On the revision of probabilistic belief states. (English) Zbl 0844.03016

Boutilier, Craig

Notre Dame J. Formal Logic 36, No. 1, 158-182 (1995).

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 Abhaya Nayak [Erkenntnis 41, 353-390 (1994)] should have been mentioned and discussed.

Reviewer: S.O.Hansson (Uppsala)

MSC:

03B60	Other nonclassical logic
68T30	Knowledge representation
03B48	Probability and inductive logic

Keywords:

belief change; conditionalization; Popper functions; revision of probabilistic states of belief; nonstandard conditional probability functions; belief revision; probabilistic ordinal conditional functions; iterated revision

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