Formalization of Evidence: A Comparative Study

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Formalization of Evidence: A Comparative Study

This article analyzes and compares several approaches of formalizing the notion of evidence in the context of general-purpose reasoning system. In each of these approaches, the notion of evidence is defined, and the evidence-based degree of belief is represented by a binary value, a number (such as a probability), or two numbers (such as an interval). The binary approaches provide simple ways to represent conclusive evidence, but cannot properly handle inconclusive evidence. The one-number approaches naturally represent inconclusive evidence as a degree of belief, but lack the information needed to revise this degree. It is argued that for systems opening to new evidence, each belief should at least have two numbers attached to indicate its evidential support. A few such approaches are discussed, including the approach used in NARS, which is designed according to the considerations of general-purpose intelligent systems, and provides novel solutions to several traditional problems on evidence.

Achinstein, P., ed. 1983. The Concept of Evidence. Oxford: Oxford University Press.

Baroni, P., and Vicig, P. 2001. On the conceptual status of belief functions with respect to coherent lower probabilities. In Bishop, C., ed., Proceedings of the 6th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty; Lecture Notes In Computer Science, Vol. 2143. London: Springer-Verlag. 328-339.

Bonissone, P. P. 1987. Summarizing and propagating uncertain information with Triangular Norms. International Journal of Approximate Reasoning 1:71-101.

Carnap, R. 1950. Logical Foundations of Probability. Chicago: The University of Chicago Press.

Cheeseman, P. 1988. An inquiry into computer understanding. Computational Intelligence 4:58-66.

Clifford, W. K. 1877. The ethics of belief. Contemporary Review. Reprinted in The Ethics of Belief and Other Essays (Prometheus Books, 1999).

DeGroot, M. H. 1970. Optimal Statistical Decisions. New York: McGraw-Hill.

Dempster, A. P. 1967. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38:325-339.

Dubois, D., and Prade, H. 1982. A class of fuzzy measures based on triangular norms. International Journal of General Systems 8:43-61.

Dubois, D., and Prade, H. 1994. Conditional objects as nonmonotonic consequence relationships. IEEE Transactions on Systems, Man, and Cybernetics 24:1724-1740.

Fitelson, B., and Hawthorne, J. 2009. How Bayesian confirmation theory handles the Paradox of the Ravens. In Eells, E., and Fetzer, J., eds., Probability in Science. Chicago: Open Court. Forthcoming.

Good, I. J. 1950. Probability and the Weighing of Evidence. London: Griffin.

Good, I. J. 1985. Weight of evidence: a brief survey. In Bernardo, J.; DeGroot, M.; Lindley, D.; and Smith, A., eds., Bayesian Statistics 2. Amsterdam: North-Holland. 249-269.

Griggs, R. A., and Cox, J. R. 1982. The elusive thematic-materials effect in Wason's selection task. British Journal of Psychology 73:407-420.

Halpern, J. Y., and Pucella, R. 2006. A logic for reasoning about evidence. Journal of Artificial Intelligence Research 26:1-34.

Hempel, C. G. 1965. Studies in the logic of confirmation. In Aspects of Scientific Explanation. New York: The Free Press. 3-46. Reprinted in The Concept of Evidence, Achinstein, P. (Ed), Oxford University Press, pp. 11-43, 1983.

Hume, D. 1748. An Enquiry Concerning Human Understanding. London.

Hutter, M. 2005. Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Berlin: Springer.

Keynes, J. M. 1921. A Treatise on Probability. London: Macmillan.

Kyburg, H. E. 1983a. Recent work in inductive logic. In Lucey, K., and Machan, T., eds., Recent Work in Philosophy. Totowa, NJ: Rowman and Allanfield. 89-150.

Kyburg, H. E. 1983b. The reference class. Philosophy of Science 50:374-397.

Kyburg, H. E. 1994. Believing on the basis of the evidence. Computational Intelligence 10:3-20.

McCarthy, J., and Hayes, P. J. 1969. Some philosophical problems from the standpoint of artificial intelligence. In Meltzer, B., and Michie, D., eds., Machine Intelligence 4. Edinburgh: Edinburgh University Press. 463-502.

McDermott, D. 1987. A critique of pure reason. Computational Intelligence 3:151-160.

Milne, P. 1997. Bruno de Finetti and the logic of conditional events. The British Journal for the Philosophy of Science 48:195-232.

