In Search of a Global Belief Model for Discrete-Time Uncertain Processes

Natan T’Joens, Jasper De Bock, Gert de Cooman
Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 103:377-385, 2019.

Abstract

To model discrete-time uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that satisfies a number of simple and intuitive axioms. We motivate these axioms on the basis of two possible interpretations for this upper expectation: a behavioural interpretation similar to that of Walley’s, and an interpretation in terms of upper envelopes of linear expectations. Subsequently, we show that the most conservative upper expectation satisfying our axioms coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications. On the one hand, it guarantees that there is a unique most conservative global belief model satisfying our axioms. On the other hand, it shows that Shafer and Vovk’s model can be given an axiomatic characterisation, thereby providing an alternative motivation for adopting this model, even outside their framework.

Cite this Paper


BibTeX
@InProceedings{pmlr-v103-t-joens19a, title = {In Search of a Global Belief Model for Discrete-Time Uncertain Processes}, author = {T'Joens, Natan and De Bock, Jasper and {de Cooman}, Gert}, booktitle = {Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications}, pages = {377--385}, year = {2019}, editor = {De Bock, Jasper and de Campos, Cassio P. and de Cooman, Gert and Quaeghebeur, Erik and Wheeler, Gregory}, volume = {103}, series = {Proceedings of Machine Learning Research}, month = {03--06 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v103/t-joens19a/t-joens19a.pdf}, url = {http://proceedings.mlr.press/v103/t-joens19a.html}, abstract = {To model discrete-time uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that satisfies a number of simple and intuitive axioms. We motivate these axioms on the basis of two possible interpretations for this upper expectation: a behavioural interpretation similar to that of Walley’s, and an interpretation in terms of upper envelopes of linear expectations. Subsequently, we show that the most conservative upper expectation satisfying our axioms coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications. On the one hand, it guarantees that there is a unique most conservative global belief model satisfying our axioms. On the other hand, it shows that Shafer and Vovk’s model can be given an axiomatic characterisation, thereby providing an alternative motivation for adopting this model, even outside their framework.} }
Endnote
%0 Conference Paper %T In Search of a Global Belief Model for Discrete-Time Uncertain Processes %A Natan T’Joens %A Jasper De Bock %A Gert de Cooman %B Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications %C Proceedings of Machine Learning Research %D 2019 %E Jasper De Bock %E Cassio P. de Campos %E Gert de Cooman %E Erik Quaeghebeur %E Gregory Wheeler %F pmlr-v103-t-joens19a %I PMLR %P 377--385 %U http://proceedings.mlr.press/v103/t-joens19a.html %V 103 %X To model discrete-time uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that satisfies a number of simple and intuitive axioms. We motivate these axioms on the basis of two possible interpretations for this upper expectation: a behavioural interpretation similar to that of Walley’s, and an interpretation in terms of upper envelopes of linear expectations. Subsequently, we show that the most conservative upper expectation satisfying our axioms coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications. On the one hand, it guarantees that there is a unique most conservative global belief model satisfying our axioms. On the other hand, it shows that Shafer and Vovk’s model can be given an axiomatic characterisation, thereby providing an alternative motivation for adopting this model, even outside their framework.
APA
T’Joens, N., De Bock, J. & de Cooman, G.. (2019). In Search of a Global Belief Model for Discrete-Time Uncertain Processes. Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, in Proceedings of Machine Learning Research 103:377-385 Available from http://proceedings.mlr.press/v103/t-joens19a.html.

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