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

  • ZALÁN GYENIS (a1) and MIKLÓS RÉDEI (a2)
Abstract
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.

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Corresponding author
*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
References
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[1] BillingsleyP. (1995). Probability and Measure (third edition). New York, Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons.
[2] BogachevV. I. (2007). Measure Theory, Vol. II. Berlin, Heidelberg, New York: Springer.
[3] BovensL. & HartmannS. (2004). Bayesian Epistemology. Oxford, UK: Oxford University Press.
[4] DiaconisP. & ZabellS. L. (1982). Updating subjective probability. Journal of the American Statistical Association, 77, 822830.
[5] DoobJ. (1996). The development of rigor in mathematical probability theory (1900–1950). American Mathematical Monthly, 103(7), 586595.
[6] DoobJ. L. (1953). Stochastic Processes. New York: John Wiley & Sons.
[7] DouglasR. G. (1965). Contractive projections on an L 1-space. Pacific Journal of Mathematics, 15(2), 443462.
[8] EarmanJ. (1992). Bayes or Bust? Cambridge, Massachusetts: MIT Press.
[9] EaswaranK. (2008). The Foundations of Conditional Probability. Ph.D. Thesis, University of California at Berkeley.
[10] EaswaranK. (2011). Bayesianism I: Introduction and arguments in favor. Philosophy Compass, 6, 312320.
[11] EaswaranK. (2011) Bayesianism II: Applications and criticisms. Philosophy Compass, 6, 321332.
[12] FellerW. (1966). An Introduction to Probability Theory and its Applications, Second Edition, Vol. 2. New York: Wiley. First edition: 1966.
[13] FremlinD. H. (2001). Measure Theory, Vol. 2. Colchester, England: Torres Fremlin.
[14] GyenisB. (2015). Bayes rules all. Submitted.
[15] GyenisZ., Hofer-SzabóG., & RédeiM. (2016). Conditioning using conditional expectations: The Borel-Kolmogorov Paradox. Synthese, forthcoming, online March 26, 2016, doi: 10.1007/s11229-016-1070-8.
[16] GyenisZ. & RédeiM. (2016). The Bayes Blind Spot of a finite Bayesian Agent is a large set. Manuscript.
[17] HájekA. (2003). What conditional probability could not be. Synthese, 137, 273333.
[18] HalmosP. (1950). Measure Theory. New York: D. Van Nostrand.
[19] HartmannS. & SprengerJ. (2010). Bayesian epistemology. In BerneckerS. and PritchardD., editors. Routledge Companion to Epistemology. London: Routledge, pp. 609620.
[20] HowsonC. (1996). Bayesian rules of updating. Erkenntnis, 45, 195208.
[21] HowsonC. (2014). Finite additivity, another lottery paradox, and conditionalization. Synthese, 191, 9891012.
[22] HowsonC. & FranklinA. (1994). Bayesian conditionalization and probability kinematics. The British Journal for the Philosophy of Science, 45, 451466.
[23] HowsonC. & UrbachP. (1989). Scientific Reasoning: The Bayesian Approach. Illinois: Open Court. Second edition: 1993.
[24] HutteggerS. M. (2015). Merging of opinions and probability kinematics. The Review of Symbolic Logic, 8, 611648.
[25] JaynesE. T. (2003). Principles and pathology of orthodox statistics. In Larry BretthorstG., editor. Probability Theory. The Logic of Science. Cambridge: Cambridge University Press, pp. 447483.
[26] JeffreyR. C. (1965). The Logic of Decision (first edition). Chicago: The University of Chicago Press.
[27] KadisonR. V. & RingroseJ. R. (1986). Fundamentals of the Theory of Operator Algebras, Vols. I. and II. Orlando: Academic Press.
[28] KalmbachG. (1983). Orthomodular Lattices. London: Academic Press.
[29] KolmogorovA. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer. English translation: Foundations of the Theory of Probability (Chelsea, New York, 1956).
[30] LoéveM. (1963). Probability Theory (third edition). Toronto, London, Melbourne: D. Van Nostrand, Princeton.
[31] MarchandJ.-P. (1977). Relative coarse-graining. Foundations of Physics, 7, 3549.
[32] MarchandJ.-P. (1981). Statistical inference in quantum mechanics. In GustafsonK. E. and ReinhardW. P., editors. Quantum Mechanics in Mathematics, Chemistry, and Physics. New York: Plenum Press, pp. 7381. 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.
[33] MarchandJ.-P. (1982). Statistical inference in non-commutative probability. Rendiconti del Seminario Matematico e Fisico di Milano, 52, 551556.
[34] MyrvoldW. (2015). You can’t always get what you want: Some considerations regarding conditional probabilities. Erkenntnis, 80, 572.
[35] PetersenK. (1989). Ergodic Theory. Cambridge: Cambridge University Press.
[36] PfanzaglJ. (1967). Characterizations of conditional expectations. The Annals of Mathematical Statistics, 38, 415421.
[37] RaoM. M. (2005). Conditional Measures and Applications (second revised and expanded edition). Boca Raton, London, New York, Singapore: Chapman & Hall/CRC.
[38] RédeiM. (1998). Quantum Logic in Algebraic Approach. Fundamental Theories of Physics, Vol. 91. Dordrecht, The Netherlands: Kluwer Academic Publisher.
[39] RescorlaM. (2015). Some epistemological ramifications of the Borel-Kolmogorov Paradox. Synthese, 192(3), 735767.
[40] RomanS. (2005). Field Theory (second edition). Graduate Texts in Mathematics, Vol. 158. New York: Springer.
[41] RosenthalJ. S. (2006). A First Look at Rigorous Probability Theory. Singapore: World Scientific.
[42] RudinW. (1987). Real and Complex Analysis (third edition). Singapore: McGraw-Hill.
[43] VillaniA. (1985). Another note on the inclusion $L^p \left( \mu \right) \subset L^q \left( \mu \right)$ . The American Mathematical Monthly, 92, 485487.
[44] WagnerC. (2002). Probability kinematics and commutativity. Philosophy of Science, 69, 266278.
[45] WeisbergJ. (2011). Varieties of Bayesianism. In GabbayD. M., HartmannS., and WoodsJ., editors. Inductive Logic. Handbook of the History of Logic, Vol. 10. Oxford: North-Holland (Elsevier), pp. 477551.
[46] WeisbergJ. (2015). You’ve come a long way, Bayesians. Journal of Philosophical Logic, 44, 817834.
[47] WilliamsonJ. (2010). In Defence of Objective Bayesianism. Oxford: Oxford University Press.
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