Abstract
The extreme points of convex probability sets play an important practical role, especially as a tool to obtain specific, easier to manipulate sets. Although this problem has been studied for many models (probability intervals, possibility distributions), it remains to be studied for imprecise cumulative distributions (a.k.a. p-boxes). This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function.
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Notes
- 1.
For the sake of simplicity, we use the terminology “extreme points of a belief function” to refer to the extreme points of the credal set associated with the belief function.
- 2.
Recall that an extreme point P of \(\mathcal {M}(\text {Bel})\) is a point such that, if \(P_1,P_2 \in \mathcal {M}(\text {Bel})\) and \(\alpha P_1 + (1-\alpha ) P_2=P\) for some \(\alpha \in (0,1)\), then \(P_1=P_2=P\).
- 3.
By interval, we mean that all elements between \(\min E\) and \(\max E\) are included in E.
References
Chateauneuf A, Jaffray JY (1989) Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Math Soc Sci 17(3):263–283
De Campos LM, Huete JF, Moral S (1994) Probability intervals: a tool for uncertain reasoning. Int J Uncertain Fuzziness Knowle-Based Syst 2(2):167–196
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38:325–339
Ferson S, Kreinovich V, Ginzburg L, Myers DS, Sentz K (2003) Constructing probability boxes and Dempster-Shafer structures. Technical Report SAND2002-4015, Sandia National Laboratories
Kriegler E (2005) Utilizing belief functions for the estimation of future climate change. Int J Approx Reason 39(2–3):185–209
Miranda E, Couso I, Gil P (2003) Extreme points of credal sets generated by 2-alternating capacities. Int J Approx Reason 33(1):95–115
Pelessoni R, Vicig P, Montes I, Miranda E (2015) Bivariate p-boxes. Int J Uncertain Fuzziness Knowl-Based Syst (accepted)
Schollmeyer G (2015) On the number and characterization of the extreme points of the core of necessity measures on finite spaces. In: Augustin T, Doria S, Miranda E, Quaeghebeur E (eds) Proceedings of the 9th ISIPTA
Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton, NJ
Troffaes MCM, Destercke S (2011) Probability boxes on totally preordered spaces for multivariate modelling. Int J Approx Reason 52(6):767–791
Walley P (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall, London
Wallner A (2007) Extreme points of coherent probabilities in finite spaces. Int J Approx Reason 44:339–357
Acknowledgments
We thank Georg Schollmeyer for insightful exchanges and discussion about this problem. The research reported in this paper has been supported by project TIN2014-59543-P and by LABEX MS2T (ANR-11-IDEX-0004-02).
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Montes, I., Destercke, S. (2017). On Extreme Points of p-Boxes and Belief Functions. In: Ferraro, M., et al. Soft Methods for Data Science. SMPS 2016. Advances in Intelligent Systems and Computing, vol 456. Springer, Cham. https://doi.org/10.1007/978-3-319-42972-4_45
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DOI: https://doi.org/10.1007/978-3-319-42972-4_45
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