Belief revision and updates in numerical formalisms âAn ... - IJCAI
Belief revision and updates in numerical formalisms —An overview, with new results for the possibilistic framework— Didier DUBOIS - Henri PRADE Institut de Recherche en Informatique de Toulouse (I.R.I.T.), Universite Paul Sabatier - CN.R.S. 118 route de Narbonne, 31062 TOULOUSE Cedex (FRANCE)
Abstract The difference between Bayesian conditioning and Lewis' imaging is somewhat similar to the one between Gardenfors' belief revision and Katsuno and Mendelzon' updating in the logical framework. Counterparts in possibility theory of these two operations are presented, including the case of conditioning upon an uncertain observation. Possibilistic conditioning satisfies all the postulates for belief revision, and possibilistic imaging all the updating postulates. Lastly, a third operation called "focusing" is naturally introduced in the setting of belief and plausibility functions.
1
Introduction
Numerical formalisms for the representation of uncertainty usually describe stales of knowledge in terms of possible states of the world These states w are supposed to be mutually exclusive and usually is assumed to gather all the possible states of the world. Both in probability theory and in possibility theory, to each state w is attached a degree < which estimates the extent to which co may represent the real state of the world. These states can be put in correspondence with the models used in logical formalisms. By convention, means that we are completely certain that co cannot be the real slate of the world. But the meaning of is completely different in probability theory where it means that co is the real state (complete knowledge), and in possibility theory where it only expresses that nothing prevents co from being the real state of the world. In these two formalisms, the change of the current state of knowledge (called 'epistemic state' in the following), upon the arrival of a new information stating that the real world is in corresponds to a modification of the assignment function d into a new assignment d\ This change should obey general principles which guarantee that i) d' is of the same nature as d (preservation of the representation principles); ii) which denotes 'not A', is excluded by d', i.e., (what is observed is held as certain after the revision or the updating); iii) some informational distance between d' and d is minimized (principle of minimal change). Counterparts to these principles are also at the basis of revision and updating in logical formalisms [12]. The probabilistic framework offers at least two ways of modifying a probability distribution upon the arrival of a new and certain information: the Bayesian conditioning, but also D. Lewis [21]'s 'imaging' which consists in translating the weights originally on models outside A to models which
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are their closest neighbours in A. This paper shows that the existence of these two modes, which can be also defined in the possibilistic framework, is analogous to the distinction between belief revision based on Alchourr6n, Gardenfors and Makinson (AGM) postulates [12] and updating based on Katsuno and Mendelzon [18]' postulates. The paper is organized in three main parts. The next section surveys basic results on probabilistic conditioning and imaging. Section 3 introduces these two operations in the possibilistic framework and provides new results and justifications for them. Section 4 briefly considers belief and plausibility functions and then emphasizes the existence of a third operation, called 'focusing', in this setting.
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Conclusion
This paper has emphasized that belief revision in the sense of Gardenfors, as well as updating in the sense of Katsuno and Mendelzon can be defined through conditioning and imaging respectively both in the probabilistic and in the possibilistic settings. The possibilistic framework leads to a more complete agreement with the two sets of postulates (first stated for propositional logic) than the probabilistic setting. The paper also has tried to relate the revision of a possibility distribution on a set of possible worlds to the revision of a knowledge base made of uncertain logical formulas. More work is needed to relate probabilistic rules to the axiomatic approaches to belief change in the logical framework, despite the existing bridges between probability and possibility theories. Namely we might consider devising revision and updating rules in logics of uncertainty different from possibilistic logic, and especially probabilistic logic. Indeed while the problem of change has been thoroughly studied for probabilistic representations of epistemic states on a set of possible worlds, nothing has been done at the syntactic level. Besides the justification of the different rules in evidence theory is in its infancy. The idea of focusing, i.e. changing the reference class as opposed to revising a body of knowledge might be worth introducing in the logical setting also. The coherence between numerical versus symbolic approaches to knowledge representation is still present in the revising and updating problems. Pushing further the consequences of such a coherence looks like a challenging task. The reader is referred to a more complete version of this paper for further discussions and proofs [10]. A c k n o w l e d g e m e n t s : This work has been partially supported by the European ESPRIT Basic Research Action n° 6156 entitled "Defeasible Reasoning and Uncertainty Management Systems (DRUMS-2)". References 1 Cox R.T. Probability, frequency and reasonable expectation. American J. of Physics 14:1-13, 1946. 2 Dempster A.P. Upper and lower probabilities induced by a multivalued mapping. Ann.Math.Stat. 38:325-339, 1967. 3 De Campos L.M.. Lamata M.T., Moral S. The concept of conditional fuzzy measure. Int. J. Intelligent Systems 237246, 1990. 4 Dubois D., Prade H. The logical view of conditioning and its application to possibility and evidence theories. Int. J. Approximate Reasoning 4:23-46, 1990.
