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A DUTCH BOOK THEOREM AND CONVERSE DUTCH BOOK THEOREM FOR KOLMOGOROV CONDITIONALIZATION

Published online by Cambridge University Press:  28 May 2018

MICHAEL RESCORLA
MICHAEL RESCORLA*
Affiliation:
Department of Philosophy, University of California, Los Angeles
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095-1555, USA E-mail: rescorla@ucla.edu
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Abstract

This article discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

Keywords

03A1003A0503A99Bayesian decision theoryformal epistemologyconditionalizationsubjective probabilityregular conditional distribution
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 705 - 735
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

BIBLIOGRAPHY

Arntzenius, F. (2003). Some problems for conditionalization and reflection. The Journal of Philosophy, 100, 356–370.CrossRefGoogle Scholar
Arntzenius, F., Elga, A., & Hawthorne, J. (2004). Bayesianism, infinite decisions, and binding. Mind, 113, 251–283.CrossRefGoogle Scholar
Bayes, T. & Price, R. (1763). An essay towards solving a problem in the doctrine of chances. By the late Rev. Mr. Bayes, F.R.S. communicated by Mr. Price, in a letter to John Canton, A.M.F.R.S. Philosophical Transactions of the Royal Society of London, 53, 418–470.Google Scholar
Billingsley, P. (1995). Probability and Measure (third edition). New York: Wiley.Google Scholar
Borel, E. (1909/1956). Elements of the Theory of Probability. Englewood Cliffs: Prentice-Hall.Google Scholar
Briggs, R. (2009). Distorted reflection. The Philosophical Review, 118, 59–85.CrossRefGoogle Scholar
Chater, N. & Oaksford, M. (2008). The Probabilistic Mind. Oxford: Oxford University Press.CrossRefGoogle Scholar
Christensen, D. (1991). Clever bookies and coherent beliefs. The Philosophical Review, 100, 229–247.CrossRefGoogle Scholar
de Finetti, B. (1937/1980). Foresight: Its logical laws, its subjective sources. In Kyburg, H. E. Jr. and Smokler, H. E., editors. Studies in Subjective Probability. Huntington: Robert E. Krieger, pp. 93–159.Google Scholar
de Finetti, B. (1972). Probability, Induction, and Statistics. New York: Wiley.Google Scholar
Durrett, R. (1991). Probability:Theory and Examples. Pacific Grove: Wadworth.Google Scholar
Easwaran, K. (2008). The Foundations of Conditional Probability. Ph.D. Thesis, University of California, Berkeley. ProQuest/UMI. (Publication No. 3331592.)Google Scholar
Easwaran, K. (2011). Varieties of conditional probability. In Bandyopadhyay, P. and Forster, M., editors. The Philosophy of Statistics. Burlington: Elsevier, pp. 137–148.CrossRefGoogle Scholar
Easwaran, K. (2013). Expected accuracy supports conditionalization—and conglomerability and reflection. Philosophy of Science, 80, 119–142.CrossRefGoogle Scholar
Fristedt, B. & Gray, L. (1997). A Modern Approach to Probability Theory. Boston: Birkhäuser.CrossRefGoogle Scholar
Fudenberg, D. & Tirole, J. (1991). Game Theory. Cambridge: MIT Press.Google Scholar
Gyenis, Z., Hofer-Szabó, G., & Rédei, M. (2017). Conditioning using conditional expectations: The Borel–Kolmogorov paradox. Synthese, 194, 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, 719–755.CrossRefGoogle Scholar
Hájek, A. (2003). What conditional probability could not be. Synthese, 137, 273–323.CrossRefGoogle Scholar
Hájek, A. (2009). Dutch book arguments. In Anand, P., Pattanaik, P., and Puppe, C., editors. The Handbook of Rationality and Social Choice. Oxford: Oxford University Press, pp. 173–195.CrossRefGoogle Scholar
Hájek, A. (2011). Conditional probability. In Bandyopadhyay, P. and Forster, M., editors. Philosophy of Statistics. Burlington: Elsevier, pp. 99–136.CrossRefGoogle Scholar
Hájek, A. (2012). Is strict coherence coherent? dialectica, 66, 411–424.CrossRefGoogle Scholar
Hill, B. (1980). On some statistical paradoxes and non-conglomerability. Trabajos de EstadIstica Y de Investigacion Operativa, 31, 39–66.CrossRefGoogle 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
