Proposition and learning of some belief function contextual correction mechanisms

Knowledge about the quality of a source can take several forms: it may for instance relate to its truthfulness or to its relevance, and may even be uncertain. Of particular interest in this paper is that such knowledge may also be contextual; for instance the reliability of a sensor may be known to depend on the actual object observed. Various tools, called correction mechanisms, have been developed within the theory of belief functions, to take into account knowledge about the quality of a source. Yet, only a single tool is available to account for contextual knowledge about the quality of a source, and precisely about the relevance of a source. There is thus some lack of flexibility since contextual knowledge about the quality of a source does not have to be restricted to its relevance. The first aim of this paper is thus to try and enlarge the set of tools available in belief function theory to deal with contextual knowledge about source quality. This aim is achieved by (1) providing an interpretation to each one of two contextual correction mechanisms introduced initially from purely formal considerations, and (2) deriving extensions essentially by uncovering contextual forms of two interesting and non-contextual correction mechanisms. The second aim of this paper is related to the origin of contextual knowledge about the quality of a source: due to the lack of dedicated approaches, it is indeed not clear how to obtain such specific knowledge in practice. A sound, easy to interpret and computationally simple method is therefore provided to learn from data contextual knowledge associated with the contextual correction mechanisms studied in this paper. An interpretation is given to contextual discounting and to contextual reinforcement.These two contextual correction mechanisms are also further extended.A contextual version of a truthfulness-based correction mechanism is introduced.A method to learn from data contextual correction mechanism parameters is studied.

[1]  Eric Lefevre,et al.  Corrigendum to "Belief functions contextual discounting and canonical decompositions" [International Journal of Approximate Reasoning 53 (2012) 146-158] , 2016, Int. J. Approx. Reason..

[2]  P. Smets,et al.  Assessing sensor reliability for multisensor data fusion within the transferable belief model , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[3]  Johan Schubert,et al.  Conflict management in Dempster-Shafer theory using the degree of falsity , 2011, Int. J. Approx. Reason..

[4]  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.

[5]  Christophe Labreuche,et al.  Multidimensional Approach to Reliability Evaluation of Information Sources , 2014 .

[6]  Frédéric Pichon,et al.  Fonctions de croyance : décompositions canoniques et règles de combinaison. (Belief functions: canonical decompositions and combination rules) , 2009 .

[7]  F. Pichon.  Interpretation and Computation of α-Junctions for Combining Belief Functions , 2009 .

[8]  Thierry Denoeux,et al.  A k-nearest neighbor classification rule based on Dempster-Shafer theory , 1995, IEEE Trans. Syst. Man Cybern..

[9]  Thierry Denoeux,et al.  Decision fusion for postal address recognition using belief functions , 2009, Expert Syst. Appl..

[10]  Thierry Denoeux,et al.  An evidence-theoretic k-NN rule with parameter optimization , 1998, IEEE Trans. Syst. Man Cybern. Part C.

[11]  Chongzhao Han,et al.  Discounted combination of unreliable evidence using degree of disagreement , 2013, Int. J. Approx. Reason..

[12]  Philippe Smets.  The alpha-junctions: Combination Operators Applicable to Belief Functions , 1997, ECSQARU-FAPR.

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

[14]  Didier Dubois,et al.  A definition of subjective possibility , 2008, Int. J. Approx. Reason..

[15]  Zied Elouedi,et al.  Discountings of a Belief Function Using a Confusion Matrix , 2010, 2010 22nd IEEE International Conference on Tools with Artificial Intelligence.

[16]  Didier Dubois,et al.  Statistical reasoning with set-valued information: Ontic vs. epistemic views , 2014, Int. J. Approx. Reason..

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

[18]  Thomas Burger,et al.  Toward an Axiomatic Definition of Conflict Between Belief Functions , 2013, IEEE Transactions on Cybernetics.

[19]  Philippe Smets,et al.  The Canonical Decomposition of a Weighted Belief , 1995, IJCAI.

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

[21]  John Klein,et al.  New distances between bodies of evidence based on Dempsterian specialization matrices and their consistency with the conjunctive combination rule , 2014, Int. J. Approx. Reason..

[22]  Olivier Colot,et al.  Automatic discounting rate computation using a dissent criterion , 2010 .

[23]  Frédéric Pichon.  On the α-Conjunctions for Combining Belief Functions , 2012, Belief Functions.

[24]  Thierry Denoeux,et al.  Relevance and truthfulness in information correction and fusion , 2012, Int. J. Approx. Reason..

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

[26]  Eric Lefevre,et al.  Belief functions contextual discounting and canonical decompositions , 2012, Int. J. Approx. Reason..

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

[28]  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).

[29]  Philippe Smets,et al.  The Transferable Belief Model , 1991, Artif. Intell..

[30]  Geoffrey K. Pullum,et al.  A Student's Introduction to English Grammar , 2021 .

[31]  Eric Lefevre,et al.  Learning Contextual Discounting and Contextual Reinforcement from Labelled Data , 2015, ECSQARU.

[32]  Quan Pan,et al.  Belief rule-based classification system: Extension of FRBCS in belief functions framework , 2015, Inf. Sci..

[33]  Thomas Burger,et al.  A Consistency-Specificity Trade-Off to Select Source Behavior in Information Fusion , 2015, IEEE Transactions on Cybernetics.

[34]  Thierry Denoeux,et al.  Refined modeling of sensor reliability in the belief function framework using contextual discounting , 2008, Inf. Fusion.

[35]  Eric Lefevre,et al.  Truthfulness in Contextual Information Correction , 2014, Belief Functions.

[36]  T. Denoeux,et al.  Interpretation and Computation of alpha-Junctions for Combining Belief Functions , 2009 .

[37]  Anne-Laure Jousselme,et al.  Distances in evidence theory: Comprehensive survey and generalizations , 2012, Int. J. Approx. Reason..

[38]  Prakash P. Shenoy,et al.  On the plausibility transformation method for translating belief function models to probability models , 2006, Int. J. Approx. Reason..

[39]  Philippe Smets,et al.  Decision making in the TBM: the necessity of the pignistic transformation , 2005, Int. J. Approx. Reason..

[40]  Eric Lefevre,et al.  A high-level application using belief functions for exchanging and managing uncertain events on the road in vehicular ad hoc networks , 2014, Ann. des Télécommunications.