Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T12:58:54.320Z Has data issue: false hasContentIssue false
  • English
  • Français

PROBABILISTIC STABILITY, AGM REVISION OPERATORS AND MAXIMUM ENTROPY

Part of: Communication, information Artificial intelligence (68Txx) Foundations of probability theory General logic

Published online by Cambridge University Press:  21 October 2020

KRZYSZTOF MIERZEWSKI
KRZYSZTOF MIERZEWSKI*
Affiliation:
LOGICAL DYNAMICS LAB, CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION STANFORD UNIVERSITYSTANFORD, CA 94308, USAE-mail: kmierzew@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb’s stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state—such as a plausibility ranking over hypotheses, or a belief set—the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generates AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of belief.

Keywords

AGMbelief revisionBayesian conditioningprobabilistic stabilityacceptance rulestrackingmaximum entropy

MSC classification

Primary: 03B42: Logics of knowledge and belief (including belief change)
Secondary: 68T27: Logic in artificial intelligence 60A05: Axioms; other general questions 94A17: Measures of information, entropy
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 553 - 590
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Alchourrón, C., Gärdenfors, P, & Makinson, D. (1985). On the logic of theory change: partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(2), 510–530.CrossRefGoogle Scholar
Arló-Costa, H. & Pedersen, A. P. (2011). Belief revision. In Horsten, L., and Pettigrew, R., editors. Continuum Companion to Philosophical Logic. London: Continuum Press.Google Scholar
Arló-Costa, H. & Pedersen, A. P. (2012). Belief and probability: A general theory of probability cores. International Journal of Approximate Reasoning, 53(3), 293–315.CrossRefGoogle Scholar
Boyd, S. & Vandenberghe, L. (2004). Convex Optimization. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Bradley, S. & Steele, K. (2016). Can free evidence be bad? value of information for the imprecise probabilist. Philosophy of Science, 83, 1–28.CrossRefGoogle Scholar
Cohen, J. (1992). An Essay on Belief and Acceptance, Oxford: Clarendon Press.Google Scholar
Foley, R. (1993). Working Without a Net. Oxford: Oxford University Press.Google Scholar
Friedman, K. & Shimony, A. (1971). Jaynes’s maximum entropy prescription and probability theory. Journal of Statistical Physics, 3(4), 620–663.CrossRefGoogle Scholar
Griffiths, T. & Lieder, F. (2019). Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources. Behavioral and Brain Sciences, 4, 1–85.Google Scholar
Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170.CrossRefGoogle Scholar
Halpern, J. (2003). Reasoning about Uncertainty. Cambridge, MA: MIT Press.Google Scholar
Icard, T. F. & Goodman, N. D. (2015). A resource-rational approach to the causal frame problem. Proceedings of the 37th Annual Conference of the Cognitive Science Society, Pasadena, CA, USA.Google Scholar
Jaynes, E. T (1957). Information theory and statistical mechanics. Physical Review Series II, 106(4), 620–663.Google Scholar
Kelly, K. & Lin, H. (2012). A geo-logical solution to the lottery-paradox, with applications to conditional logic. Synthese, 186(2), 531–575.Google Scholar
Kelly, K. & Lin, H. (2012). Propositional reasoning that tracks probabilistic reasoning. Journal of Philosophical Logic, 41(6), 957–981.Google Scholar
Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1-2), 167–207.CrossRefGoogle Scholar
Kyburg, H. (1970). Probability and Inductive Logic. Toronto: Palgrave Macmillan.Google Scholar
Leitgeb, H. (2013). Reducing belief simpliciter to degrees of belief. Annals of Pure and Applied Logic, 164, 1338–1389.CrossRefGoogle Scholar
Leitgeb, H. (2014). Belief as a simplification of probability, and what it entails. In Baltag, A., & Smets, S., editors. Johan van Benthem on Logic and Information Dynamics. New York: Springer-Verlag, pp. 405–417.CrossRefGoogle Scholar
Leitgeb, H. (2014). The stability theory of belief. Philosophical Review, 123(2), 131–171.CrossRefGoogle Scholar
Leitgeb, H. (2017). The Stability of Belief. Oxford: Oxford University Press.CrossRefGoogle Scholar
Levi, I. (1980). The Enterprise of Knowledge: An Essay on Knowledge, Credal Probability, and Chances. Cambridge, MA: MIT Press.Google Scholar
Lin, H. (2013). Foundations of everyday practical reasoning. Journal of Philosophical Logic, 42, 831–862.CrossRefGoogle Scholar
Lin, H. (2016). Bridging the logic-based and probability-based approaches to artificial intelligence. In Hung, T.-W., editor. Rationality: Constraints and Contexts. Amsterdam: Elsevier, pp. 215–225.Google Scholar
Mierzewski, K. (2018). Probabilistic Stability: Dynamics, Nonmonotonic Logics, and Stable Revision. MSc Thesis, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
Paris, J. B. (1994). The Uncertain Reasoner’s Companion: A Mathematical Perspective. Cambridge Tracts in Theoretical Computer Science, Vol. 39. Cambridge: Cambridge University Press.Google Scholar
Roman, S. (1997). Coding and Information Theory, Graduate Texts in Mathematics. New York: Springer-Verlag.Google Scholar
Savage, L. J. (1954). The Foundations of Statistics. New York: John Wiley & Sons Inc.Google Scholar
Seidenfeld, T. (1986). Entropy and uncertainty. Philosophy of Science, 53, 467–491.CrossRefGoogle Scholar
Shear, T. & Fitelson, B. (2019). Two approaches to belief revision. Erkenntnis, 84, 487–518.CrossRefGoogle Scholar
Skyrms, B. (1977). Resiliency, propensities, and causal necessity. The Journal of Philosophy, 74(11), 704–711.CrossRefGoogle Scholar
Skyrms, B. (1987). Updating, Supposing, and Maxent. Theory and Decision, 22(3), 225–246.CrossRefGoogle Scholar
Sundaram, R. K. (1996). A First Course in Optimization Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
van Fraassen, B. (1980). The Scientific Image. Oxford: Oxford University Press.CrossRefGoogle Scholar
Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Monographs on Statistics and Applied Probability, Vol. 42. London: Chapman and Hall.CrossRefGoogle Scholar