Errors Bounds for Finite Approximations of Coherent Lower Previsions

Damjan Škulj
Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 62:289-300, 2017.

Abstract

Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. The goal of this paper is to provide such a method. The proposed method allows calculating an upper bound for the error of a finite approximation of coherent lower prevision on a finite underlying sample space. An estimate of the maximal error is especially useful in the cases where calculating assessments is computationally demanding. Our method is based on convex analysis applied to credal sets, which in the case of finite sample spaces correspond to convex polyhedra.

Cite this Paper

BibTeX
@InProceedings{pmlr-v62-Škulj17a, title = {Errors Bounds for Finite Approximations of Coherent Lower Previsions}, author = {Škulj, Damjan}, booktitle = {Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {289--300}, 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/Škulj17a/Škulj17a.pdf}, url = {https://proceedings.mlr.press/v62/%C5%A0kulj17a.html}, abstract = {Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. The goal of this paper is to provide such a method. The proposed method allows calculating an upper bound for the error of a finite approximation of coherent lower prevision on a finite underlying sample space. An estimate of the maximal error is especially useful in the cases where calculating assessments is computationally demanding. Our method is based on convex analysis applied to credal sets, which in the case of finite sample spaces correspond to convex polyhedra.} }
Endnote
%0 Conference Paper %T Errors Bounds for Finite Approximations of Coherent Lower Previsions %A Damjan Škulj %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-Škulj17a %I PMLR %P 289--300 %U https://proceedings.mlr.press/v62/%C5%A0kulj17a.html %V 62 %X Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. The goal of this paper is to provide such a method. The proposed method allows calculating an upper bound for the error of a finite approximation of coherent lower prevision on a finite underlying sample space. An estimate of the maximal error is especially useful in the cases where calculating assessments is computationally demanding. Our method is based on convex analysis applied to credal sets, which in the case of finite sample spaces correspond to convex polyhedra.
APA
Škulj, D.. (2017). Errors Bounds for Finite Approximations of Coherent Lower Previsions. Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 62:289-300 Available from https://proceedings.mlr.press/v62/%C5%A0kulj17a.html.

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