Jeffrey, R. (1983). The Logic of Decision (second edition). Chicago: University of Chicago Press.Google Scholar
Jeffrey, R. (1992). Probability and the Art of Judgment. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kadane, J., Schervish, M., & Seidenfeld, T. (1986). Statistical implications of finitely additive probability. In Goel, P. and Zellner, A., editors. Bayesian Inference and Decision Techniques. Amsterdam: North-Holland, pp. 59–76.Google Scholar
Kemeny, J. (1955). Fair bets and inductive probabilities. The Journal of Symbolic Logic, 20, 263–273.CrossRefGoogle Scholar
Knill, D. C. & Richards, W. (editors) (1996). Perception as Bayesian Inference. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kolmogorov, A. N. (1933/1956). Foundations of the Theory of Probability (second english edition). New York: Chelsea.Google Scholar
Levi, I. (1987). The demons of decision. Monist, 70, 193–211.CrossRefGoogle Scholar
Lewis, D. (1999). Why conditionalize? In Papers in Metaphysics and Epistemology, vol. 2. Cambridge: Cambridge University Press, pp. 403–407.CrossRefGoogle Scholar
Lindley, D. V. (1982). The Bayesian approach to statistics. In de Oliveria, J. and Epstein, B., editors. Some Recent Advances in Statistics. New York: Academic Press, pp. 65–87.Google Scholar
Madl, T., Franklin, S., Chen, K., Montaldi, D., & Trappl, R. (2014). Bayesian integration of information in hippocampal place cells. PLoS one, 9, e89762.CrossRefGoogle ScholarPubMed
Maher, P. (1992). Diachronic rationality. Philosophy of Science, 59, 120–141.CrossRefGoogle Scholar
Mahtani, A. (2015). Dutch books, coherence, and logical consistency. Noûs, 49, 522–537.CrossRefGoogle Scholar
McGee, V. (1994). Learning the impossible. In Eells, E. and Skyrms, B., editors. Probability and Conditionals. Cambridge: Cambridge University Press, pp. 179–199.Google Scholar
McGee, V. (1999). An airtight dutch book. Analysis, 59, 257–265.CrossRefGoogle Scholar
Myrvold, W. (2015). You can’t always get what you want: Some considerations regarding conditional probabilities. Erkenntnis, 80, 573–603.CrossRefGoogle Scholar
Pfanzagl, J. (1979). Conditional distributions as derivatives. The Annals of Probability, 7, 1046–1050.CrossRefGoogle Scholar
Popper, K. (1959). The Logic of Scientific Discovery. London: Hutchinson.Google Scholar
Ramsey, F. P. (1931). Truth and probability. In Braithwaite, R., editor. The Foundations of Mathematics and Other Logical Essays. London: Routledge and Kegan, pp. 156–198.Google Scholar
Rao, M. M. (2005). Conditional Measures and Applications (second edition). Boca Raton: CRC Press.CrossRefGoogle Scholar
Rényi, A. (1955). On a new axiomatic theory of probability. Acta Mathematica Academiae Scientiarum Hungarica, 6, 285–335.CrossRefGoogle Scholar
Rescorla, M. (2015). Some epistemological ramifications of the Borel–Kolmogorov paradox. Synthese, 192, 735–767.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., Pedersen, S., and 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
Shimony, A. (1955). Coherence and the axioms of confirmation. The Journal of Symbolic Logic, 20, 1–28.CrossRefGoogle Scholar
Skyrms, B. (1980). Causal Necessity. New Haven: Yale University Press.Google Scholar
Skyrms, B. (1987). Dynamic coherence and probability kinematics. Philosophy of Science, 54, 1–20.CrossRefGoogle Scholar
Skyrms, B. (1992). Coherence, probability and induction. Philosophical Issues, 2, 215–226.CrossRefGoogle Scholar
Skyrms, B. (1993). A mistake in dynamic coherence arguments? Philosophy of Science, 60, 320–328.CrossRefGoogle Scholar
Stalnaker, R. (1970). Probabilities and conditionals. Philosophy of Science, 37, 64–80.CrossRefGoogle Scholar
Teller, P. (1973). Conditionalization and observation. Synthese, 26, 218–258.CrossRefGoogle Scholar
Thrun, S., Burgard, W., & Fox, D. (2006). Probabilistic Robotics. Cambridge: MIT Press.Google Scholar
van Fraassen, B. (1984). Belief and the will. The Journal of Philosophy, 81, 235–256.CrossRefGoogle Scholar