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Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria

In many practical situations, we only know the interval containing the quantity of interest, we have no information about the probabilities of different values within this interval. In contrast to the cases when we know the distributions and can thus use Monte-Carlo simulations, processing such inte...

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Autores principales: Beer, Michael, Urenda, Julio, Kosheleva, Olga, Kreinovich, Vladik
Formato: Online Artículo Texto
Lenguaje:English
Publicado: 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7274317/
http://dx.doi.org/10.1007/978-3-030-50146-4_6
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author Beer, Michael
Urenda, Julio
Kosheleva, Olga
Kreinovich, Vladik
author_facet Beer, Michael
Urenda, Julio
Kosheleva, Olga
Kreinovich, Vladik
author_sort Beer, Michael
collection PubMed
description In many practical situations, we only know the interval containing the quantity of interest, we have no information about the probabilities of different values within this interval. In contrast to the cases when we know the distributions and can thus use Monte-Carlo simulations, processing such interval uncertainty is difficult – crudely speaking, because we need to try all possible distributions on this interval. Sometimes, the problem can be simplified: namely, for estimating the range of values of some characteristics of the distribution, it is possible to select a single distribution (or a small family of distributions) whose analysis provides a good understanding of the situation. The most known case is when we are estimating the largest possible value of Shannon’s entropy: in this case, it is sufficient to consider the uniform distribution on the interval. Interesting, estimating other characteristics leads to the selection of the same uniform distribution: e.g., estimating the largest possible values of generalized entropy or of some sensitivity-related characteristics. In this paper, we provide a general explanation of why uniform distribution appears in different situations – namely, it appears every time we have a permutation-invariant optimization problem with the unique optimum. We also discuss what happens if we have an optimization problem that attains its optimum at several different distributions – this happens, e.g., when we are estimating the smallest possible value of Shannon’s entropy (or of its generalizations).
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spelling pubmed-72743172020-06-05 Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria Beer, Michael Urenda, Julio Kosheleva, Olga Kreinovich, Vladik Information Processing and Management of Uncertainty in Knowledge-Based Systems Article In many practical situations, we only know the interval containing the quantity of interest, we have no information about the probabilities of different values within this interval. In contrast to the cases when we know the distributions and can thus use Monte-Carlo simulations, processing such interval uncertainty is difficult – crudely speaking, because we need to try all possible distributions on this interval. Sometimes, the problem can be simplified: namely, for estimating the range of values of some characteristics of the distribution, it is possible to select a single distribution (or a small family of distributions) whose analysis provides a good understanding of the situation. The most known case is when we are estimating the largest possible value of Shannon’s entropy: in this case, it is sufficient to consider the uniform distribution on the interval. Interesting, estimating other characteristics leads to the selection of the same uniform distribution: e.g., estimating the largest possible values of generalized entropy or of some sensitivity-related characteristics. In this paper, we provide a general explanation of why uniform distribution appears in different situations – namely, it appears every time we have a permutation-invariant optimization problem with the unique optimum. We also discuss what happens if we have an optimization problem that attains its optimum at several different distributions – this happens, e.g., when we are estimating the smallest possible value of Shannon’s entropy (or of its generalizations). 2020-05-18 /pmc/articles/PMC7274317/ http://dx.doi.org/10.1007/978-3-030-50146-4_6 Text en © Springer Nature Switzerland AG 2020 This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.
spellingShingle Article
Beer, Michael
Urenda, Julio
Kosheleva, Olga
Kreinovich, Vladik
Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title_full Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title_fullStr Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title_full_unstemmed Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title_short Which Distributions (or Families of Distributions) Best Represent Interval Uncertainty: Case of Permutation-Invariant Criteria
title_sort which distributions (or families of distributions) best represent interval uncertainty: case of permutation-invariant criteria
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7274317/
http://dx.doi.org/10.1007/978-3-030-50146-4_6
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