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REWARD VERSUS RISK IN UNCERTAIN INFERENCE: THEOREMS AND SIMULATIONS

Published online by Cambridge University Press:  04 July 2012

GERHARD SCHURZ  and
PAUL D. THORN
GERHARD SCHURZ*
Affiliation:
Department of Philosophy, University of Duesseldorf
PAUL D. THORN*
Affiliation:
Department of Philosophy, University of Duesseldorf
*
*DEPARTMENT OF PHILOSOPHY, HEINRICH HEINE UNIVERSITY OF DUESSELDORF, GEB. 23.21., UNIVERSITAETSTRASSE 140225 DUESSELDORF, GERMANY E-mail: [email protected], [email protected]
*DEPARTMENT OF PHILOSOPHY, HEINRICH HEINE UNIVERSITY OF DUESSELDORF, GEB. 23.21., UNIVERSITAETSTRASSE 140225 DUESSELDORF, GERMANY E-mail: [email protected], [email protected]
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Abstract

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems O, P, Z, and QC. These systems differ in the number of inferences they licence (O ⊂ P ⊂ Z ⊂ QC). LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of reward versus risk. Our results suggest that system Z offers the best balance.

Type
Research Articles
Information
The Review of Symbolic Logic , Volume 5 , Issue 4 , December 2012 , pp. 574 - 612
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Adams, E. W. (1965). The logic of conditionals. Inquiry, 8, 166–197.CrossRefGoogle Scholar
Adams, E. W. (1971). A note on comparing probabilistic and modal logics of conditionals. Theoria, 43, 186–194.Google Scholar
Adams, E. W. (1974). On the logic of ‘almost all’. Journal of Philosophical Logic, 3, 3–17.CrossRefGoogle Scholar
Adams, E. W. (1975). The Logic of Conditionals. Dordrecht, The Netherlands: Reidel.CrossRefGoogle Scholar
Adams, E. W. (1986). On the logic of high probability. Journal of Philosophical Logic, 15, 255–279.CrossRefGoogle Scholar
Adams, E. W. (1996). Four probability-preserving properties of inferences. Journal of Philosophical Logic, 25, 1–24.CrossRefGoogle Scholar
Bamber, D. (2000). Entailment with near surety of scaled assertions of high conditional probability. Journal of Philosophical Logic, 29, 1–74.Google Scholar
Bennett, J. (2003). A Philosophical Guide to Conditionals. New York: Oxford University Press.CrossRefGoogle Scholar
Bourne, R., & Parsons, S. (1998). Propagating probabilities in system P. In Proceedings of the 11th International FLAIRS Conference. pp. 440–445.Google Scholar
Brewka, G. (1991). Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge, UK: Cambridge University Press.Google Scholar
Carnap, R. (1950). Logical Foundations of Probability. Chicago, IL: University of Chicago.Google Scholar
Carnap, R. (1971). Inductive logic and rational decisions. In Carnap, R., and Jeffrey, R., editors. Studies in Inductive Logic and Probability I. Los Angeles, CA: University of California Press, pp. 5–32.Google Scholar
Delgrande, J. P. (1988). An approach to default reasoning based on a 1st order conditional logic: Revised report. Artificial Intelligence, 36, 63–90.Google Scholar
Edgington, D. (1995). On conditionals. Mind, 104, 235–329.CrossRefGoogle Scholar
Edgington, D. (2001). Conditionals. In Goble, L., editor. The Blackwell Guide to Philosophical Logic. Oxford, UK: Blackwell, pp. 385–414.Google Scholar
Evans, J. S., Simon, J. H., & Over, D. E. (2003). Conditionals and conditional probability. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 321–335.Google ScholarPubMed
Gabbay, D. (1984). Theoretical foundations for non-monotonic reasoning in expert systems. In Apt, K. R., editor. Logics and Models for Concurrent Systems. Berlin: Springer, pp. 439–458.Google Scholar
Gabbay, D., Hogger, C., & Robinson, J., editors. (1994). Handbook of Logic in Artificial Intelligence, Vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning. Oxford, UK: Clarendon Press.Google Scholar
Gärdenfors, P., & Makinson, D. (1994). Nonmonotonic inference based on expectation orderings. Artificial Intelligence, 65, 197–245.Google Scholar
Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction. Psychological Review, 102, 684–704.CrossRefGoogle Scholar
Gilio, A. (2002). Precise propagation of upper and lower probability bounds in system P. Annals of Mathematics and Artificial Intelligence, 34, 5–34.CrossRefGoogle Scholar
