pp 1–13 | Cite as

Regression to the mean and Judy Benjamin

  • Randall G. McCutcheon


Van Fraassen’s Judy Benjamin problem asks how one ought to update one’s credence in A upon receiving evidence of the sort “A may or may not obtain, but B is k times likelier than C”, where \(\{A,B,C\}\) is a partition. Van Fraassen’s solution, in the limiting case \(k\rightarrow \infty \), recommends a posterior converging to \(P(A|A\cup B)\) (where P is one’s prior probability function). Grove and Halpern, and more recently Douven and Romeijn, have argued that one ought to leave credence in A unchanged, i.e. fixed at P(A). We argue that while the former approach is superior, it brings about a reflection violation due in part to neglect of a “regression to the mean” phenomenon, whereby when C is eliminated by random evidence that leaves A and B alive, the ratio P(A) : P(B) ought to drift in the direction of 1 : 1.


Conditionalization Regression Reflection Judy Benjamin problem 


  1. Douven, I., & Romeijn, J.-W. (2011). A new resolution of the Judy Benjamin problem. Mind, 120, 637–670.CrossRefGoogle Scholar
  2. Ferguson, T. Gambler’s ruin in three dimensions. Manuscript. Accessed February 13, 2018.
  3. Goodman, I. R., & Nguyen, H. T. (1999). Probability updating using second order probabilities and conditional event algebra. Information Sciences, 121, 295–347.CrossRefGoogle Scholar
  4. Grove, A. J., & Halpern, J. Y. (1997). Probability: Conditioning vs. cross-entropy. In Proceedings of the 13th annual conference on uncertainty in artificial intelligence (pp. 208–214). San Francisco: Morgan Kaufmann.Google Scholar
  5. Jaynes, E. T. (2003). The Borel-Kolmogorov paradox. Probability theory: The logic of science (pp. 467–470). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  6. Jeffrey, R. (1965). The logic of decision. University of Chicago Press.Google Scholar
  7. Schervish, M. J., Seidenfeld, T., & Kadane, J. B. (2004). Stopping to reflect. Journal of Philosophy, 101, 315–322.CrossRefGoogle Scholar
  8. Seidenfeld, T. (1986). Entropy and uncertainty. Philosophy of Science, 53, 467–491.CrossRefGoogle Scholar
  9. van Fraassen, B. C. (1981). A problem for relative information minimizers in probability kinematics. The British Journal for the Philosophy of Science, 1981, 375–379.CrossRefGoogle Scholar
  10. van Fraassen, B. C. (1984). Belief and the will. Journal of Philosophy, 81, 235–256.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations