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Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional

Author

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  • Tanita Botha

    (Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South Africa
    These authors contributed equally to this work.)

  • Johannes Ferreira

    (Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South Africa
    Centre of Excellence in Mathematical and Statistical Science, University of Witwatersrand, Johannesburg 2050, South Africa
    These authors contributed equally to this work.)

  • Andriette Bekker

    (Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0028 , South Africa
    Centre of Excellence in Mathematical and Statistical Science, University of Witwatersrand, Johannesburg 2050, South Africa
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Tanita Botha & Johannes Ferreira & Andriette Bekker, 2021. "Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional," Mathematics, MDPI, vol. 9(13), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1493-:d:582286
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    References listed on IDEAS

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    1. Mathai, A.M. & Haubold, H.J., 2007. "On generalized entropy measures and pathways," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(2), pages 493-500.
    2. Andrea Ongaro & Carlo Orsi, 2015. "Some results on non-central beta distributions," Statistica, Department of Statistics, University of Bologna, vol. 75(1), pages 85-100.
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