Dubois D., Prade H. Epistemic entrenchment and possibilistic logic. Artif. Intelligence 50:223-239, 1991. 6 Dubois D., Prade H. Updating with belief functions, ordinal conditional functions and possibility measures. I n : Uncertainty in Artificial Intelligence 6 (P.P. Bonissone et al., eds.), North-Holland, 311-329, 1991. 7 Dubois D., Prade H. Possibilistic logic, preferential models, non-monotonicity and related issues. IJCAI'91, 419-424. 8 Dubois D., Prade H. Evidence, knowledge and belief functions. Int. J. Approx. Reasoning 6:295-319, 1992. 9 Dubois D., Prade H. Belief change and possibility theory. In: Belief Revision (P. Gardenfors, ed.), Cambridge University Press, 142-182. 1992. 10 Dubois D., Prade H. A survey of belief revision and updating rules in numerical uncertainty models. Forthcoming, 11 Fagin R., Halpern J.Y. A new approach to updating beliefs. In: Research Report n° RJ 7222 (67989), I B M Research Division, San Jose, CA, 1989. 12 Gardenfors P. Knowledge in Flux: Modeling the Dynamics of Epistemic States. M I T Press, 1988. 13 Gilboa I., Schmeidler D. Updating ambiguous beliefs. J. of Economic Theory, TARK'92. 14 Heckerman D.E. An axiomatic framework for belief updates. In: Uncertainty in Artificial Intelligence 2 (J.F. Lemmer, L.N. Kanal, eds.). North-Holland, 11-22, 1988. 15 Hisdal E. Conditional possibilities - Independence and non-interactivity. Fuzzy Sets & Syst. 1:283-297, 1978. 16 Jaffray J.Y. Bayesian updating and belief functions. IEEE Trans, on Systems, Man and Cybern. 22:1144-1152, 1992. 17 Jeffrey R. The Logic of Decision. McGraw-Hill, 1965. 1 8 Katsuno H., Mendelzon A.O. On the difference between updating a knowledge base and revising it. KR'91, 387-394. 19 Lehrer K., Wagner C.G. Rational Consensus in Science and Society. D. Reidel Publ. Co., Boston, 1981. 20 Lewis D.K. Counterfactuals and comparative possibility. J. Philos. Logic 2:418-446, 1973. 21 Lewis D.K. Probabilities of conditionals and conditional probabilities. The Philosophical Rev. 85:297-315, 1976. 22 Moral S., De Campos L.M. Updating uncertain information. In: Uncertainty in Knowledge Bases (B. BouchonMeunier et al., eds.), Springer Verlag, 58-67, 1991. 23 Pearl J. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan-Kaufmann,1988. 24 Ruspini E.H. Approximate deduction in single evidential bodies. Proc. 2nd Workshop on Uncertainty in Artificial Intelligence, Pennsylvania, 1986, 215-222. 25 Shafer G. A Mathematical Theory of Evidence. Princeton University Press, Princeton, 1976. 26 Smets P. Belief functions. I n : Non-Standard Logics for Automated Reasoning (P. Smets et al., eds.), Academic Press, 253-286, 1988. 27 Spohn W. Ordinal conditional functions: a dynamic theory of epistemic states. I n : Causation in Decision, Belief Change, and Statistics, Vol. 2 (W.L. Harper, B. Skyrms, eds.), D. Reidel, Dordrecht, 105-134, 1988. 28 Suppes P., Zanotti M. On using random relations to generate upper and lower probabilities. Synthese 36:427440. 1977. 29 Williams P. Bayesian conditionalization and the principle of minimum information. British J. of the Philosophy of Science 31:131-144, 1980. 30 Yager R.R. Measuring tranquility and anxiety in decisionmaking: an application of fuzzy sets. Int. J. General Systems 8:139-146, 1982. 31 Zadeh L.A. Possibility theory and soft data analysis. In: Mathematical Frontiers of the Social and Policy Sciences (L. Cobb, R.M. Thrall, eds.). West View Press Boulder Co., 69-127, 1981.
