Abstract
In this chapter, we review the basic concepts of the theory of belief functions and sketch a brief history of its conceptual development. We then provide an overview of the classic works and examine how they established a body of knowledge on belief functions, transformed the theory into a computational tool for evidential reasoning in artificial intelligence, opened up new avenues for applications, and became authoritative resources for anyone who is interested in gaining further insight into and understanding of belief functions.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bernoulli, J. The Art of Conjecturing: Together with His Letter to a Friend on Sets in Court Tennis. Johns Hopkins University Press, 2006. Translated by Edith Dudley Sylla.
Boole, G. An Investigation into the Laws of Thought. Walton and Maberly, London, 1854. Reprinted 1951, Dover, NY.
Carnap, R. Meaning and Necessity. University of Chicago Press, Chicago, Illinois, 1956.
Choquet, G. Theory of capacities. Ann. Inst. Fourier 5 (1953), 131–295.
Dempster, A. P On direct probabilities. Journal of the Royal Statistical Society Series B 25 (1962), 100–110.
Dempster, A. P Further examples of inconsistencies in the fiducial argument. Annals of Mathematical Statistics 34 (1963), 884–891.
Dempster, A. P On the difficulties inherent in Fisher’s fiducial argument. J. Amer. Statist. Assoc. 59 (1964), 56–66.
Dempster, A. P New methods for reasoning towards posterior distributions based on sample data. Annals of Mathematical Statistics 37 (1966), 355–374.
Dempster, A. P Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38 (1967), 325–339.
Dempster, A. P A generalization of Bayesian inference (with discussion). Journal of the Royal Statistical Society Series B 30 (1968), 205–247.
Dempster, A. P Belief functions in the 21st century: A statistical perspective. In Proceedings of Insitute for Operations Research and Management Science Annual Meeting (INFORMS-2001) (Miami Beach, FL, 2001).
Dempster, A. P The Dempster-Shafer calculus for statisticians. International Journal of Approximate Reasoning (2006), in press.
Drucker, H., Schapire, R. E., and Simard, P. Y. Boosting performance in neural networks. International Journal of Pattern Recognition and Artificial Intelligence 7 (1993), 705–719.
Einstein, A., and Infeld, L. The Evolution of Physics. Simon and Schuster, New York, 1961.
Fagin, R., and Halpern, J. Y. A new approach to updating beliefs. In Uncertainty in Artificial Intelligence 6, P. P. Bonissone, M. Henrion, L. N. Kanal, and J. F. Lemmer, Eds. Morgan Kaufmann, San Mateo, CA, 1991, pp. 317–325.
Fishburn, P. Decision and Value Theory. Wiley, New York, 1964.
Fisher, R. A Inverse probability. Proc. Camb. Phil. Soc. 26 (1930), 154–57, 172–173. Reprinted in Bennett, J. H. (1971). Collected Papers of R. A. Fisher 2, Univ. of Adelaide.
Good, I. The measure of a non-measurable set. In Logic, Methodology and Philosophy of Science, E. Nagel, P. Suppes, and A. Tarski, Eds. Stanford University Press, Stanford, 1962, pp. 319–329.
Hacking, I. The Emergence of Probability. Cambridge University Press, New York, 1975.
Kearns, M., and Valiant, L. Cryptographic limitations on learning Boolean formulae and finite automata. Journal of the ACM 41 (1994), 67–95.
Kohlas, J., and Monney, P.-A. A Mathematical Theory of Hints. Springer, 1995.
Kong, A. Multivariate Belief Functions and Graphical Models. PhD thesis, Department of Statistics, Harvard University, Cambridge, MA, 1986.
Laskey, K. B., and Lehner, P. E. Assumption, belief and probabilities. Artificial Intelligence 41 (1989), 65–77.
Lauritzen, S. L., and Spiegelhalter, D. J. Local computations with probabilities on graphical structures and their application to expert systems (with discussion). Journal of the Royal Statistical Society Series B 50 (1988), 157–224.
Liu, L. A note on Luce-Fishburn axiomatization of rank-dependent utility. Journal of Risk and Uncertainty 28, 1 (2004), 55–71.
Luce, R. D., and Fishburn, P. C. A note on deriving rank-dependent linear utility using additive joint receipts. Journal of Risk and Uncertainty 11 (1995), 5–16.
Pal, N. R., Bezdek, J. C., and Hemasinha, R. Uncertainty measures for evidential reasoning II: A new measure of total uncertainty. International Journal of Approximate Reasoning 8 (1993), 1–16.
Pearl, J. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo, CA, 1988.
Pearl, J. Reasoning with belief functions: An analysis of compatibility. International Journal of Approximate Reasoning 4 (1990), 363–389.
Quiggin, J. A theory of anticipated utility. Journal of Economic Behavior and Organization 3 (1982), 323–343.
Ruspini, E. H The logical foundations of evidential reasoning. Tech. rep., SRI International, Menlo Park, California, 1986.
Savage, L. J The Foundations of Statistics. Wiley, New York, NY, 1954.
Schapire, R. E The strength of weak learnability. Machine Learning 5 (1990), 197–227.
Shafer, G. A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ, 1976.
Shafer, G. Belief functions and possibility measures. In The Analysis of Fuzzy Information, J. Bezdek, Ed., vol. 1. CRC Press, Boca Raton, FL, 1987, pp. 51–84.
Smets, P. Un mod童 math謡tico-statistique simulant le processus du diagnostic m裩cal. PhD thesis, Universit矌ibre de Bruxelles, Bruxelles, Belgium, 1978.
Smets, P. Probability of provability and belief functions. Logique et Analyse 133-134 (1993), 177–195.
Smith, C. A. B. Consistency in statistical inference and decision (with discussion). Journal of the Royal Statistical Society Series B 23 (1961), 1–25.
Smith, C. A. B. Personal probability and statistical analysis (with discussion). Journal of the Royal Statistical Society Series A 128 (1965), 469–499.
Srivastava, R. R., and Shafer, G. Belief-function formulas for audit risk. The Accounting Review 67, 2 (1992), 249–283.
Sung, B. Translations from James Bernoulli (with a preface by A. P. Dempster). Department of Statistics, Harvard University, Cambridge, Massachusetts, 1966.
Zadeh, L. A Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1 (1978), 3–28.
Zadeh, L. A Review of A Mathematical Theory of Evidence. AI Magazine 5 (1984), 81.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Liu, L., Yager, R.R. (2008). Classic Works of the Dempster-Shafer Theory of Belief Functions: An Introduction. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-44792-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25381-5
Online ISBN: 978-3-540-44792-4
eBook Packages: EngineeringEngineering (R0)