New Distributions for Modeling Subjective Lower and Upper Probabilities

Michael Smithson
Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 62:301-312, 2017.

Abstract

This paper presents an investigation of approaches to modeling lower and upper subjective probabilities. A relatively unexplored approach is introduced, based on the fact that every cumulative distribution function (CDF) with support (0,1) has a “dual” CDF that obeys the conjugacy relation between coherent lower and upper probabilities. A new 2-parameter family of “CDF-Quantile” distributions with support (0,1) is extended via a third parameter for the purpose of modeling lower-upper probabilities. The extension exploits certain properties of the CDF-Quantile family, and the fact that continuous CDFs on (0,1) random variables form an algebraic group that is closed under composition. This extension also yields models for testing specific models of lower-upper probability assignments. Finally, the new models are applied to a real data-set, and compared with the alternative approaches for their relative advantages and drawbacks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v62-smithson17a, title = {New Distributions for Modeling Subjective Lower and Upper Probabilities}, author = {Smithson, Michael}, booktitle = {Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {301--312}, 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/smithson17a/smithson17a.pdf}, url = {https://proceedings.mlr.press/v62/smithson17a.html}, abstract = {This paper presents an investigation of approaches to modeling lower and upper subjective probabilities. A relatively unexplored approach is introduced, based on the fact that every cumulative distribution function (CDF) with support (0,1) has a “dual” CDF that obeys the conjugacy relation between coherent lower and upper probabilities. A new 2-parameter family of “CDF-Quantile” distributions with support (0,1) is extended via a third parameter for the purpose of modeling lower-upper probabilities. The extension exploits certain properties of the CDF-Quantile family, and the fact that continuous CDFs on (0,1) random variables form an algebraic group that is closed under composition. This extension also yields models for testing specific models of lower-upper probability assignments. Finally, the new models are applied to a real data-set, and compared with the alternative approaches for their relative advantages and drawbacks.} }
Endnote
%0 Conference Paper %T New Distributions for Modeling Subjective Lower and Upper Probabilities %A Michael Smithson %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-smithson17a %I PMLR %P 301--312 %U https://proceedings.mlr.press/v62/smithson17a.html %V 62 %X This paper presents an investigation of approaches to modeling lower and upper subjective probabilities. A relatively unexplored approach is introduced, based on the fact that every cumulative distribution function (CDF) with support (0,1) has a “dual” CDF that obeys the conjugacy relation between coherent lower and upper probabilities. A new 2-parameter family of “CDF-Quantile” distributions with support (0,1) is extended via a third parameter for the purpose of modeling lower-upper probabilities. The extension exploits certain properties of the CDF-Quantile family, and the fact that continuous CDFs on (0,1) random variables form an algebraic group that is closed under composition. This extension also yields models for testing specific models of lower-upper probability assignments. Finally, the new models are applied to a real data-set, and compared with the alternative approaches for their relative advantages and drawbacks.
APA
Smithson, M.. (2017). New Distributions for Modeling Subjective Lower and Upper Probabilities. Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 62:301-312 Available from https://proceedings.mlr.press/v62/smithson17a.html.

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