On the revision of probabilistic belief states
type: Item
instance of: scholarly article
title: On the revision of probabilistic belief states
zbMATH Open document ID: 0844.03016
author: Craig Boutilier
published in: Notre Dame Journal of Formal Logic
publication date: 1996-08-27T00:00:00Z
review text: The author first investigates revision of probabilistic states of belief that are represented by nonstandard conditional probability functions (Popper functions). Although plausible accounts of belief revision are available in this framework, iterated belief revision does not seem possible to achieve. Let \(P(/)\) be a Popper function. Then the revised factual probability function \(P^*_A\) is given by \(P^*_A(B)= P(B/A)\). However, no revised conditional probability function corresponding to \(P^*_A\) is obtainable, and therefore it is not possible to revise \(P^*_A\). The remedy proposed for this is to introduce probabilistic ordinal conditional functions, POCFs (an extension of Spohn's ordinal conditional functions, OCFs). \(\langle \kappa, P\rangle\) is a POCF if and only if (1) \(\kappa\) is a function assigning a non-negative integer to each world, and assigning \(0\) to at least one world, and (2) \(P\) is a function that assigns to each world a real number in the interval \((0, 1]\). (Worlds assigned the value \(0\) by \(\kappa\) are those considered possible by the agent.) Each POCF determines a Popper function as follows: \[ P(B/A)= {\sum \{P(w)\mid w\in \min(\kappa, A)\text{ and } w\models B\}\over \sum\{P(w)\mid w\in \min(\kappa, A)\}}. \] Many different POCFs may induce the same Popper function. Instead of conditionalizing by a sentence \(A\), conditionalization refers to a pair \(\langle A, k\rangle\), where \(k\) is an integer representing the strength with which \(A\) is given. The outcome \(\kappa_{A, k}\) of \(A, k\)-conditionalizing \(\kappa\) is determined in the way proposed by Spohn for OCFs. This model has the advantage of allowing for iterated revision. A series of results are proved that confirm its intuitive plausibility. Its major disadvantage, acknowledged by the author, is that it requires more information than most other belief revision models. Related work by \textit{Abhaya Nayak} [Erkenntnis 41, 353-390 (1994)] should have been mentio…
reviewed by: Sven Ove Hansson
Mathematics Subject Classification ID: 03B60
Mathematics Subject Classification ID: 68T30
Mathematics Subject Classification ID: 03B48
zbMATH DE Number: 824597
zbMATH Keywords: belief change
zbMATH Keywords: conditionalization
zbMATH Keywords: Popper functions
zbMATH Keywords: revision of probabilistic states of belief
zbMATH Keywords: nonstandard conditional probability functions
zbMATH Keywords: belief revision
zbMATH Keywords: probabilistic ordinal conditional functions
zbMATH Keywords: iterated revision
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1040308833
OpenAlex ID: W2047748361
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DOI: 10.1305/NDJFL/1040308833
description: scientific article; zbMATH DE number 824597