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Synthese

, Volume 156, Issue 3, pp 563–585 | Cite as

The reference class problem is your problem too

  • Alan Hájek
Original Paper

Abstract

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions.

I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains.

Keywords

Probability Conditional probability Reference class problem Frequentist Classical Logical Propensity Subjectivist interpretations of probability Kolmogorov Popper 

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References

  1. Ayer A.J. (1963). Two notes on probability. In: Ayer A.J. (eds). The concept of a person and other essays. MacMillan, London, pp. 188–208Google Scholar
  2. Bertrand, J. (1889). Calcul des Probabilités (1st ed.). Gauthier-Villars.Google Scholar
  3. Carnap, R. (1950). Logical foundations of probability. University of Chicago Press.Google Scholar
  4. Carnap R. (1963). Replies and systematic expositions. In: Schilpp P.A. (eds). The philosophy of Rudolf Carnap. Open Court, LaSalle, IL, pp. 966–998Google Scholar
  5. Church A. (1940). On the concept of a random sequence. Bulletin of the American Mathematical Society 46, 130–135CrossRefGoogle Scholar
  6. de Finetti, B. (1937). Foresight: Its logical laws, its subjective sources, translated in Kyburg and Smokler (1964).Google Scholar
  7. Fetzer, J. (1977). Reichenbach, reference classes, and single case ‘Probabilities’. Synthese, 34, 185–217; Errata, 37, 113–114.Google Scholar
  8. Fetzer J. (1982). Probabilistic explanations. PSA 2, 194–207Google Scholar
  9. Fine, T. (1973). Theories of probability. Academic Press.Google Scholar
  10. Frieden, B. R. (1991). Probability, statistical optics, and data testing. Springer-Verlag.Google Scholar
  11. Gaifman, H. (1988). A theory of higher order probabilities. In B. Skyrms, & W. L. Harper (Eds.), Causation, chance, and credence. Kluwer.Google Scholar
  12. Giere, R. N. (1973). Objective single-case probabilities and the foundations of statistics. In P. Suppes et al. (Eds.), Logic, methodology and philosophy of science IV (pp. 467–483). North Holland.Google Scholar
  13. Gillies D. (2000). Varieties of propensity. British Journal of Philosophy of Science 51, 807–835CrossRefGoogle Scholar
  14. Goldstein M. (1983). The prevision of a prevision. Journal of the American Statistical Association 77, 817–819CrossRefGoogle Scholar
  15. Hacking, I. (1965). Logic of statistical inference. Cambridge University Press.Google Scholar
  16. Hájek, A. (1997). ‘Mises Redux’—Redux: Fifteen arguments against finite frequentism. In Erkenntnis, Vol. 45 (pp. 209–227). Reprinted in Probability, dynamics and causality—Essays in Honor of R. C. Jeffrey, D. Costantini, & M. Galavotti (Eds.), Kluwer.Google Scholar
  17. Hájek, A. (2003a). Conditional probability is the very guide of life. In H. Kyburg Jr., & M. Thalos (Eds.), Probability is the very guide of life: The philosophical uses of chance (pp. 183–203). Open Court. Abridged version in Proceedings of the International Society for Bayesian Analysis 2002.Google Scholar
  18. Hájek A. (2003b). What conditional probability could not be. Synthese 137(3): 273–323CrossRefGoogle Scholar
  19. Hild, M. (in preparation a). An introduction to induction.Google Scholar
  20. Hild, M. (in preparation b). Introduction to the concept of probability: A reader.Google Scholar
  21. Jeffrey, R. (1992). Probability and the art of judgment. Cambridge University Press.Google Scholar
  22. Jeffreys, H. (1939). Theory of probability. Reprinted in Oxford Classics in the Physical Sciences series, Oxford University Press, 1998.Google Scholar
  23. Johnson W.E. (1932). Probability: The deductive and inductive problems. Mind 49, 409–423CrossRefGoogle Scholar
  24. Keynes, J. M. (1921). Treatise on probability. London. Macmillan. Reprinted 1962, New York: Harper and Row.Google Scholar
  25. Laplace, P. S. (1814). Essai Philosophique sur les Probabilités. Paris. Translated into English as A philosophical essay on probabilities, New York, 1952.Google Scholar
  26. Levi I. (1990). Chance. Philosophical Topics 18(2): 117–149Google Scholar
  27. Lewis, D. (1980). A subjectivist’s guide to objective chance. In Philosophical papers Volume II. Oxford University Press.Google Scholar
  28. Mellor D.H. (1971). The matter of chance. Cambridge University Press, CambridgeGoogle Scholar
  29. Miller D.W. (1994). Critical rationalism: A restatement and defence. Open Court, La Salle, IllinoisGoogle Scholar
  30. Miller, D. W. (1996). Propensities and indeterminism. In A. O’Hear (Ed.), Karl Popper: Philosophy and problems (pp. 121–147). Cambridge University Press.Google Scholar
  31. Peirce, C. S. (1867): Review of Venn (1866), reprinted in Writings of Charles S. Peirce, Vol. 2, pp. 98–102.Google Scholar
  32. Popper K. (1959a). The propensity interpretation of probability. British Journal of Philosophy of Science 10, 25–42CrossRefGoogle Scholar
  33. Popper, K. (1959b). The logic of scientific discovery. Basic Books.Google Scholar
  34. Reichenbach, H. (1949). The theory of probability. University of California Press.Google Scholar
  35. Renyi, A. (1970). Foundations of probability. Holden-Day, Inc.Google Scholar
  36. Roeper, P., & Leblanc, H. (1999). Probability theory and probability semantics. Toronto Studies in Philosophy.Google Scholar
  37. Sober, E. (2000). Philosophy of biology (2nd ed.). Westview Press.Google Scholar
  38. Spohn W. (1986). The representation of Popper measures. Topoi 5, 69–74CrossRefGoogle Scholar
  39. Strevens M. (1998). Inferring probabilities from symmetries. Noûs 32(2): 231–246Google Scholar
  40. Strevens, M. (2003). Bigger than chaos. Harvard University Press.Google Scholar
  41. van Fraassen B. (1984). Belief and the will. Journal of Philosophy 81, 235–256CrossRefGoogle Scholar
  42. van Fraassen B. (1989). Laws and symmetry. Clarendon Press, OxfordGoogle Scholar
  43. van Fraassen B. (1995). Belief and the problem of Ulysses and the Sirens. Philosophical Studies 77, 7–37CrossRefGoogle Scholar
  44. Venn, J. (1876). The logic of chance (2nd ed.). Macmillan and Co; originally published 1866.Google Scholar
  45. von Mises, R. (1957). Probability, statistics and truth, revised English edition, New York.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Research School of the Social SciencesAustralian National UniversityCanberraAustralia

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