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GENERAL PROPERTIES OF BAYESIAN LEARNING AS STATISTICAL INFERENCE DETERMINED BY CONDITIONAL EXPECTATIONS

Published online by Cambridge University Press:  27 February 2017

ZALÁN GYENIS  and
MIKLÓS RÉDEI
ZALÁN GYENIS*
Affiliation:
Department of Algebra, Budapest University of Technology and Economics
MIKLÓS RÉDEI*
Affiliation:
Department of Philosophy, Logic and Scientific Method London School of Economics and Political Science
*
*DEPARTMENT OF ALGEBRA BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS BUDAPEST, HUNGARY E-mail: gyz@renyi.hu
†DEPARTMENT OF PHILOSOPHY, LOGIC AND SCIENTIFIC METHOD LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE HOUGHTON STREET, LONDON WC2A 2AE, UK E-mail: m.redei@lse.ac.uk
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Abstract

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric, and nontransitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization.

Keywords

03A1003A0503A99Bayesian learningconditionalizationBayes accessibility
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 4 , December 2017 , pp. 719 - 755
Copyright
Copyright © Association for Symbolic Logic 2017 

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References

BIBLIOGRAPHY

Billingsley, P. (1995). Probability and Measure (third edition). New York, Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons.Google Scholar
Bogachev, V. I. (2007). Measure Theory, Vol. II. Berlin, Heidelberg, New York: Springer.Google Scholar
Bovens, L. & Hartmann, S. (2004). Bayesian Epistemology. Oxford, UK: Oxford University Press.Google Scholar
Diaconis, P. & Zabell, S. L. (1982). Updating subjective probability. Journal of the American Statistical Association, 77, 822–830.Google Scholar
Doob, J. (1996). The development of rigor in mathematical probability theory (1900–1950). American Mathematical Monthly, 103(7), 586–595.Google Scholar
Doob, J. L. (1953). Stochastic Processes. New York: John Wiley & Sons.Google Scholar
Douglas, R. G. (1965). Contractive projections on an L 1-space. Pacific Journal of Mathematics, 15(2), 443–462.Google Scholar
Earman, J. (1992). Bayes or Bust? Cambridge, Massachusetts: MIT Press.Google Scholar
Easwaran, K. (2008). The Foundations of Conditional Probability. Ph.D. Thesis, University of California at Berkeley.Google Scholar
Easwaran, K. (2011). Bayesianism I: Introduction and arguments in favor. Philosophy Compass, 6, 312–320.Google Scholar
Easwaran, K. (2011) Bayesianism II: Applications and criticisms. Philosophy Compass, 6, 321–332.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and its Applications, Second Edition, Vol. 2. New York: Wiley. First edition: 1966.Google Scholar
Fremlin, D. H. (2001). Measure Theory, Vol. 2. Colchester, England: Torres Fremlin.Google Scholar
Gyenis, B. (2015). Bayes rules all. Submitted.Google Scholar
Gyenis, Z., Hofer-Szabó, G., & Rédei, M. (2016). Conditioning using conditional expectations: The Borel-Kolmogorov Paradox. Synthese, forthcoming, online March 26, 2016, doi: 10.1007/s11229-016-1070-8.Google Scholar
Gyenis, Z. & Rédei, M. (2016). The Bayes Blind Spot of a finite Bayesian Agent is a large set. Manuscript.Google Scholar
Hájek, A. (2003). What conditional probability could not be. Synthese, 137, 273–333.CrossRefGoogle Scholar
Halmos, P. (1950). Measure Theory. New York: D. Van Nostrand.Google Scholar
Hartmann, S. & Sprenger, J. (2010). Bayesian epistemology. In Bernecker, S. and Pritchard, D., editors. Routledge Companion to Epistemology. London: Routledge, pp. 609–620.Google Scholar
Howson, C. (1996). Bayesian rules of updating. Erkenntnis, 45, 195–208.CrossRefGoogle Scholar
Howson, C. (2014). Finite additivity, another lottery paradox, and conditionalization. Synthese, 191, 989–1012.Google Scholar
