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A Causal Calculus for Statistical Research

  • Judea Pearl
Part of the Lecture Notes in Statistics book series (LNS, volume 112)

Abstract

A calculus is proposed that admits two conditioning operators: ordinary Bayes conditioning, P (y|X = x), and causal conditioning, P (y|set(X = x)), that is, conditioning P (y) on holding X constant (at x) by external intervention. This distinction, which will be supported by three rules of inference, will permit us to derive probability expressions for the combined effect of observations and interventions. The resulting calculus yields simple solutions to a number of interesting problems in causal inference and should allow rank-and-file researchers to tackle practical problems that are generally considered too hard, or impossible. Examples are:
  1. 1.

    Deciding whether the information available in a given observational study is sufficient for obtaining consistent estimates of causal effects.

     
  2. 2.

    Deriving algebraic expressions for causal effect estimands.

     
  3. 3.

    Selecting measurements that would render randomized experiments unnecessary.

     
  4. 4.

    Selecting a set of indirect (randomized) experiments to replace direct experiments that are either infeasible or too expensive.

     
  5. 5.

    Predicting (or bounding) the efficacy of treatments from randomized trials with imperfect compliance.

     
Starting with nonparametric specification of structural equations, the paper establishes the semantics necessary for a theory of interventions, presents the three rules of inference, and proposes an operational definition of structural equations.

Keywords

Causal Effect Causal Theory Manipulate Variable Causal Information Causal Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Balke & Pearl, 1994a]
    Balke, A. and Pearl, J., “Probabilistic evaluation of counterfactual queries,” in Proceedings of the Twelfth National Conference on Artificial Intelligence (AAAI-94), Seattle, WA, Volume I, 230–237, July 31 - August 4, 1994.Google Scholar
  2. [Cox, 1992]
    Cox, D.R.,“Some statistical aspects,” Journal of the Royal Statistical Society, Series A, 155, 291–301, 1992.zbMATHGoogle Scholar
  3. [Cox & Wermuth, 1993]
    Cox, D.R. and Wermuth, N., “Linear dependencies represented by chain graphs,” Statistical Society, 8, 204–218, 1993MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Fisher, 1970]
    Fisher, F.M., “A correspondence principle for simultaneous equation models,” Econometrica, 38, 73–92, 1970.CrossRefGoogle Scholar
  5. [Freedman, 1987]
    Freedman, D., “As others see us: A case study in path analysis” (with discussion), Journal of Educational Statistics, 12, 101–223, 1987.CrossRefGoogle Scholar
  6. [Galles & Pearl, 1995]
    Galles, D. and Pearl, J., “Testing Identifiability of Causal Effects,” in P. Besnard and S. Hanks (Eds.), Uncertainty in Artificial Intelligence 11, Morgan Kaufmann, San Francisco, CA, 185–195, 1995.Google Scholar
  7. [Goldszmidt & Pearl, 1992]
    Goldszmidt, M. and Pearl, J., “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 Mateo, CA, 661–672, October 1992.Google Scholar
  8. [Pearl, 1988]
    Pearl, J., Probabilistic Reasoning in Intelligence Systems, Morgan Kaufmann, San Mateo, CA, 1988.Google Scholar
  9. [Pearl, 1993a]
    Pearl, J., “From Conditional Oughts to Qualitative Decision Theory” in D. Heckerman and A. Mamdani (Eds.), Proceedings of the Ninth Conference on Uncertainty in Artificial Intelligence, Washington, D.C., Morgan Kaufmann, San Mateo, CA, 12–20, July 1993.Google Scholar
  10. [Pearl, 1993b]
    Pearl, J., “Graphical models, causality, and intervention,” Statistical Science, 8 (3), 266–273, 1993.CrossRefGoogle Scholar
  11. [Pearl, 1994b]
    Pearl, J., “A probabilistic calculus of actions,” in R. Lopez de Mantaras and D. Poole (Eds.), Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence (UAI-94), Morgan Kaufmann, San Mateo, CA, 454–462, 1994.Google Scholar
  12. [Pearl, 1995]
    Pearl, J., “Causal diagrams for experimental research, (with discussion),” Biometrika, 82 (4), 669–709, December 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Pratt & Schlaifer, 1988]
    Pratt, J.W. and Schlaifer, R., “On the interpretation and observation of laws,” Journal of Econometrics, 39, 23–52, 1988.MathSciNetCrossRefGoogle Scholar
  14. [Robins, 1986]
    Robins, J., “A new approach to causal inference in mortality studies with a sustained exposure period–applications to control of the healthy workers survivor effect,” Mathematical Modelling, 7, 1393–1512, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Sobel 1990]
    Sobel, M.E., “Effect analysis and causation in linear structural equation models,” Psychometrika, 55 (3), 495–515, 1990.MathSciNetCrossRefGoogle Scholar
  16. [Spirtes et al., 1993]
    Spirtes, P., Glymour, C., and Schienes, R., Causation, Prediction, and Search, Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  17. [Strotz & Wold, 1960]
    Strotz, R.H. and Wold, H.O.A., “Recursive versus nonrecursive systems: An attempt at synthesis,” Econometrica 28, 417–427, 1960.MathSciNetCrossRefGoogle Scholar
  18. [Wermuth, 1992]
    Wermuth, N., “On block-recursive regression equations” (with discus- sion), Brazilian Journal of Probability and Statistics, 6, 1–56, 1992.MathSciNetzbMATHGoogle Scholar
  19. [Whittaker, 1990]
    Whittaker, J., Graphical Models in Applied Multivariate Statistics, John Wiley and Sons, Chinchester, England, 1990.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

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

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