Extending stochastic ordering to belief functions on the real line

In this paper, the concept of stochastic ordering is extended to belief functions on the real line defined by random closed intervals. In this context, the usual stochastic ordering is shown to break down into four distinct ordering relations, called credal orderings, which correspond to the four basic ordering structures between intervals. These orderings are characterized in terms of lower and upper expectations. We then derive the expressions of the least committed (least informative) belief function credally less (respectively, greater) than or equal to a given belief function. In each case, the solution is a consonant belief function that can be described by a possibility distribution. A simple application to reliability analysis is used as an example throughout the paper.

[1]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[2]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[3]  Jürg Kohlas,et al.  An algebraic theory for statistical information based on the theory of hints , 2008, Int. J. Approx. Reason..

[4]  Thierry Denoeux,et al.  Fusion of one-class classifiers in the belief function framework , 2007, 2007 10th International Conference on Information Fusion.

[5]  P. Sevastianov.  Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory , 2007 .

[6]  Radko Mesiar,et al.  Fuzzy Interval Analysis , 2000 .

[7]  Ronald R. Yager,et al.  The entailment principle for dempster—shafer granules , 1986, Int. J. Intell. Syst..

[8]  L. Wasserman.  Belief functions and statistical inference , 1990 .

[9]  D. Dubois,et al.  A set-theoretic view of belief functions: Logical operations and approximations by fuzzy sets , 1986 .

[10]  Scott Ferson,et al.  Constructing Probability Boxes and Dempster-Shafer Structures , 2003 .

[11]  Philippe Smets,et al.  Classification Using Belief Functions: Relationship Between Case-Based and Model-Based Approaches , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[12]  Ronald R. Yager.  Cumulative distribution functions from Dempster-Shafer belief structures , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[13]  Jürg Kohlas,et al.  Handbook of Defeasible Reasoning and Uncertainty Management Systems , 2000 .

[14]  Alessandro Saffiotti,et al.  The Transferable Belief Model , 1991, ECSQARU.

[15]  Suez Environnement.  Novelty detection in the belief functions framework , 2006 .

[16]  Thierry Denoeux,et al.  Nonparametric regression analysis of uncertain and imprecise data using belief functions , 2004, Int. J. Approx. Reason..

[17]  Arthur P. Dempster,et al.  The Dempster-Shafer calculus for statisticians , 2008, Int. J. Approx. Reason..

[18]  E. Lehmann.  Ordered Families of Distributions , 1955 .

[19]  Russell G. Almond.  Graphical belief modeling , 1995 .

[20]  Arthur P. Dempster,et al.  New Methods for Reasoning Towards PosteriorDistributions Based on Sample Data , 1966, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[21]  T. Denœux.  Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence , 2008 .

[22]  Jian-Bo Yang,et al.  On the combination and normalization of interval-valued belief structures , 2007, Information Sciences.

[23]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

[24]  Michael Friendly,et al.  Discrete Distributions , 2005, Probability and Bayesian Modeling.

[25]  Thierry Denoeux,et al.  Pairwise classifier combination using belief functions , 2007, Pattern Recognit. Lett..

[26]  Wei-Zhi Wu,et al.  Attribute reduction based on evidence theory in incomplete decision systems , 2008, Inf. Sci..

[27]  Minh Ha-Duong.  Hierarchical fusion of expert opinions in the Transferable Belief Model, application to climate sensitivity , 2008, Int. J. Approx. Reason..

[28]  G. Klir,et al.  Uncertainty-based information: Elements of generalized information theory (studies in fuzziness and soft computing). , 1998 .

[29]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[30]  Philippe Smets,et al.  Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem , 1993, Int. J. Approx. Reason..

[31]  A. Dempster.  Upper and Lower Probabilities Generated by a Random Closed Interval , 1968 .

[32]  Thierry Denoeux,et al.  Learning from partially supervised data using mixture models and belief functions , 2009, Pattern Recognit..

[33]  Philippe Smets,et al.  The Transferable Belief Model for Quantified Belief Representation , 1998 .

[34]  Ronald R. Yager,et al.  Classic Works of the Dempster-Shafer Theory of Belief Functions , 2010, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[35]  Thierry Denoeux,et al.  ECM: An evidential version of the fuzzy c , 2008, Pattern Recognit..

[36]  Thierry Denoeux,et al.  EVCLUS: evidential clustering of proximity data , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[37]  Philippe Smets,et al.  Belief functions on real numbers , 2005, Int. J. Approx. Reason..

[38]  George J. Klir,et al.  Uncertainty-Based Information , 1999 .