Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional
Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of e...
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1493 |
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| author | Tanita Botha Johannes Ferreira Andriette Bekker |
| author_facet | Tanita Botha Johannes Ferreira Andriette Bekker |
| author_sort | Tanita Botha |
| collection | DOAJ |
| description | Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios. |
| first_indexed | 2024-03-10T10:03:14Z |
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| language | English |
| last_indexed | 2024-03-10T10:03:14Z |
| publishDate | 2021-06-01 |
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| spelling | doaj.art-055f392196c04edf9d9dc386724f61782023-11-22T01:45:35ZengMDPI AGMathematics2227-73902021-06-01913149310.3390/math9131493Alternative Dirichlet Priors for Estimating Entropy via a Power Sum FunctionalTanita Botha0Johannes Ferreira1Andriette Bekker2Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South AfricaDepartment of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South AfricaDepartment of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South AfricaEntropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios.https://www.mdpi.com/2227-7390/9/13/1493generatormultinomialnoncentralPoissonpower sumTsallis |
| spellingShingle | Tanita Botha Johannes Ferreira Andriette Bekker Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional Mathematics generator multinomial noncentral Poisson power sum Tsallis |
| title | Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional |
| title_full | Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional |
| title_fullStr | Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional |
| title_full_unstemmed | Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional |
| title_short | Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional |
| title_sort | alternative dirichlet priors for estimating entropy via a power sum functional |
| topic | generator multinomial noncentral Poisson power sum Tsallis |
| url | https://www.mdpi.com/2227-7390/9/13/1493 |
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