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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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