Abstract
In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or “consistent probabilities”. We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Haenni, R., Romeijn, J., Wheeler, G., Williamson, J.: Possible semantics for a common framework of probabilistic logics. In: Huynh, V.N., Nakamori, Y., Ono, H., Lawry, J., Kreinovich, V., Nguyen, H.T. (eds.) UncLog 2008, International Workshop on Interval/Probabilistic Uncertainty and Non-Classical Logics, Ishikawa, Japan. Advances in Soft Computing, vol. 46, pp. 268–279 (2008)
Mercier, D., Denoeux, T., Masson, M.: Refined sensor tuning in the belief function framework using contextual discounting. In: IPMU (2006)
Demotier, S., Schon, W., Denoeux, T.: Risk assessment based on weak information using belief functions: a case study in water treatment. IEEE Transactions on Systems, Man and Cybernetics, Part C 36(3), 382–396 (2006)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Dubois, D., Prade, H.: Possibility theory. Plenum Press, New York (1988)
Smets, P.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate Reasoning 9, 1–35 (1993)
Denoeux, T.: A new justification of the unnormalized Dempster’s rule of combination from the Least Commitment Principle. In: Proceedings of FLAIRS 2008, Special Track on Uncertaint Reasoning (2008)
Yager, R.R.: The entailment principle Dempster-Shafer granules. International Journal of Intelligent Systems 1, 247–262 (1986)
Dubois, D., Prade, H.: A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets. Int. J. of General Systems 12, 193–226 (1986)
Levi, I.: The enterprise of knowledge: An essay on knowledge, credal probability, and chance. The MIT Press, Cambridge (1980)
Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences 17, 263–283 (1989)
Ha, V., Haddawy, P.: Geometric foundations for interval-based probabilities. In: Cohn, A.G., Schubert, L., Shapiro, S.C. (eds.) KR 1998: Principles of Knowledge Representation and Reasoning, pp. 582–593. Morgan Kaufmann, San Francisco (1998)
Smets, P.: The nature of the unnormalized beliefs encountered in the transferable belief model. In: Proceedings of the 8th Annual Conference on Uncertainty in Artificial Intelligence (UAI-92), San Mateo, CA, Morgan Kaufmann (1992) 292–29
Dubois, D., Grabisch, M., Prade, H., Smets, P.: Using the transferable belief model and a qualitative possibility theory approach on an illustrative example: the assessment of the value of a candidate. Intern. J. Intell. Systems (2001)
Snow, P.: The vulnerability of the transferable belief model to dutch books. Artificial Intelligence 105, 345–354 (1998)
Smets, P., Kennes, R.: The transferable belief model. AI 66(2), 191–234 (1994)
Melkonyan, T., Chambers, R.: Degree of imprecision: Geometric and algebraic approaches. International Journal of Approximate Reasoning (2006)
Cozman, F.G.: Calculation of posterior bounds given convex sets of prior probability measures and likelihood functions. Journal of Computational and Graphical Statistics 8(4), 824–838 (1999)
Seidenfeld, T., Wasserman, L.: Dilation for convex sets of probabilities. Annals of Statistics 21, 1139–1154 (1993)
Seidenfeld, T., Schervish, M., Kadane, J.: Coherent choice functions under uncertainty. In: Proceedings of ISIPTA 2007 (2007)
Baroni, P.: Extending consonant approximations to capacities. In: Proceedings of IPMU, pp. 1127–1134 (2004)
Dubois, D., Prade, H.: Consonant approximations of belief functions. International Journal of Approximate Reasoning 4, 419–449 (1990)
Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Trans. Systems, Man, and Cybernetics C 38(3) (in press, 2008)
Denoeux, T.: Conjunctive and disjunctive combination of belief functions induced by non distinct bodies of evidence. Artificial Intelligence (2007)
Dubois, D., Prade, H., Smets, P.: New semantics for quantitative possibility theory. In: ISIPTA, pp. 152–161 (2001)
Joslyn, C.: Towards an empirical semantics of possibility through maximum uncertainty. In: Lowen, R., Roubens, M. (eds.) Proc. IFSA 1991, pp. 86–89 (1991)
Cuzzolin, F.: The geometry of consonant belief functions: Simplicial complexes of possibility measures. Fuzzy Sets and Systems (under review) (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cuzzolin, F. (2008). On the Credal Structure of Consistent Probabilities. In: Hölldobler, S., Lutz, C., Wansing, H. (eds) Logics in Artificial Intelligence. JELIA 2008. Lecture Notes in Computer Science(), vol 5293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87803-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-87803-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87802-5
Online ISBN: 978-3-540-87803-2
eBook Packages: Computer ScienceComputer Science (R0)