Abstract
As a general formalism for uncertain reasoning, the theory of belief functions extends the logical and probabilistic approaches to uncertainty: a belief function (or a completely monotone Choquet capacity) can be seen both as a non additive measure and as a generalized set. In this paper, the theory of belief functions is argued to be a suitable framework for statistical analysis of low quality, i.e., imprecise and/or partially reliable data. After a reminder of general concepts of the theory, we show how this approach can be applied to statistical inference by viewing the normalized likelihood function as defining a consonant belief function. The links with likelihood-based and Bayesian inference are discussed.We then show how this method can be extended to the analysis of uncertain data. The approach is illustrated using a running example.
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
Aickin, M.: Connecting Dempster-Shafer belief functions with likelihood-based inference. Synthese 123, 347–364 (2000)
Barnard, G.A., Jenkins, G.M., Winsten, C.B.: Likelihood inference and time series. Journal of the Royal Statistical Society 125(3), 321–372 (1962)
Birnbaum, A.: On the foundations of statistical inference. Journal of the American Statistical Association 57(298), 269–306 (1962)
Côme, E., Oukhellou, L., Denœux, T., Aknin, P.: Learning from partially supervised data using mixture models and belief functions. Pattern Recognition 42(3), 334–348 (2009)
Cox, D.R.: Some remarks on statistical aspects of econometrics. In: Panaretos, J. (ed.) Stochastics Musings. Lawrence Erlbaum, Mahwah (2003)
Dempster, A.P.: New methods for reasoning towards posterior distributions based on sample data. Annals of Mathematical Statistics 37, 355–374 (1966)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38, 325–339 (1967)
Dempster, A.P.: A generalization of Bayesian inference (with discussion). J. R. Statistical Society B 30, 205–247 (1968)
Dempster, A.P.: Upper and lower probabilities generated by a random closed interval. Annals of Mathematical Statistics 39(3), 957–966 (1968)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B 39, 1–38 (1977)
Denœux, T.: Extending stochastic ordering to belief functions on the real line. Information Sciences 179, 1362–1376 (2009)
Denœux, T.: Maximum likelihood from evidential data: an extension of the EM algorithm. In: Borgelt, C., et al. (eds.) Combining Soft Computing and Statistical Methods in Data Analysis (Proceedings of SMPS 2010), Oviedeo, Spain. AISC, pp. 181–188. Springer (2010)
Denœux, T.: Maximum likelihood estimation from fuzzy data using the fuzzy EM algorithm. Fuzzy Sets and Systems 18(1), 72–91 (2011)
Denœux, T.: Maximum likelihood estimation from uncertain data in the belief function framework. IEEE Transactions on Knowledge and Data Engineering (to appear, 2012), doi:10.1109/TKDE.2011.201
Denœux, T., Masson, M.-H.: EVCLUS: Evidential clustering of proximity data. IEEE Trans. on Systems, Man and Cybernetics B 34(1), 95–109 (2004)
Dubois, D.: Possibility theory and statistical reasoning. Computational Statistics and Data Analysis 51(1), 47–69 (2006)
Dubois, D., Denœux, T.: Statistical inference with belief functions and possibility measures: a discussion of basic assumptions. In: Borgelt, C., et al. (eds.) Combining Soft Computing and Statistical Methods in Data Analysis (Proceedings of SMPS 2010), Oviedo, Spain. AISC, pp. 217–225. Springer (2010)
Dubois, D., Prade, H.: A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets. International Journal of General Systems 12(3), 193–226 (1986)
Dubois, D., Prade, H.: Possibility Theory: An approach to computerized processing of uncertainty. Plenum Press, New York (1988)
Edwards, A.W.F.: Likelihood (expanded edition). The John Hopkins University Press, Baltimore (1992)
Ferraro, M.B., Coppi, R., González Rodríguez, G., Colubi, A.: A linear regression model for imprecise response. International Journal of Approximate Reasoning 51(7), 759–770 (2010)
Heckman, J.J.: Econometrics and empirical economics. Journal of Econometrics 100, 3–5 (2001)
Hudson, D.J.: Interval estimation from the likelihood function. J. R. Statistical Society B 33(2), 256–262 (1973)
Nguyen, H.T.: An Introduction to Random Sets. Chapman and Hall/CRC Press, Boca Raton, Florida (2006)
Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)
Shafer, G.: Allocations of probability. Annals of Probability 7(5), 827–839 (1979)
Shafer, G.: Constructive probability. Synthese 48(1), 1–60 (1981)
Shafer, G.: Belief functions and parametric models (with discussion). J. Roy. Statist. Soc. Ser. B 44, 322–352 (1982)
Smets, P.: Resolving misunderstandings about belief functions. International Journal of Approximate Reasoning 6, 321–344 (1990)
Smets, P.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate Reasoning 9, 1–35 (1993)
Smets, P.: The Transferable Belief Model for quantified belief representation. In: Gabbay, D.M., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, pp. 267–301. Kluwer Academic Publishers, Dordrecht (1998)
Smets, P.: Belief functions on real numbers. International Journal of Approximate Reasoning 40(3), 181–223 (2005)
Smets, P., Kennes, R.: The Transferable Belief Model. Artificial Intelligence 66, 191–243 (1994)
Sprott, D.A.: Statistical Inference in Science. Springer, Berlin (2000)
Walley, P.: Belief function representations of statistical evidence. The Annals of Statistics 15(4), 1439–1465 (1987)
Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)
Wasserman, L.A.: Belief functions and statistical evidence. The Canadian Journal of Statistics 18(3), 183–196 (1990)
Yager, R.R.: The entailment principle for Dempster-Shafer granules. Int. J. of Intelligent Systems 1, 247–262 (1986)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28 (1978)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Denœux, T. (2013). Statistical Inference from Ill-known Data Using Belief Functions. In: Huynh, VN., Kreinovich, V., Sriboonchitta, S., Suriya, K. (eds) Uncertainty Analysis in Econometrics with Applications. Advances in Intelligent Systems and Computing, vol 200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35443-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-35443-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35442-7
Online ISBN: 978-3-642-35443-4
eBook Packages: EngineeringEngineering (R0)