Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-29T17:14:40.287Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW ARGUMENT FOR KOLOMOGOROV CONDITIONALIZATION

Part of: Foundations of probability theory

Published online by Cambridge University Press:  21 July 2020

MICHAEL NIELSEN
MICHAEL NIELSEN*
Affiliation:
DEPARTMENT OF PHILOSOPHY THE UNIVERSITY OF SYDNEYNSW2006, AUSTRALIA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper contributes to a recent research program that extends arguments supporting elementary conditionalization to arguments supporting conditionalization with general, measure-theoretic conditional probabilities. I begin by suggesting an amendment to the framework that Rescorla (2018) has used to characterize regular conditional probabilities in terms of avoiding Dutch book. If we wish to model learning scenarios in which an agent gains complete membership knowledge about some subcollection of the events of interest to her, then we should focus on updating policies that are what I shall call proper. I go on to characterize regular conditional probabilities in proper learning scenarios using what van Fraassen (1999) calls The General Reflection Principle.

Keywords

conditionalizationKolmogorovregular conditional probabilityreflection principle

MSC classification

Primary: 60A05: Axioms; other general questions
Secondary: 60A10: Probabilistic measure theory
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 4 , December 2021 , pp. 930 - 945
Check for updates
Copyright
© Association for Symbolic Logic, 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Aliprantis, C. D., & Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Berlin: Springer-Verlag.Google Scholar
Blackwell, D., & Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. The Annals of Probability, 3(5), 741–752.CrossRefGoogle Scholar
Blackwell, D., & Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. Annals of Mathematical Statistics, 34(1), 223–235.CrossRefGoogle Scholar
de Finetti, B. (1974). Theory of Probability, Vol. 1. Chichester: John Wiley & Sons.Google Scholar
Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. The Annals of Probability, 3(1), 89–99.CrossRefGoogle Scholar
Durrett, R. (2010). Probability: Theory and Examples. New York: Cambridge University Press.CrossRefGoogle Scholar
Good, I. J. (1967). On the principle of total evidence. The British Journal for the Philosophy of Science, 17(4), 319–321.CrossRefGoogle Scholar
Greaves, H., & Wallace, D. (2006). Justifying conditionalization: conditionalization maximizes expected epistemic utility. Mind, 115(459), 607–632.CrossRefGoogle Scholar
Gyenis, Z., Hofer-Szabo, G., & Rédei, M. (2017). Conditioning using conditional expectations: the Borel–Kolmogorov paradox. Synthese, 194(7), 2595–2630.CrossRefGoogle Scholar
Gyenis, Z., & Rédei, M. (2017). General properties of Bayesian learning as statistical inference determined by conditional expectations. The Review of Symbolic Logic, 10(4), 719–755.CrossRefGoogle Scholar
Hájek, A. (2009). Dutch book arguments. In Anand, P., Pattanaik, P., & Puppe, C., editors. The Handbook of Rationality and Social Choice. Oxford: Oxford University Press, pp. 173–195.CrossRefGoogle Scholar
Harsanyi, J. C. (1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy, 63(4), 309–321.CrossRefGoogle Scholar
Huttegger, S. M. (2013). In defense of reflection. Philosophy of Science, 80(3), 413–433.CrossRefGoogle Scholar
Huttegger, S. M. (2014). Learning experiences and the value of knowledge. Philosophical Studies, 171(2), 279–288.CrossRefGoogle Scholar
Huttegger, S. M. (2015). Merging of opinions and probability kinematics. The Review of Symbolic Logic, 8(4), 611–648.CrossRefGoogle Scholar
Huttegger, S. M. (2017). The Probabilistic Foundations of Rational Learning. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Huttegger, S. M., & Nielsen, M. (2019). Generalized learning and conditional expectation. Philosophy of Science, forthcoming.Google Scholar
Jeffrey, R. (1965). The Logic of Decision. Chicago: University of Chicago Press.Google Scholar
Jeffrey, R. (1992). Probability and the Art of Judgment. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kolmogorov, A. N. (1950). Foundations of the Theory of Probability. New York: Chelsea Publishing Company.Google Scholar
Leitgeb, H., & Pettigrew, R. (2010a). An objective justification of bayesianism i: measuring inaccuracy. Philosophy of Science, 77(2), 201–235.CrossRefGoogle Scholar
Leitgeb, H., & Pettigrew, R. (2010b). An objective justification of bayesianism ii: the consequences of minimizing inaccuracy. Philosophy of Science, 77(2), 236–272.CrossRefGoogle Scholar
Levi, I. (1980). The Enterprise of Knowledge. Cambridge, MA: MIT Press.Google Scholar
Ramsey, F. P. (1931). Truth and probability. In Braithwaite, R. B., editor. The Foundations of Mathematics and Other Essays. New York: Kegan, Paul, Trench, Trubner, & Co., pp. 156–198.Google Scholar
Rao, K. B., & Rao, M. B. (1983). Theory of Charges: A Study of Finitely Additive Measures. Pure and Applied Mathematics, Vol. 109. London: Academic Press.Google Scholar
Rescorla, M. (2018). A Dutch book theorem and converse Dutch book theorem for Kolmogorov conditionalization. The Review of Symbolic Logic, 11(4), 705–735.CrossRefGoogle Scholar
Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley.Google Scholar
Seidenfeld, T. (2001). Remarks on the theory of conditional probability: Some issues of finite versus countable additivity. In Hendricks, V. F., Pedersen, S. A., & Jørgensen, K. F., editors. Probability Theory. Dordrecht, The Netherlands: Klewer, pp. 167–178.CrossRefGoogle Scholar
Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (2001). Improper regular conditional distributions. The Annals of Probability, 29(4), 1612–1624.Google Scholar
Skyrms, B. (1990a). The Dynamics of Rational Deliberation. Cambridge, MA: Harvard University Press.Google Scholar
Skyrms, B. (1990b). The value of knowledge. Minnesota Studies in the Philosophy of Science, 14, 245–266.Google Scholar
Skyrms, B. (1996). The structure of radical probabilism. Erkenntnis, 45(2–3), 285–297.Google Scholar
Teller, P. (1973). Conditionalization and observation. Synthese, 26(2), 218–258.CrossRefGoogle Scholar
van Fraassen, B. C. (1984). Belief and the will. The Journal of Philosophy, 81(5), 235–256.CrossRefGoogle Scholar
van Fraassen, B. C. (1989). Laws and Symmetry. Oxford: Clarendon Press.CrossRefGoogle Scholar
van Fraassen, B. C. (1999). Conditionalization, a new argument for. Topoi, 18(2), 93–96.CrossRefGoogle Scholar
Zhou, L. (1997). Harsanyi’s utilitarianism theorems: general societies. Journal of Economic Theory, 72(1), 198–207.CrossRefGoogle Scholar