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Erkenntnis

, Volume 83, Issue 3, pp 369–389 | Cite as

Learning and Pooling, Pooling and Learning

  • Rush T. Stewart
  • Ignacio Ojea Quintana
Original Research

Abstract

We explore which types of probabilistic updating commute with convex IP pooling (Stewart and Ojea Quintana 2017). Positive results are stated for Bayesian conditionalization (and a mild generalization of it), imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of (precise) externally Bayesian pooling operators due to Wagner (Log J IGPL 18(2):336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

Notes

Acknowledgements

The bulk of this work was done while we were on a Junior Group Visiting Fellowship at the Munich Center for Mathematical Philosophy. The paper benefited from conversations with Stephan Hartmann and Hannes Leitgeb. We would especially like to thank Greg Wheeler for feedback, numerous relevant discussions, and support. We are grate- ful to Matt Duncan, Robby Finley, Arthur Heller, Isaac Levi, Michael Nielsen, Rohit Parikh, Paul Pedersen, Teddy Seidenfeld, and Reuben Stern for their excellent comments on drafts or presentations of the pa- per. Finally, thanks to an anonymous referee for his or her meticulous and valuable review.

References

  1. Arló-Costa, H. (2007). The logic of conditionals. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2014 ed.). Stanford University: Metaphysics Research Lab.Google Scholar
  2. Baratgin, J., & Politzer, G. (2010). Updating: A psychologically basic situation of probability revision. Thinking & Reasoning, 16(4), 253–287.CrossRefGoogle Scholar
  3. Blackwell, D., & Dubins, L. (1962). Merging of opinions with increasing information. The Annals of Mathematical Statistics, 33, 882–886.CrossRefGoogle Scholar
  4. Christensen, D. (2009). Disagreement as evidence: The epistemology of controversy. Philosophy Compass, 4(5), 756–767.CrossRefGoogle Scholar
  5. de Finetti, B. (1964). Foresight: Its logical laws, its subjective sources. In H. E. Kyburg & H. E. Smoklery (Eds.), Studies in Subjective Probability. Hoboken: Wiley.Google Scholar
  6. Diaconis, P., & Zabell, S. L. (1982). Updating subjective probability. Journal of the American Statistical Association, 77(380), 822–830.CrossRefGoogle Scholar
  7. Dietrich, F., & List, C. (2014). Probabilistic opinion pooling. In A. Hájek & C. Hitchcock (Eds.), Oxford Handbook of Probability and Philosophy. Oxford: Oxford University Press.Google Scholar
  8. Elga, A. (2007). Reflection and disagreement. Noûs, 41(3), 478–502.CrossRefGoogle Scholar
  9. Elkin, L., & Wheeler, G. (2016). Resolving peer disagreements through imprecise probabilities. Noûs. doi: 10.1111/nous.12143.Google Scholar
  10. Field, H. (1978). A note on jeffrey conditionalization. Philosophy of Science, 45, 361–367.CrossRefGoogle Scholar
  11. Gaifman, H., & Snir, M. (1982). Probabilities over rich languages, testing and randomness. The Journal of Symbolic Logic, 47(03), 495–548.CrossRefGoogle Scholar
  12. Gaifman, H., & Vasudevan, A. (2012). Deceptive updating and minimal information methods. Synthese, 187(1), 147–178.CrossRefGoogle Scholar
  13. Gärdenfors, P. (1982). Imaging and conditionalization. The Journal of Philosophy, 79, 747–760.CrossRefGoogle Scholar
  14. Genest, C. (1984). A characterization theorem for externally bayesian groups. The Annals of Statistics, 12, 1100–1105.CrossRefGoogle Scholar
  15. Genest, C., McConway, K. J., & Schervish, M. J. (1986). Characterization of externally bayesian pooling operators. The Annals of Statistics,14, 487–501.CrossRefGoogle Scholar
  16. Genest, C., & Wagner, C. G. (1987). Further evidence against independence preservation in expert judgement synthesis. Aequationes Mathematicae, 32(1), 74–86.CrossRefGoogle Scholar
  17. Genest, C., & Zidek, J. V. (1986). Combining probability distributions: A critique and an annotated bibliography. Statistical Science, 1, 114–135.CrossRefGoogle Scholar
  18. Girón, F. J., & Ríos, S. (1980). Quasi-bayesian behaviour: A more realistic approach to decision making? Trabajos de Estadística y de Investigación Operativa, 31(1), 17–38.CrossRefGoogle Scholar
  19. Good, I. J. (1983). Good Thinking: The Foundations of Probability and Its Applications. Minneapolis: U of Minnesota Press.Google Scholar
  20. Hájek, A., & Hall, N. (1994). The hypothesis of the conditional construal of conditional probability. In E. Eells & B. Skyrms (Eds.), Probability and conditionals: Belief revision and rational decision (pp. 75–112). Cambridge: Cambridge University Press.Google Scholar
  21. Hartmann, S. (2014). A new solution to the problem of old evidence. In Philosophy of Science Association 24th Biennial Meeting, Chicago, IL.Google Scholar
  22. Herron, T., Seidenfeld, T., & Wasserman, L. (1997). Divisive conditioning: Further results on dilation. Philosophy of Science, 64, 411–444.CrossRefGoogle Scholar