Dubois and Prade
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Abstract The difference between Bayesian conditioning and Lewis' imaging is somewhat similar to the one between Gardenfors' belief revision and Katsuno and Mendelzon' updating in the logical framework. Counterparts in possibility theory of these two operations are presented, including the case of conditioning upon an uncertain observation. Possibilistic conditioning satisfies all the postulates for belief revision, and possibilistic imaging all the updating postulates. Lastly, a third operation called "focusing" is naturally introduced in the setting of belief and plausibility functions.
1
Introduction
Numerical formalisms for the representation of uncertainty usually describe stales of knowledge in terms of possible states of the world These states w are supposed to be mutually exclusive and usually is assumed to gather all the possible states of the world. Both in probability theory and in possibility theory, to each state w is attached a degree < which estimates the extent to which co may represent the real state of the world. These states can be put in correspondence with the models used in logical formalisms. By convention, means that we are completely certain that co cannot be the real slate of the world. But the meaning of is completely different in probability theory where it means that co is the real state (complete knowledge), and in possibility theory where it only expresses that nothing prevents co from being the real state of the world. In these two formalisms, the change of the current state of knowledge (called 'epistemic state' in the following), upon the arrival of a new information stating that the real world is in corresponds to a modification of the assignment function d into a new assignment d\ This change should obey general principles which guarantee that i) d' is of the same nature as d (preservation of the representation principles); ii) which denotes 'not A', is excluded by d', i.e., (what is observed is held as certain after the revision or the updating); iii) some informational distance between d' and d is minimized (principle of minimal change). Counterparts to these principles are also at the basis of revision and updating in logical formalisms [12]. The probabilistic framework offers at least two ways of modifying a probability distribution upon the arrival of a new and certain information: the Bayesian conditioning, but also D. Lewis [21]'s 'imaging' which consists in translating the weights originally on models outside A to models which
620
Knowledge Representation
are their closest neighbours in A. This paper shows that the existence of these two modes, which can be also defined in the possibilistic framework, is analogous to the distinction between belief revision based on Alchourr6n, Gardenfors and Makinson (AGM) postulates [12] and updating based on Katsuno and Mendelzon [18]' postulates. The paper is organized in three main parts. The next section surveys basic results on probabilistic conditioning and imaging. Section 3 introduces these two operations in the possibilistic framework and provides new results and justifications for them. Section 4 briefly considers belief and plausibility functions and then emphasizes the existence of a third operation, called 'focusing', in this setting.
Dubois and Prade
621
622
Knowledge Representation
Dubois and Prade
623
624
Knowledge Representation
5
5
Conclusion
This paper has emphasized that belief revision in the sense of Gardenfors, as well as updating in the sense of Katsuno and Mendelzon can be defined through conditioning and imaging respectively both in the probabilistic and in the possibilistic settings. The possibilistic framework leads to a more complete agreement with the two sets of postulates (first stated for propositional logic) than the probabilistic setting. The paper also has tried to relate the revision of a possibility distribution on a set of possible worlds to the revision of a knowledge base made of uncertain logical formulas. More work is needed to relate probabilistic rules to the axiomatic approaches to belief change in the logical framework, despite the existing bridges between probability and possibility theories. Namely we might consider devising revision and updating rules in logics of uncertainty different from possibilistic logic, and especially probabilistic logic. Indeed while the problem of change has been thoroughly studied for probabilistic representations of epistemic states on a set of possible worlds, nothing has been done at the syntactic level. Besides the justification of the different rules in evidence theory is in its infancy. The idea of focusing, i.e. changing the reference class as opposed to revising a body of knowledge might be worth introducing in the logical setting also. The coherence between numerical versus symbolic approaches to knowledge representation is still present in the revising and updating problems. Pushing further the consequences of such a coherence looks like a challenging task. The reader is referred to a more complete version of this paper for further discussions and proofs [10]. A c k n o w l e d g e m e n t s : This work has been partially supported by the European ESPRIT Basic Research Action n° 6156 entitled "Defeasible Reasoning and Uncertainty Management Systems (DRUMS-2)". References 1 Cox R.T. Probability, frequency and reasonable expectation. American J. of Physics 14:1-13, 1946. 2 Dempster A.P. Upper and lower probabilities induced by a multivalued mapping. Ann.Math.Stat. 38:325-339, 1967. 3 De Campos L.M.. Lamata M.T., Moral S. The concept of conditional fuzzy measure. Int. J. Intelligent Systems 237246, 1990. 4 Dubois D., Prade H. The logical view of conditioning and its application to possibility and evidence theories. Int. J. Approximate Reasoning 4:23-46, 1990.