Goldszmidt, M., & Pearl, J. (1996). Qualitative probabilities for default reasoning, belief revision and causal modeling. Artificial Intelligence, 84, 57–112.Google Scholar
Good, I. J. (1983). Good Thinking. Minneapolis, MN: University of Minnesota Press.Google Scholar
Halpern, J. (2003). Reasoning about Uncertainty. Cambridge, MA: MIT Press.Google Scholar
Halpern, J., & Koller, D. (2004). Representation dependence in probabilistic inference. Journal of Artificial Intelligence Research, 21, 319–356.Google Scholar
Harper, W. (1976). Rational belief change, Popper-functions and counterfactuals. In Harper, W., and Hooker, C. A., editors. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Dordrecht, The Netherlands: Reidel, pp. 73–112.Google Scholar
Harper, W., Stalnaker, R., & Pearce, G., editors. (1981). Ifs. Dordrecht, The Netherlands: Reidel.Google Scholar
Hawthorne, J. (1996). On the logic of non-monotonic conditionals and conditional probabilities. Journal of Philosophical Logic, 25, 185–218.Google Scholar
Hawthorne, J. (2005). Degree-of-belief and degree-of-support: Why Bayesians need both notions. Mind, 114, 277–320.Google Scholar
Hawthorne, J., & Makinson, D. (2007). The quantitative/qualitative watershed for rules of uncertain inference. Studia Logica, 86, 247–297.Google Scholar
Horty, J. (2002). Skepticism and floating conclusions. Artificial Intelligence, 135, 55–72.CrossRefGoogle Scholar
Horty, J. (2007). Defaults with priorities. Journal of Philosophical Logic, 36, 367–413.Google Scholar
Hill, L. H., & Paris, J. B. (2003). When maximizing entropy gives the rational closure. Journal of Logic and Computation, 13, 51–68.Google Scholar
Howson, C., & Urbach, P. (2006). Scientific Reasoning: The Bayesian Approach (third edition). Chicago, IL: Open Court Publishing.Google Scholar
Jaynes, E. (1968). Prior probabilities, IEEE Transactions On Systems Science and Cybernetics, 4, 227–241.Google Scholar
Johnson, M. P., & Parikh, R. (2008). Probabilistic conditionals are almost monotonic. The Review of Symbolic Logic, 1, 1–7.CrossRefGoogle Scholar
Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.Google Scholar
Kyburg, H. E. (1974). The Logical Foundations of Statistical Inference. Dordrecht, The Netherlands: Reidel.CrossRefGoogle Scholar
Lehmann, D., & Magidor, M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55, 1–60.Google Scholar
Leitgeb, H. (2004). Inference on the Low Level. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
Levi, I. (1977). Direct inference. The Journal of Philosophy, 74, 5–29.Google Scholar
Lewis, D. (1973). Counterfactuals. Oxford, UK: Blackwell.Google Scholar
Lewis, D. (1976). Probabilities of conditionals and conditional probabilities. The Philosophical Review, 85, 297–315.Google Scholar
Lukasiewicz, T. (1999). Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10, 199–241.Google Scholar
Makinson, D. (2005). Bridges from Classical to Nonmonotonic Logic. London: College Publications.Google Scholar
Makinson, D. (2011). Conditional probability logic in the light of qualitative belief change. Journal of Philosophical Logic, 40, 125–153.Google Scholar
McDermott, D., & Doyle, J. (1980). Non-monotonic logic I. Artificial Intelligence, 25, 41–72.Google Scholar
McGee, V. (1989). Conditional probabilities and compounds of conditionals. The Philosophical Review, 98, 485–541.Google Scholar
Moore, R. C. (1985). Semantical considerations on nonmonotonic logic. Artificial Intelligence, 25, 75–94.Google Scholar
Oberauer, K., & Wilhelm, O. (2003). The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 680–693.Google Scholar
Over, D. E. (2003a). From massive modularity to metarepresentation: The evolution of higher cognition. In Over, D. E., editor. Evolution and the Psychology of Thinking: The Debate, Hove, UK: Psychology Press, pp. 121–144.Google Scholar
Over, D. E., editor. (2003b). Evolution and the Psychology of Thinking: The Debate. Hove, UK: Psychology Press.Google Scholar
Paris, J. B. (1994). The Uncertain Reasoner’s Companion. Cambridge: Cambridge University Press.Google Scholar
Paris, J. B., & Simmonds, R. (2009). O is not enough. Review of Symbolic Logic, 2, 298–309.Google Scholar
Paris, J. B., & Vencovská, A. (1997). In defence of the maximum entropy inference process. International Journal of Approximate Reasoning, 17, 77–103Google Scholar
