Betting Schemes for Assessing Coherent Numerical and Comparative Conditional Possibilities

Davide Petturiti, Barbara Vantaggi
Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, PMLR 147:254-263, 2021.

Abstract

We introduce coherence conditions having a betting scheme interpretation both for a numerical and a comparative conditional possibility assessment. The conditional bets are considered under partially resolving uncertainty and assuming consonance. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. Further, he/she can only consider families of nested non-impossible events in computing the gain and has a systematically optimistic behavior. Both conditions are proved to be equivalent to the existence of a conditional possibility agreeing with an axiomatic definition based on the algebraic product t-norm, that extends, either numerically or comparatively (through the induced comparative conditional possibility relation), the given assessment.

Cite this Paper

BibTeX
@InProceedings{pmlr-v147-petturiti21a, title = {Betting Schemes for Assessing Coherent Numerical and Comparative Conditional Possibilities}, author = {Petturiti, Davide and Vantaggi, Barbara}, booktitle = {Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications}, pages = {254--263}, 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/petturiti21a/petturiti21a.pdf}, url = {https://proceedings.mlr.press/v147/petturiti21a.html}, abstract = {We introduce coherence conditions having a betting scheme interpretation both for a numerical and a comparative conditional possibility assessment. The conditional bets are considered under partially resolving uncertainty and assuming consonance. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. Further, he/she can only consider families of nested non-impossible events in computing the gain and has a systematically optimistic behavior. Both conditions are proved to be equivalent to the existence of a conditional possibility agreeing with an axiomatic definition based on the algebraic product t-norm, that extends, either numerically or comparatively (through the induced comparative conditional possibility relation), the given assessment.} }
Endnote
%0 Conference Paper %T Betting Schemes for Assessing Coherent Numerical and Comparative Conditional Possibilities %A Davide Petturiti %A Barbara Vantaggi %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-petturiti21a %I PMLR %P 254--263 %U https://proceedings.mlr.press/v147/petturiti21a.html %V 147 %X We introduce coherence conditions having a betting scheme interpretation both for a numerical and a comparative conditional possibility assessment. The conditional bets are considered under partially resolving uncertainty and assuming consonance. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. Further, he/she can only consider families of nested non-impossible events in computing the gain and has a systematically optimistic behavior. Both conditions are proved to be equivalent to the existence of a conditional possibility agreeing with an axiomatic definition based on the algebraic product t-norm, that extends, either numerically or comparatively (through the induced comparative conditional possibility relation), the given assessment.
APA
Petturiti, D. & Vantaggi, B.. (2021). Betting Schemes for Assessing Coherent Numerical and Comparative Conditional Possibilities. Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 147:254-263 Available from https://proceedings.mlr.press/v147/petturiti21a.html.

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