Constructing belief functions from sample data using multinomial confidence regions

International Journal of Approximate Reasoning

Volumn 42, Issue 3, 2006, Pages 228-252

  Denœux, Thierry ^a  

^a UNIVERSITÉ DE TECHNOLOGIE DE COMPIÈGNE   (France)

Author keywords

Confidence intervals; Dempster Shafer theory; Evidence theory; Multinomial proportions; Transferable belief model

Indexed keywords

APPROXIMATION THEORY; DECISION THEORY; MATHEMATICAL MODELS; PROBABILITY DISTRIBUTIONS; PROBLEM SOLVING;

CONFIDENCE INTERVALS; DEMPSTER-SHAFER THEORY; EVIDENCE THEORY; MULTINOMIAL PROPORTIONS; TRANSFERABLE BELIEF MODEL;

UNCERTAIN SYSTEMS;

EID: 33745394396     PISSN: 0888613X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijar.2006.01.001     Document Type: Article

References (41)

1. Almond R.G. Graphical Belief Models (1995), Chapman and Hall, London
2. Baroni P., and Vicig P. An uncertainty interchange format with imprecise probabilities. International Journal of Approximate Reasoning 40 3 (2005) 147-180
3. Bernard J.-M. An introduction to the imprecise Dirichlet model for multinomial data. International Journal of Approximate Reasoning 39 2-3 (2005) 123-150
4. Bryson N., and Mobolurin A. A process for generating quantitative belief functions. European Journal of Operational Research 115 (1999) 624-633
5. de Campos L.M., Huete J.F., and Moral S. Probability intervals: a tool for uncertain reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 2 (1994) 167-196
6. Dempster A.P. New methods of reasoning towards posterior distributions based on sample data. Annals of Mathematical Statistics 37 (1966) 355-374
7. Dempster A.P. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38 (1967) 325-339
8. Dempster A.P. A generalization of Bayesian inference (with discussion). Journal of the Royal Statistical Society B 30 (1968) 205-247
9. Dempster A.P. Upper and lower probabilities generated by a random closed interval. Annals of Mathematical Statistics 39 3 (1968) 957-966
10. Dempster A.P. A class of random convex polytopes. Annals of Mathematical Statistics 43 1 (1972) 260-272
11. D. Dubois, H. Prade, Ph. Smets. New semantics for quantitative possibility theory, in: 2nd International Symposium on Imprecise Probabilities and their Applications, Ithaca, NY, 2001.
12. Fitzpatrick S., and Scott A. Quick simultaneous confidence intervals for multinomial proportions. Journal of the American Statistical Association 82 399 (1987) 875-878
13. Glaz J., and Sison C.P. Simultaneous confidence intervals for multinomial proportions. Journal of Statistical Planning and Management 82 (1999) 251-262
14. Goodman L.A. On simultaneous confidence intervals for multinomial proportions. Technometrics 7 2 (1965) 247-254
15. Hacking I. Logic of Statistical Inference (1965), Cambridge University Press, Cambridge
16. Hall J.W., and Lawry J. Generation, combination and extension of random set approximations to coherent lower and upper probabilities. Reliability Engineering and System Safety 85 (2004) 89-101
17. Hou C.-D., Chiang J., and Tai J.J. A family of simultaneous confidence intervals for multinomial proportions. Computational Statistics and Data Analysis 43 (2003) 23-45
18. Jhun M., and Jeong H.-C. Application of bootstrap methods for categorical data analysis. Computational Statistics and Data Analysis 35 (2000) 83-91
19. Kohlas J., and Monney P.-A. A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence (1995), Springer-Verlag, Berlin
20. Masson M., and Denœux T. Inferring a possibility distribution from empirical data. Fuzzy Sets and Systems 157 3 (2006) 319-340
21. May W.L., and Johnson W.D. A SAS macro for constructing simultaneous confidence intervals for multinomial proportions. Computer methods and Programs in Biomedicine 53 (1997) 153-162
22. Quesenberry C.P., and Hurst D.C. Large sample simultaneous confidence intervals for multinomial proportions. Technometrics 6 2 (1964) 191-195
23. Shafer G. A Mathematical Theory of Evidence (1976), Princeton University Press, Princeton, N.J.
24. Shafer G. Belief functions and parametric models (with discussion). Journal of the Royal Statistical Society Series B 44 (1982) 322-352
25. Shafer G. Perspectives in the theory and practice of belief functions. International Journal of Approximate Reasoning 4 (1990) 323-362
26. Shafer G., Shenoy P.P., and Mellouli K. Propagating belief functions in qualitative Markov trees. International Journal of Approximate Reasoning 1 (1987) 349-400
27. Shenoy P.P. Binary joint trees for computing marginals in the Shenoy-Shafer architecture. International Journal of Approximate Reasoning 17 (1997) 239-263
28. Sison C.P., and Glaz J. Simultaneous confidence intervals and sample size determination for multinomial proportions. Journal of the American Statistical Association 90 429 (1995) 366-369
29. Ph. Smets, Un modèle mathématico-statistique simulant le processus du diagnostic médical, Ph.D. thesis, Université Libre de Bruxelles, Brussels, Belgium, 1978 (in French).
30. Smets Ph. Resolving misunderstandings about belief functions. International Journal of Approximate Reasoning 6 (1990) 321-344
31. Smets Ph. Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate Reasoning 9 (1993) 1-35
32. Smets Ph. Belief induced by the partial knowledge of the probabilities. In: Heckerman D., et al. (Ed). Uncertainty in AI'94 (1994), Morgan Kaufmann, San Mateo 523-530
33. Smets Ph. The transferable belief model for quantified belief representation. In: Gabbay D.M., and Smets Ph. (Eds). Handbook of Defeasible Reasoning and Uncertainty Management Systems vol. 1 (1998), Kluwer Academic Publishers, Dordrecht 267-301
34. Smets Ph. Belief functions on real numbers. International Journal of Approximate Reasoning 40 3 (2005) 181-223
35. Smets Ph., and Kennes R. The transferable belief model. Artificial Intelligence 66 (1994) 191-243
36. Utkin L.V. Extensions of belief functions and possibility distributions by using the imprecise Dirichlet model. Fuzzy Sets and Systems 154 3 (2005) 413-431
37. Walley P. Belief function representations of statistical evidence. The Annals of Statistics 15 4 (1987) 1439-1465
38. Walley P. Inferences from multinomial data: learning about a bag of marbles. Journal of the Royal Statistical Society B 58 1 (1996) 3-57
39. Walley P. Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24 2-3 (2000) 125-148
40. Wasserman L.A. Belief functions and statistical evidence. The Canadian Journal of Statistics 18 3 (1990) 183-196
41. Wong S.K.M., and Lingras P. Representation of qualitative user preference by quantitative belief functions. IEEE Transactions on Knowledge and Data Engineering 6 1 (1994) 72-78