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NON-FACTIVE KOLMOGOROV CONDITIONALIZATION

Part of: Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  31 October 2023

MICHAEL RESCORLA
MICHAEL RESCORLA*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA, LOS ANGELES 390 PORTOLA PLAZA LOS ANGELES, CA 90095, USA
*
E-mail: rescorla@ucla.edu
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Abstract

Kolmogorov conditionalization is a strategy for updating credences based on propositions that have initial probability 0. I explore the connection between Kolmogorov conditionalization and Dutch books. Previous discussions of the connection rely crucially upon a factivity assumption: they assume that the agent updates credences based on true propositions. The factivity assumption discounts cases of misplaced certainty, i.e., cases where the agent invests credence 1 in a falsehood. Yet misplaced certainty arises routinely in scientific and philosophical applications of Bayesian decision theory. I prove a non-factive Dutch book theorem and converse Dutch book theorem for Kolmogorov conditionalization. The theorems do not rely upon the factivity assumption, so they establish that Kolmogorov conditionalization has unique pragmatic virtues that persist even in cases of misplaced certainty.

Keywords

conditionalizationregular conditional distributionsubjective probabilityBayesian decision theoryDutch books

MSC classification

Primary: 03A05: Philosophical and critical 03A10: Logic in the philosophy of science
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 27
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Bennett, B., Hoffman, D., & Prakash, C. (1996). Observer theory, Bayes theory, and psychophysics. In Knill, D. & Richards, W., editors. Perception as Bayesian Inference. Cambridge: Cambridge University Press, pp. 229–235.Google Scholar
Billingsley, P. (1995). Probability and Measure (third edition). New York: Wiley.Google Scholar
Blackwell, D., & Dubins, L. (1975). On existence and non-existence of proper, regular conditional distributions. The Annals of Probability, 3, 741–752.CrossRefGoogle Scholar
Blackwell, D., & Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. The Annals of Mathematical Statistics, 34, 223–225.CrossRefGoogle Scholar
Christensen, D. (1991). Clever bookies and coherent beliefs. Philosophical Review, 100, 229–247.CrossRefGoogle Scholar
de Finetti, B. (1937/1980). Foresight: Its logical laws, its subjective sources. In KyburgJr., H. E. & Smokler, H. E., editors. Rpt. in Studies in Subjective Probability . Huntington: Robert E. Krieger, pp. 53–118.Google Scholar
de Finetti, B. (1972). Probability, Induction, and Statistics. New York: Wiley.Google Scholar
Easwaran, K. (2008). The Foundations of Conditional Probability. Ph.D. Thesis, University of California, Berkeley; Ann Arbor: ProQuest/UMI (Publication No. 3331592).Google Scholar
Easwaran, K. (2011). Varieties of conditional probability. In Bandyopadhyay, P. & Forster, M., editors. Philosophy of Statistics. Burlington: Elsevier, pp. 137–148.CrossRefGoogle Scholar
Easwaran, K. (2019). Conditional probabilities. In Pettigrew, R. & Weisberg, J., editors. The Open Handbook of Formal Epistemology. PhilPapers, pp. 131–198.Google Scholar
Florens, J.-P., Mouchart, M., & Rolin, J.-M. (1990). Elements of Bayesian Statistics. New York: Marcel Dekker.Google Scholar
Fristedt, B., & Gray, L. (1997). A Modern Approach to Probability Theory. Boston: Birkhäuser.CrossRefGoogle Scholar
Gaifman, H. (1988). A theory of higher-order probabilities. In Skyrms, B. & Harper, W., editors. Causation, Chance, and Credence: Proceedings of the Irvine Conference on Probability and Causation. Boston: Kluwer.Google Scholar
Ghosal, S., & van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Gyenis, Z., Hofer-Szabó, G., & Rédei, M. (2017). Conditioning using conditional expectations: The Borel–Kolmogorov paradox. Synthese, 194, 2595–2630.CrossRefGoogle Scholar
Hájek, A. (2003) What conditional probability could not be. Synthese, 137, 273–323.CrossRefGoogle Scholar
Hájek, A. (2008). 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.Google Scholar
Hájek, A. (2011). Conditional probability. In Bandyopadhyay, P. & Forster, M., editors. Philosophy of Statistics. Burlington: Elsevier.Google Scholar