Dubois D., Prade H. Epistemic entrenchment and possibilistic logic. Artif. Intelligence 50:223-239, 1991. 6 Dubois D., Prade H. Updating with belief functions, ordinal conditional functions and possibility measures. I n : Uncertainty in Artificial Intelligence 6 (P.P. Bonissone et al., eds.), North-Holland, 311-329, 1991. 7 Dubois D., Prade H. Possibilistic logic, preferential models, non-monotonicity and related issues. IJCAI'91, 419-424. 8 Dubois D., Prade H. Evidence, knowledge and belief functions. Int. J. Approx. Reasoning 6:295-319, 1992. 9 Dubois D., Prade H. Belief change and possibility theory. In: Belief Revision (P. Gardenfors, ed.), Cambridge University Press, 142-182. 1992. 10 Dubois D., Prade H. A survey of belief revision and updating rules in numerical uncertainty models. Forthcoming, 11 Fagin R., Halpern J.Y. A new approach to updating beliefs. In: Research Report n° RJ 7222 (67989), I B M Research Division, San Jose, CA, 1989. 12 Gardenfors P. Knowledge in Flux: Modeling the Dynamics of Epistemic States. M I T Press, 1988. 13 Gilboa I., Schmeidler D. Updating ambiguous beliefs. J. of Economic Theory, TARK'92. 14 Heckerman D.E. An axiomatic framework for belief updates. In: Uncertainty in Artificial Intelligence 2 (J.F. Lemmer, L.N. Kanal, eds.). North-Holland, 11-22, 1988. 15 Hisdal E. Conditional possibilities - Independence and non-interactivity. Fuzzy Sets & Syst. 1:283-297, 1978. 16 Jaffray J.Y. Bayesian updating and belief functions. IEEE Trans, on Systems, Man and Cybern. 22:1144-1152, 1992. 17 Jeffrey R. The Logic of Decision. McGraw-Hill, 1965. 1 8 Katsuno H., Mendelzon A.O. On the difference between updating a knowledge base and revising it. KR'91, 387-394. 19 Lehrer K., Wagner C.G. Rational Consensus in Science and Society. D. Reidel Publ. Co., Boston, 1981. 20 Lewis D.K. Counterfactuals and comparative possibility. J. Philos. Logic 2:418-446, 1973. 21 Lewis D.K. Probabilities of conditionals and conditional probabilities. The Philosophical Rev. 85:297-315, 1976. 22 Moral S., De Campos L.M. Updating uncertain information. In: Uncertainty in Knowledge Bases (B. BouchonMeunier et al., eds.), Springer Verlag, 58-67, 1991. 23 Pearl J. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan-Kaufmann,1988. 24 Ruspini E.H. Approximate deduction in single evidential bodies. Proc. 2nd Workshop on Uncertainty in Artificial Intelligence, Pennsylvania, 1986, 215-222. 25 Shafer G. A Mathematical Theory of Evidence. Princeton University Press, Princeton, 1976. 26 Smets P. Belief functions. I n : Non-Standard Logics for Automated Reasoning (P. Smets et al., eds.), Academic Press, 253-286, 1988. 27 Spohn W. Ordinal conditional functions: a dynamic theory of epistemic states. I n : Causation in Decision, Belief Change, and Statistics, Vol. 2 (W.L. Harper, B. Skyrms, eds.), D. Reidel, Dordrecht, 105-134, 1988. 28 Suppes P., Zanotti M. On using random relations to generate upper and lower probabilities. Synthese 36:427440. 1977. 29 Williams P. Bayesian conditionalization and the principle of minimum information. British J. of the Philosophy of Science 31:131-144, 1980. 30 Yager R.R. Measuring tranquility and anxiety in decisionmaking: an application of fuzzy sets. Int. J. General Systems 8:139-146, 1982. 31 Zadeh L.A. Possibility theory and soft data analysis. In: Mathematical Frontiers of the Social and Policy Sciences (L. Cobb, R.M. Thrall, eds.). West View Press Boulder Co., 69-127, 1981.
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