  23. Huttegger, S. M. (2015). Merging of opinions and probability kinematics. The Review of Symbolic Logic, 8(04), 611–648.CrossRefGoogle Scholar
  24. Jeffrey, R. (2004). Subjective Probability: The Real Thing. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  25. Joyce, J. M. (1999). The Foundations of Causal Decision Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  26. Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22, 79–86.CrossRefGoogle Scholar
  27. Kyburg, H. E. (1987). Bayesian and non-bayesian evidential updating. Artificial Intelligence, 31(3), 271–293.CrossRefGoogle Scholar
  28. Kyburg, H.E., Pittarelli, M. (1992). Some problems for convex bayesians. In Proceedings of the Eighth International Conference on Uncertainty in Artificial Intelligence, pp. 149–154. Morgan Kaufmann Publishers Inc.Google Scholar
  29. Leitgeb, H. (2016). Imaging all the people. Episteme. doi: 10.1017/epi.2016.14.
  30. Levi, I. (1967). Probability kinematics. British Journal for the Philosophy of Science, 18(3), 197–209.CrossRefGoogle Scholar
  31. Levi, I. (1970). Probability and evidence. In M. Swain (Ed.), Induction, Acceptance, and Rational Belief (pp. 134–156). New York: Humanities Press.CrossRefGoogle Scholar
  32. Levi, I. (1978). Irrelevance. In C. Hooker, J. Leach, & E. McClennen (Eds.), Foundations and Applications of Decision Theory (Vol. 1, pp. 263–273). Boston: Springer.Google Scholar
  33. Levi, I. (1980). The Enterprise of Knowledge. Cambridge, MA: MIT Press.Google Scholar
  34. Levi, I. (1985). Consensus as shared agreement and outcome of inquiry. Synthese, 62(1), 3–11.CrossRefGoogle Scholar
  35. Levi, I. (1990). Pareto unanimity and consensus. The Journal of Philosophy, 87(9), 481–492.CrossRefGoogle Scholar
  36. Levi, I. (1996). For the Sake of the Argument: Ramsey Test Conditionals, Inductive Inference and Nonmonotonic Reasoning. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  37. Levi, I. (2009). Why indeterminate probability is rational. Journal of Applied Logic, 7(4), 364–376.CrossRefGoogle Scholar
  38. Lewis, D. (1976). Probabilities of conditionals and conditional probabilities. The Philosophical Review, 85, 297–315.CrossRefGoogle Scholar
  39. Madansky, A. (1964). Externally Bayesian Groups. Santa Monica, CA: RAND Corporation.Google Scholar
  40. Nau, R. F. (2002). The aggregation of imprecise probabilities. Journal of Statistical Planning and Inference, 105(1), 265–282.CrossRefGoogle Scholar
  41. Pedersen, A. P., & Wheeler, G. (2014). Demystifying dilation. Erkenntnis, 79(6), 1305–1342.CrossRefGoogle Scholar
  42. Raiffa, H. (1968). Decision analysis: Introductory lectures on choices under uncertainty. Random House.Google Scholar
  43. Ramsey, F. P. (1990). Truth and probability. In D. H. Mellor (Ed.), Philosophical Papers (pp. 52–109). Cambridge University Press.Google Scholar
  44. Russell, J. S., Hawthorne, J., & Buchak, L. (2015). Groupthink. Philosophical Studies, 172(5), 1287–1309.CrossRefGoogle Scholar
  45. Savage, L. (1972, originally published in 1954). The Foundations of Statistics. New York: Wiley.Google Scholar
  46. Schervish, M., & Seidenfeld, T. (1990). An approach to consensus and certainty with increasing evidence. Journal of Statistical Planning and Inference, 25(3), 401–414.CrossRefGoogle Scholar
  47. Seidenfeld, T. (1986). Entropy and uncertainty. Philosophy of Science, 53, 467–491.CrossRefGoogle Scholar
  48. Seidenfeld, T., Kadane, J. B., & Schervish, M. J. (1989). On the shared preferences of two bayesian decision makers. The Journal of Philosophy, 86(5), 225–244.CrossRefGoogle Scholar
  49. Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (2010). Coherent choice functions under uncertainty. Synthese, 172(1), 157–176.CrossRefGoogle Scholar
  50. Seidenfeld, T., & Wasserman, L. (1993). Dilation for sets of probabilities. The Annals of Statistics, 21(3), 1139–1154.CrossRefGoogle Scholar
  51. Skyrms, B. (1986). Choice and Chance: An Introduction to Inductive Logic (3rd ed.). Belmont: Wadsworth Publishing Company.Google Scholar
  52. Spohn, W. (2012). The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford: Oxford University Press.CrossRefGoogle Scholar
  53. Stewart, R. T. & Ojea Quintana, I. (2017). Probabilistic opinion pooling with imprecise probabilities. Journal of Philosophical Logic. doi: 10.1007/s10992-016-9415-9.Google Scholar
  54. van Fraassen, B. C. (1989). Laws and Symmetry. Oxford: Clarendon Press.CrossRefGoogle Scholar
  55. Wagner, C. (2002). Probability kinematics and commutativity. Philosophy of Science, 69(2), 266–278.CrossRefGoogle Scholar
  56. Wagner, C. (2009). Jeffrey conditioning and external bayesianity. Logic Journal of IGPL, 18(2), 336–345.CrossRefGoogle Scholar
  57. Williams, P. M. (1980). Bayesian conditionalisation and the principle of minimum information. British Journal for the Philosophy of Science, 31, 131–144.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of PhilosophyColumbia UniversityNew YorkUSA

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