Howson, C. (2014). Finite additivity, another lottery paradox, and conditionalization. Synthese, 191, 989–1012.CrossRefGoogle Scholar
Huttegger, S. (2015). Merging of opinions and probability kinematics. The Review of Symbolic Logic, 8, 611–648.CrossRefGoogle Scholar
Huttegger, S. (2017). The Probabilistic Foundations of Rational Learning. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Huttegger, S., & Nielsen, M. (2020). Generalized learning and conditional expectation. Philosophy of Science, 87, 868–883.CrossRefGoogle Scholar
Kiefer, N., & Nyarko, Y. (1995). Savage–Bayesian models of economics. In Kirman, A. & Salmon, M., editors. Learning and Rationality in Economics. Oxford: Blackwell, pp. 40–62.Google Scholar
Kolmogorov, A. N. (1933/1956). Foundations of the Theory of Probability (second English edition). N. Morrison, translator. New York: Chelsea.Google Scholar
Lee, J. J. (2018). Formalization of information: Knowledge and belief. Economic Theory, 66, 1007–1022.CrossRefGoogle Scholar
Lewis, D. (1999). Why conditionalize?. In Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, pp. 403–407.CrossRefGoogle Scholar
Meehan, A., & Zhang, S. (2020). Jeffrey meets Kolmogorov: A general theory of conditioning. Journal of Philosophical Logic, 49, 941–979.CrossRefGoogle Scholar
Meehan, A., & Zhang, S. (2022). Kolmogorov conditionalizers can be Dutch booked (if and only if they are evidentially uncertain). The Review of Symbolic Logic, 15, 722–757.CrossRefGoogle Scholar
Mertens, J.-F., & Zamir, S. (1985). Formulation of Bayesian analysis for games of incomplete information. International Journal of Game Theory, 14, 1–29.CrossRefGoogle Scholar
Myrvold, W. (2015). You can’t always get what you want: Some considerations regarding conditional probabilities. Erkenntnis, 80, 573–603.CrossRefGoogle Scholar
Nielsen, M. (2021). A new argument for Kolmogorov conditionalization. The Review of Symbolic Logic, 14, 930–945.CrossRefGoogle Scholar
Ramachandran, D. (1979). Existence of independent complements in regular conditional probability spaces. Annals of Probability, 7, 433–443.CrossRefGoogle Scholar
Ramsey, F. P. (1931). Truth and probability. In Braithwaite, R. B., editor. The Foundations of Mathematics and Other Logical Essays. London: Routledge and Kegan, pp. 156–199.Google Scholar
Rao, M. M. (2005). Conditional Measures and Applications (second edition). Boca Raton: CRC Press.CrossRefGoogle Scholar
Rescorla, M. (2015). Some epistemological ramifications of the Borel–Kolmogorov paradox. Synthese, 192, 735–767.CrossRefGoogle Scholar
Rescorla, M. (2018). A Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The Review of Symbolic Logic, 11, 705–735.CrossRefGoogle Scholar
Rescorla, M. (2021). On the proper formulation of Conditionalization. Synthese, 198, 1935–1965.CrossRefGoogle Scholar
Rescorla, M. (2022). An improved Dutch book theorem for Conditionalization. Erkenntnis, 87, 1013–1041.CrossRefGoogle Scholar
Rescorla, M. (2023). Reflecting on diachronic Dutch books. Noûs, 57, 511–538.CrossRefGoogle Scholar
Rescorla, M. (2024). Bayesian defeat of certainties. Synthese, 203, 50.CrossRefGoogle Scholar
Seidenfeld, T. (2001). Remarks on the theory of conditional probability: Some issues of finite versus countable additivity. In Hendricks, V., Pedersen, S., & Jørgensen, K., editors. Probability Theory: Philosophy, Recent History, and Relations to Science. Dordrecht: Kluwer, pp. 167–178.CrossRefGoogle Scholar
Seidenfeld, T., Schervish, M., & Kadane, J. (2001). Improper regular conditional distributions. Annals of Probability, 29, 1612–1624.Google Scholar
Skyrms, B. (1987). Dynamic coherence and probability kinematics. Philosophy of Science, 54, 1–20.CrossRefGoogle Scholar
Sokal, A. (1981). Existence of compatible families of proper regular conditional probabilities. Zeitschrift für Wahrscheinliehkeitstheorie und verwandte Gebiete, 56, 537–548.CrossRefGoogle Scholar
Teller, P. (1973). Conditionalization and observation. Synthese, 26, 218–258.CrossRefGoogle Scholar
Thrun, S., Burgard, W., & Fox, D. (2005). Probabilistic Robotics. Cambridge: MIT Press.Google Scholar
Trotta, R. (2008). Bayes in the sky: Bayesian inference and model selection in cosmology. Contemporary Physics, 49, 71–104.CrossRefGoogle Scholar