Abstract
Monotone capacities (on finite sets) of finite or infinite order (lower probabilities) are characterized by properties of their Möbius inverses. A necessary property of probabilities dominating a given capacity is demonstrated through the use of Gale’s theorem for the transshipment problem. This property is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities of order 2 is also proved.
This is a preview of subscription content, log in via an institution to check access.
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
B. Anger, Approximation of Capacities by Measures, in: Lecture Notes in Mathematics 226 (Springer, Berlin, 1971) pp. 152–170.
B. Anger, Representation of capacities, Math. Ann. 229 (1977) 245–258.
C. Berge, Espaces Topologiques, Fonctions Multivoques (Dunod, Paris, 1965).
R.E. Bixby, W.H. Cunningham and D.M. Tokpis, The partial order of a polymatroïd extreme point, Math. Oper. Res. 10 (1985) 367–378.
G. Choquet, Théorie des capacités, Ann. Inst. Fourier (Grenoble) (1953) V. 131–295.
M. Cohen and J.Y. Jaffray, Decision making in a case of mixed uncertainty: A normative model, J. Math. Psych. 29 (1985) 428–442.
C. Dellacherie, Quelques commentaires sur les prolongements de capacités, Lect. Notes Math. 191 (Sem. Prob. V) (1971) 77–81.
A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325–339.
J. Edmonds, Submodular functions, matroïds and certain polyhedra, Combinatorial structures and their applications (Proc. Calgary Int. Conf. 1969). R.L. Guy et al., eds (Gordon and Breach, New York, 1970) pp. 69–87.
D. Gale, The Theory of Linear Economic Models (Mc Graw Hill, New York, 1960).
P.J. Huber, The use of Choquet capacities in statistics, Bull. Int. Statist. Inst. XLV, Book 4 (1973) 181–188.
P.J. Huber, Kapazitäten statt Wahrscheinlichkeiten. Gedanken zur Grundlegung der Statistik, J. der Dt. Math. Verein. 78 (1976) 81–92.
P.J. Huber and V. Strassen, Minimax tests and the Neyman-Pearson lemma for capacities, Ann. Statist. 1 (1973) 251–263.
T. Ishiishi, Super-modularity: applications to convex games and to the greedy algorithm for LP, J. Econom. Theory 25 (1981) 283–286.
S. Karlin, Mathematical Methods and Theory in Games, Programming and Economics, Vol. 1 (Pergamon Press, London, Paris, 1959).
H. Kyburg, The Logical Foundations of Statistical Inference (Reidel, Dordrecht, 1974).
I. Levi, The Enterprise of Knowledge (MIT Press, Cambridge, 1980).
A. Papamarcou and T.L. Fine, A note on undominated lower probabilities, The Annals of Probab. 14 (1986) 710–723.
A. Revuz, Fonctions croissantes et mesures sur les espaces topologiques ordonnés, Ann. Instit. Fourier (Grenoble VI) (1955) 187–269.
G.C. Rota, Theory of Möbius functions, Z. fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 2 (1964) 340–368.
G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, New Jersey, 1976).
G. Shafer, Allocations of probability, Ann. Prob. 7 (1979) 827–839.
G. Shafer, Constructive probability, Synthese 48 (1981) 1–59.
L.S. Shapley, Cores of convex games, Int. J. Game Theory 1 (1971) 11–26.
A. Wald, Statistical Decision Functions (Chelsea Publishing Company, Bronx, New York, 1971).
P. Walley and T.L. Fine, Towards a frequentist theory of upper and lower probability, Ann. Statist. 10 (1982) 741–761.
P. Walley and T.L. Fine, Varieties of modal (classificatory) and comparative probability, Synthese, 41 (1979) 321–374.
M. Wolfenson and T.L. Fine, Bayes-like decision making with upper and lower probabilities, J. Amer. Statist. Assoc. 77 (1982) 80–88.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Chateauneuf, A., Jaffray, JY. (2008). Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-44792-4_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25381-5
Online ISBN: 978-3-540-44792-4
eBook Packages: EngineeringEngineering (R0)