A Polarity Theory for Sets of Desirable Gambles

Alessio Benavoli, Alessandro Facchini, Marco Zaffalon, José Vicente-Pérez
Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 62:37-48, 2017.

Abstract

Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities.

Cite this Paper

BibTeX
@InProceedings{pmlr-v62-benavoli17b, title = {A Polarity Theory for Sets of Desirable Gambles}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco and Vicente-Pérez, José}, booktitle = {Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {37--48}, year = {2017}, editor = {Antonucci, Alessandro and Corani, Giorgio and Couso, Inés and Destercke, Sébastien}, volume = {62}, series = {Proceedings of Machine Learning Research}, month = {10--14 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v62/benavoli17b/benavoli17b.pdf}, url = {https://proceedings.mlr.press/v62/benavoli17b.html}, abstract = {Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities.} }
Endnote
%0 Conference Paper %T A Polarity Theory for Sets of Desirable Gambles %A Alessio Benavoli %A Alessandro Facchini %A Marco Zaffalon %A José Vicente-Pérez %B Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2017 %E Alessandro Antonucci %E Giorgio Corani %E Inés Couso %E Sébastien Destercke %F pmlr-v62-benavoli17b %I PMLR %P 37--48 %U https://proceedings.mlr.press/v62/benavoli17b.html %V 62 %X Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities.
APA
Benavoli, A., Facchini, A., Zaffalon, M. & Vicente-Pérez, J.. (2017). A Polarity Theory for Sets of Desirable Gambles. Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 62:37-48 Available from https://proceedings.mlr.press/v62/benavoli17b.html.

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