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. Santa Mateo, CA: Morgan Kaufmann.Google Scholar
Pearl, J. (1990). System Z. In Proceedings of Theoretical Aspects of Reasoning about Knowledge, Santa Mateo, CA. pp. 21–135.Google Scholar
Pelletier, F. J., & Elio, R. (1997). What should default reasoning be, by default? Computational Intelligence, 13, 165–187.Google Scholar
Pollock, J. (1990). Nomic Probability and the Foundations of Induction. Oxford: Oxford University Press.Google Scholar
Pollock, J. (1994). Justification and defeat. Artificial Intelligence, 67, 377–407.Google Scholar
Pollock, J. (1995). Cognitive carpentry: A blueprint for how to build a person. Cambridge, MA: MIT Press.Google Scholar
Poole, D. (1988). A logical framework for default reasoning. Artificial Intelligence, 36, 27–47.Google Scholar
Poole, D. (1991). The Effect of Knowledge on Belief. Artificial Intelligence, 49, 281–307.Google Scholar
Reichenbach, H. (1949). The Theory of Probability. Berkeley: University of California Press.Google Scholar
Reiter, R. (1980). A logic for default reasoning. Artificial Intelligence, 13, 81–132.Google Scholar
Schurz, G. (1997). Probabilistic default reasoning based on relevance and irrelevance assumptions. In Gabbay, D., et al. ., editors. Qualitative and Quantitative Practical Reasoning (LNAI 1244). Berlin: Springer, pp. 536–553.Google Scholar
Schurz, G. (1998). Probabilistic semantics for Delgrande’s conditional logic and a counterexample to his default logic. Artificial Intelligence, 102, 81–95.Google Scholar
Schurz, G. (2001). What is ‘normal’? An evolution-theoretic foundation of normic laws and their relation to statistical normality. Philosophy of Science, 68, 476–497.Google Scholar
Schurz, G. (2005a). Non-monotonic reasoning from an evolutionary viewpoint. Synthese, 146, 37–51.Google Scholar
Schurz, G. (2005b). Logic, matter of form, and closure under substitution. In Behounek, L., and Bilkova, M., editors. The Logica Yearbook 2004. Prague, Czech Republic: Filosofia, pp. 33–46.Google Scholar
Schurz, G. (2007). Human conditional reasoning explained by non-monotonicity and probability. In Vosniadou, S., et al. ., editors. Proceedings of EuroCogSci07. The European Cognitive Science Conference 2007. New York: Lawrence Erlbaum Assoc., pp. 628–633.Google Scholar
Schurz, G. (2012). Tweety, or why probabilism (and even Bayesianism) need objective and evidential probabilities. In Dieks, D., et al. ., editors. Probabilities, Laws and Structures. New York: Springer, pp. 57–74.Google Scholar
Schurz, G., & Leitgeb, H. (2008). Finitistic and frequentistic approximations of probability measures with or without sigma-additivity. Studia Logica, 89/2, 258–283.Google Scholar
Segerberg, K. (1989). Notes on conditional logic. Studia Logica, 48, 157–168.CrossRefGoogle Scholar
Skyrms, B. (1980). Causal Necessity. New Haven, CT: Yale University Press.Google Scholar
Spohn, W. (1980). Stochastic independence, causal independence, and shieldability. Journal of Philosophical Logic, 9, 73–99.Google Scholar
Spohn, W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In Harper, W. L., and Skyrms, B., editors. Causation in Decision, Belief Change and Statistics. Dordrecht, The Netherlands: Reidel, pp. 105–134.Google Scholar
Stalnaker, R. C. (1970). Probability and conditionals. Philosophy of Science, 37, 64–80.Google Scholar
Strevens, M. (2000). Do large probabilities explain better? Philosophy of Science, 67, 366–390.Google Scholar
Suppes, P. (1966). Probabilistic inference and the concept of total evidence. In Hintikka, J., and Suppes, P. editors. Aspects of Inductive Logic. Amsterdam: North-Holland Publ. Comp., pp. 49–65.Google Scholar
Touretzky, D., Horty, J., & Thomason, R. (1987). A clash of intuitions: The current state of monotonic multiple inheritance systems. In Proceedings of the Tenth international Joint Conference on Artificial Intelligence. pp. 476–482.Google Scholar
Van Fraassen, B. (1989). Laws and Symmetry. Oxford: Oxford University Press.CrossRefGoogle Scholar
Williamson, J. (2007). Motivating objective Bayesianism: From empirical constraints to objective probabilities. In Harper, W. L., and Wheeler, G. R., editors. Probability and Inference: Essays in Honor of Henry E. Kyburg Jr. London: College Publications, pp. 155–183.Google Scholar