam.Google Scholar
  68. Thomason, R. and A. Gupta: 1980, 'A Theory of Conditionals in the Context of Branching Time', Philosophical Review 88, 65–90.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Judea Pearl
    • 1
  1. 1.Cognitive Systems Laboratory Computer Science DepartmentUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations