Aggregation of Epistemic Uncertainty in Forms of Possibility and Certainty Factors
Nagaoka University of Technology
1603-1 Kami-Tomioka, Nagaoka, Niigata 940-2188, Japan
Uncertainty aggregation is an important reasoning for making decisions in the real world, which is full of uncertainty. The paper proposes an information source model for aggregating epistemic uncertainties about truth and discusses uncertainty aggregation in the form of possibility distributions. A new combination rule of possibilities for truth is proposed. Then, this paper proceeds to discussion about a traditional but seemingly forgotten representation of uncertainty (i.e., certainty factors (CFs)) and proposes a new interpretation based on possibility theory. CFs have been criticized because of their lack of sound mathematical interpretation from the viewpoint of probability. Thus, this paper first establishes a theory for a sound interpretation using possibility theory. Then it examines aggregation of CFs based on the interpretation and some combination rules of possibility distributions. The paper proposes several combination rules for CFs having sound theoretical basis, one of which is exactly the same as the oft-criticized combination.
-  D. Dubois and H. Prade, “A review of fuzzy set aggregation operators,” Information Sciences, Vol.36, No.1-2, pp. 85-121, 1985.
-  S. Parsons and A. Hunter, “A review of uncertainty handling formalisms,” A. Hunter and S. Parsons (Eds.), “Applications of Uncertainty Formalisms,” Lecture Notes in Computer Science (LNCS), Vol.1455, pp. 8-37, Springer, 1998.
-  D. Dubois and H. Prade, “On the use of aggregation operators in information fusion processes,” Fuzzy Sets and Systems, Vol.142, No.1, pp. 143-161, 2004.
-  J.-L. Marichal, “Aggregation functions for decision making,” D. Bouyssou, D. Dubois, M. Pirlot, and H. Prade (Eds.), “Decision Making Process – Concepts and Methods,” pp. 673-721, Wiley, 2009.
-  B. Khaleghi, A. Khamis, F. O. Karray, and S. N. Razavi, “Multisensor data fusion: A review of the-state-of-the-art,” Information Fusion, Vol.14, No.1, pp. 28-44, 2013.
-  C. Y. Wang, “Generalized aggregation of fuzzy truth values,” Information Sciences, Vol.324, pp. 208-216, 2015.
-  D. Dubois, W. Liu, J. Ma, and H. Prade, “The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks,” Information Fusion, Vol.32, Part A, pp. 12-39, 2016.
-  S. Marrara, G. Pasi, and M. Viviani, “Aggregation operators in information retrieval,” Fuzzy Sets and Systems, Vol.324, pp. 3-19, 2017.
-  A. Mardani, M. Nilashi, E. K. Zavadskas, S. R. Awang, H. Zare, and N. M. Jamal, “Decision making methods based on fuzzy aggregation operators: Three decades review from 1986 to 2017,” Int. J. of Information Technology & Decision Making, Vol.17, No.2, pp. 391-466, 2018.
-  A. K. Tsadiras and K. G. Margaritis, “The MYCIN certainty factor handling functions as uninorm operator and its use as a threshold function in artificial neurons,” Fuzzy Sets and Systems, Vol.93, No.3, pp. 263-274, 1998.
-  D. Dubois and H. Prade, “Representation and combination of uncertainty with belief functions and possibility measures,” Computational Intelligence, Vol.4, No.3, pp. 244-264, 1988.
-  D. Dubois and H. Prade, “Possibility theory and data fusion in poorly informed environments,” Control Engineering Practice, Vol.2, No.5, pp. 811-823, 1994.
-  B. G. Buchanan and E. H. Shortliffe, “Rule-based expert systems – The MYCIN experiments of the Stanford Heuristic Programming Project,” Addison-Wesley, 1984.
-  E. H. Shortliffe and B. G. Buchanan, “A model of inexact reasoning in medicine,” Mathematical Biosciences, Vol.23, No.3-4, pp. 351-379, 1975.
-  A. P. Dempster, “Upper and lower probabilities induced by multivalued mapping,” The Annals of Mathematical Statistics, Vol.38, No.2, pp. 325-339, 1967.
-  G. Shafer, “A Mathematical Theory of Evidence,” Princeton University Press, 1976.
-  R. R. Yager, “On the relationship of methods of aggregating evidence in expert systems,” Cybernetics and Systems, Vol.16, No.16, pp. 1-21, 1985.
-  R. R. Yager, “On the Dempster-Shafer framework and new combination rules,” Information Sciences, Vol.41, No.2, pp. 93-137, 1987.
-  D. I. Blockley and J. F. Baldwin, “Uncertain inference in knowledge-based systems,” J. of Engineering Mechanics, Vol.113, No.4, pp. 467-481, 1987.
-  H. Y. Hau and R. L. Kashyap, “Belief combination and propagation in a lattice-structured inference network,” IEEE Trans. on on Systems, Man, and Cybernetics (SMC), Vol.20, No.1, pp. 45-57, 1990.
-  P. Smets, “The combination of evidence in the transferable belief model,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.12, No.5, pp. 447-458, 1990.
-  L. Zhang, “Representation independence and combination of evidence in the Dempster-Shafer theory,” R. R. Yager, J. Kacprzyk, and M. Fedrizzi (Eds.), “Advances in the Dempster-Shafer Theory of Evidence,” pp. 51-69, Wiley, 1994.
-  C. K. Murphy, “Combining belief functions when evidence conflicts,” Decision Support Systems, Vol.29, No.1, pp. 1-9, 2000.
-  A. Jøsang, “The consensus operator for combining beliefs,” Artificial Intelligence, Vol.141, No.1-2, pp. 157-170, 2002.
-  E. Lefevre, O. Colot, and P. Vannoorenberghe, “Belief function combination and conflict management,” Information Fusion, Vol.3, No.2, pp. 149-162, 2002.
-  K. Yamada, “A new combination of evidence based on compromise,” Fuzzy Sets and Systems, Vol.159, No.13, pp. 1689-1708, 2008.
-  D. Dubois and H. Prade, “Possibility theory in information fusion,” G. D. Riccia, H.-J. Lenz, and R. Kruse (Eds.), “Data Fusion and Perception,” pp. 53-76, Springer, 2001.
-  M. Oussalah, H. Maaref, and C. Barret, “From adaptive to progressive combination of possibility distributions,” Fuzzy Sets and Systems, Vol.139, No.3, pp. 559-582, 2003.
-  D. Heckerman, “Probabilistic Interpretations for MYCIN’s Certainty Factors,” Machine Intelligence and Pattern Recognition, Vol.4, pp. 167-196, 1986.
-  G. P. Amaya Cruz and G. Beliakov, “On the interpretation of certainty factors in expert systems,” Artificial Intelligence in Medicine, Vol.8, No.1, pp. 1-14, 1996.
-  P. Walley, “Belief function representations of statistical evidence,” The Annals of Statistics, Vol.15, No.4, pp. 1439-1465, 1987.
-  P. Walley, “Measures of uncertainty in expert systems,” Artificial Intelligence, Vol.83, No.1, pp. 1-58, 1996.
-  F. Voorbraak, “On the justification of Dempster’s rule of combination,” Artificial Intelligence, Vol.48, pp. 171-197, 1991.
-  K. Yamada, “Aggregation of epistemic uncertainty: A new interpretation of the Certainty Factor with possibility theory and causation events,” Proc. of 2018 Joint 10th Int. Conf. on Soft Computing and Intelligent Systems (SCIS) and 19th Int. Symp. on Advanced Intelligent Systems (ISIS), pp. 377-382, 2018.
-  G. Shafer, “Perspectives on the theory and practice of belief functions,” Int. J. of Approximate Reasoning, Vol.4, No.5-6, pp. 323-362, 1990.
-  Y. Peng and J. A. Reggia, “A probabilistic causal model for diagnostic problem solving Part I: integrating symbolic causal inference with numeric probabilistic inference,” IEEE Trans. on Systems, Man, and Cybernetics, Vol.17, No.2, pp. 146-162, 1987.
-  Y. Peng and J. A. Reggia, “Abductive Inference Models for Diagnostic Problem-Solving,” Springer, 1990.
-  K. Yamada, “Possibilistic causality consistency problem based on asymmetrically-valued causal model,” Fuzzy Sets and Systems, Vol.132, No.1, pp. 33-48, 2002.
-  K. Yamada, “Diagnosis under compound effects and multiple causes by means of the conditional causal possibility approach,” Fuzzy Sets and Systems, Vol.145, No.2, pp. 183-212, 2004.
-  L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, Vol.1, pp. 3-28, 1978.
-  E. Hisdal, “Conditional possibilities independence and noninteraction,” Fuzzy Sets and Systems, Vol.1, No.4, pp. 283-297, 1978.
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