Dependent Possibilistic Arithmetic using Copulas

Ander Gray, Dominik Hose, Marco De Angelis, Michael Hanss, Scott Ferson
Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, PMLR 147:169-179, 2021.

Abstract

We describe two functions on possibility distributions which allow one to compute binary operations with dependence either specified by a copula or partially defined by an imprecise copula. We use the fact that possibility distributions are consonant belief functions to aggregate two possibility distributions into a bivariate belief function using a version of Sklar’s theorem for minitive belief functions, i.e. necessity measures. The results generalise previously published independent and Fréchet methods, allowing for any stochastic dependence to be specified in the form of a (imprecise) copula. This new method produces tighter extensions than previous methods when a precise copula is used. These latest additions to possibilistic arithmetic give it the same capabilities as p-box arithmetic, and provides a basis for a p-box/possibility hybrid arithmetic. This combined arithmetic provides tighter bounds on the exact upper and lower probabilities than either method alone for the propagation of general belief functions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v147-gray21a, title = {Dependent Possibilistic Arithmetic using Copulas}, author = {Gray, Ander and Hose, Dominik and De Angelis, Marco and Hanss, Michael and Ferson, Scott}, booktitle = {Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications}, pages = {169--179}, year = {2021}, editor = {Cano, Andrés and De Bock, Jasper and Miranda, Enrique and Moral, Serafı́n}, volume = {147}, series = {Proceedings of Machine Learning Research}, month = {06--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v147/gray21a/gray21a.pdf}, url = {https://proceedings.mlr.press/v147/gray21a.html}, abstract = {We describe two functions on possibility distributions which allow one to compute binary operations with dependence either specified by a copula or partially defined by an imprecise copula. We use the fact that possibility distributions are consonant belief functions to aggregate two possibility distributions into a bivariate belief function using a version of Sklar’s theorem for minitive belief functions, i.e. necessity measures. The results generalise previously published independent and Fréchet methods, allowing for any stochastic dependence to be specified in the form of a (imprecise) copula. This new method produces tighter extensions than previous methods when a precise copula is used. These latest additions to possibilistic arithmetic give it the same capabilities as p-box arithmetic, and provides a basis for a p-box/possibility hybrid arithmetic. This combined arithmetic provides tighter bounds on the exact upper and lower probabilities than either method alone for the propagation of general belief functions.} }
Endnote
%0 Conference Paper %T Dependent Possibilistic Arithmetic using Copulas %A Ander Gray %A Dominik Hose %A Marco De Angelis %A Michael Hanss %A Scott Ferson %B Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2021 %E Andrés Cano %E Jasper De Bock %E Enrique Miranda %E Serafı́n Moral %F pmlr-v147-gray21a %I PMLR %P 169--179 %U https://proceedings.mlr.press/v147/gray21a.html %V 147 %X We describe two functions on possibility distributions which allow one to compute binary operations with dependence either specified by a copula or partially defined by an imprecise copula. We use the fact that possibility distributions are consonant belief functions to aggregate two possibility distributions into a bivariate belief function using a version of Sklar’s theorem for minitive belief functions, i.e. necessity measures. The results generalise previously published independent and Fréchet methods, allowing for any stochastic dependence to be specified in the form of a (imprecise) copula. This new method produces tighter extensions than previous methods when a precise copula is used. These latest additions to possibilistic arithmetic give it the same capabilities as p-box arithmetic, and provides a basis for a p-box/possibility hybrid arithmetic. This combined arithmetic provides tighter bounds on the exact upper and lower probabilities than either method alone for the propagation of general belief functions.
APA
Gray, A., Hose, D., De Angelis, M., Hanss, M. & Ferson, S.. (2021). Dependent Possibilistic Arithmetic using Copulas. Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 147:169-179 Available from https://proceedings.mlr.press/v147/gray21a.html.

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