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General properties of general Bayesian learning

Rédei, Miklós and Gyenis, Zalán (2015) General properties of general Bayesian learning. [Preprint]

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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 non-transitive. 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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Item Type: Preprint
Creators:
CreatorsEmailORCID
Rédei, MiklósM.Redei@lse.ac.uk
Gyenis, Zalángyz@renyi.hu
Additional Information: submitted for publication
Keywords: Bayesianism, conditional probability, Bayesian learning
Subjects: General Issues > Formal Learning Theory
Specific Sciences > Probability/Statistics
Depositing User: Dr Miklos Redei
Date Deposited: 23 Aug 2015 14:15
Last Modified: 23 Aug 2015 14:15
Item ID: 11632
Subjects: General Issues > Formal Learning Theory
Specific Sciences > Probability/Statistics
Date: 22 August 2015
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/11632

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