Oaksford, M., and Chater, N. 1994. A rational analysis of the selection task as optimal data selection. Psychological Review 101:608-631.

Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. San Mateo, California: Morgan Kaufmann Publishers.

Pearl, J. 1990. Jeffrey's rule, passage of experience, and Neo-Bayesianism. In Kyburg, H. E.; Loui, R. P.; and N., C. G., eds., Knowledge Representation and Defeasible Reasoning. Amsterdam: Kluwer Academic Publishers. 245-265.

Peirce, C. S. 1878. The probability of induction. Popular Science Monthly 12:705-718. Reprinted in The Essential Peirce, Vol. 1, N. Houser and C. Kloesel, eds., Bloomington, IN: Indiana University Press (1992), 155-169.

Popper, K. R. 1959. The Logic of Scientific Discovery. New York: Basic Books.

Reiter, R. 1987. Nonmonotonic reasoning. Annual Review of Computer Science 2:147-186.

Rescher, N. 1958. A theory of evidence. Philosophy of Science 25(1):83-94.

Shafer, G. 1976. A Mathematical Theory of Evidence. Princeton, New Jersey: Princeton University Press.

Smets, P., and Kennes, R. 1994. The transferable belief model. Artificial Intelligence 66:191-234.

Smets, P. 1991. The transferable belief model and other interpretations of Dempster-Shafer's model. In Bonissone, P. P.; Henrion, M.; Kanal, L. N.; and Lemmer, J. F., eds., Uncertainty in Artificial Intelligence 6. Amsterdam: North-Holland. 375-383.

Solomonoff, R. J. 1964. A formal theory of inductive inference. Part I and II. Information and Control 7(1-2):1-22,224-254.

Tversky, A., and Kahneman, D. 1974. Judgment under uncertainty: heuristics and biases. Science 185:1124-1131.

Tversky, A., and Kahneman, D. 1983. Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment. Psychological Review 90:293-315.

Walley, P. 1991. Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall.

Walley, P. 1996a. Inferences from multinomial data: learning about a bag of marbles. Journal of the Royal Statistical Society, Series B 58:3-57.

Walley, P. 1996b. Measures of uncertainty in expert systems. Artificial Intelligence 83:1-58.

Wang, P. 1993. Belief revision in probability theory. In Proceedings of the Ninth Conference on Uncertainty in Artificial Intelligence, 519-526. Morgan Kaufmann Publishers, San Mateo, California.

Wang, P. 1994a. A defect in Dempster-Shafer Theory. In Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence, 560-566. Morgan Kaufmann Publishers, San Mateo, California.

Wang, P. 1994b. From inheritance relation to nonaxiomatic logic. International Journal of Approximate Reasoning 11(4):281-319.

Wang, P. 1995a. Non-Axiomatic Reasoning System: Exploring the Essence of Intelligence. Ph.D. Dissertation, Indiana University.

Wang, P. 1995b. Reference classes and multiple inheritances. International Journal of Uncertainty, Fuzziness and and Knowledge-based Systems 3(1):79-91.

Wang, P. 1996a. Heuristics and normative models of judgment under uncertainty. International Journal of Approximate Reasoning 14(4):221-235.

Wang, P. 1996b. The interpretation of fuzziness. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 26(4):321-326.

Wang, P. 2001a. Abduction in non-axiomatic logic. In Working Notes of the IJCAI workshop on Abductive Reasoning, 56-63.

Wang, P. 2001b. Confidence as higher-order uncertainty. In Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, 352-361.

Wang, P. 2001c. Wason's cards: what is wrong? In Proceedings of the Third International Conference on Cognitive Science, 371-375.

Wang, P. 2004. The limitation of Bayesianism. Artificial Intelligence 158(1):97-106.

Wang, P. 2005. Experience-grounded semantics: a theory for intelligent systems. Cognitive Systems Research 6(4):282-302.

Wang, P. 2006. Rigid Flexibility: The Logic of Intelligence. Dordrecht: Springer.

Wason, P. C., and Johnson-Laird, P. N. 1972. Psychology of Reasoning: Structure and Content. Cambridge, Massachusetts: Harvard University Press.

Zadeh, L. A. 1975. The concept of a linguistic variable and its application to approximate reasoning. Information Sciences 8:199-249, 8:301-357, 9:43-80.