Howson, C. & Franklin, A. (1994). Bayesian conditionalization and probability kinematics. The British Journal for the Philosophy of Science, 45, 451–466.Google Scholar
Howson, C. & Urbach, P. (1989). Scientific Reasoning: The Bayesian Approach. Illinois: Open Court. Second edition: 1993.Google Scholar
Huttegger, S. M. (2015). Merging of opinions and probability kinematics. The Review of Symbolic Logic, 8, 611–648.Google Scholar
Jaynes, E. T. (2003). Principles and pathology of orthodox statistics. In Larry Bretthorst, G., editor. Probability Theory. The Logic of Science. Cambridge: Cambridge University Press, pp. 447–483.Google Scholar
Jeffrey, R. C. (1965). The Logic of Decision (first edition). Chicago: The University of Chicago Press.Google Scholar
Kadison, R. V. & Ringrose, J. R. (1986). Fundamentals of the Theory of Operator Algebras, Vols. I. and II. Orlando: Academic Press.Google Scholar
Kalmbach, G. (1983). Orthomodular Lattices. London: Academic Press.Google Scholar
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer. English translation: Foundations of the Theory of Probability (Chelsea, New York, 1956).Google Scholar
Loéve, M. (1963). Probability Theory (third edition). Toronto, London, Melbourne: D. Van Nostrand, Princeton.Google Scholar
Marchand, J.-P. (1977). Relative coarse-graining. Foundations of Physics, 7, 35–49.CrossRefGoogle Scholar
Marchand, J.-P. (1981). Statistical inference in quantum mechanics. In Gustafson, K. E. and Reinhard, W. P., editors. Quantum Mechanics in Mathematics, Chemistry, and Physics. New York: Plenum Press, pp. 73–81. Proceedings of a special session in mathematical physics organized as a part of the 774th meeting of the American Mathematical Society, held March 27–29, 1980, in Boulder, Colorado.CrossRefGoogle Scholar
Marchand, J.-P. (1982). Statistical inference in non-commutative probability. Rendiconti del Seminario Matematico e Fisico di Milano, 52, 551–556.Google Scholar
Myrvold, W. (2015). You can’t always get what you want: Some considerations regarding conditional probabilities. Erkenntnis, 80, 572.CrossRefGoogle Scholar
Petersen, K. (1989). Ergodic Theory. Cambridge: Cambridge University Press.Google Scholar
Pfanzagl, J. (1967). Characterizations of conditional expectations. The Annals of Mathematical Statistics, 38, 415–421.Google Scholar
Rao, M. M. (2005). Conditional Measures and Applications (second revised and expanded edition). Boca Raton, London, New York, Singapore: Chapman & Hall/CRC.CrossRefGoogle Scholar
Rédei, M. (1998). Quantum Logic in Algebraic Approach. Fundamental Theories of Physics, Vol. 91. Dordrecht, The Netherlands: Kluwer Academic Publisher.Google Scholar
Rescorla, M. (2015). Some epistemological ramifications of the Borel-Kolmogorov Paradox. Synthese, 192(3), 735–767.Google Scholar
Roman, S. (2005). Field Theory (second edition). Graduate Texts in Mathematics, Vol. 158. New York: Springer.Google Scholar
Rosenthal, J. S. (2006). A First Look at Rigorous Probability Theory. Singapore: World Scientific.Google Scholar
Rudin, W. (1987). Real and Complex Analysis (third edition). Singapore: McGraw-Hill.Google Scholar
Villani, A. (1985). Another note on the inclusion $L^p \left( \mu \right) \subset L^q \left( \mu \right)$ . The American Mathematical Monthly, 92, 485–487.Google Scholar
Wagner, C. (2002). Probability kinematics and commutativity. Philosophy of Science, 69, 266–278.Google Scholar
Weisberg, J. (2011). Varieties of Bayesianism. In Gabbay, D. M., Hartmann, S., and Woods, J., editors. Inductive Logic. Handbook of the History of Logic, Vol. 10. Oxford: North-Holland (Elsevier), pp. 477–551.Google Scholar
Weisberg, J. (2015). You’ve come a long way, Bayesians. Journal of Philosophical Logic, 44, 817–834.Google Scholar
Williamson, J. (2010). In Defence of Objective Bayesianism. Oxford: Oxford University Press.Google Scholar