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Entropy differential metric, distance and divergence measures in probability spaces: A unified approach

Author

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  • Burbea, Jacob
  • Rao, C. Radhakrishna

Abstract

The paper is devoted to metrization of probability spaces through the introduction of a quadratic differential metric in the parameter space of the probability distributions. For this purpose, a [phi]-entropy functional is defined on the probability space and its Hessian along a direction of the tangent space of the parameter space is taken as the metric. The distance between two probability distributions is computed as the geodesic distance induced by the metric. The paper also deals with three measures of divergence between probability distributions and their interrelationships.

Suggested Citation

  • Burbea, Jacob & Rao, C. Radhakrishna, 1982. "Entropy differential metric, distance and divergence measures in probability spaces: A unified approach," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 575-596, December.
    • Handle: RePEc:eee:jmvana:v:12:y:1982:i:4:p:575-596
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    Citations

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    Cited by:

    1. Remuzgo, Lorena & Trueba, Carmen & Sarabia, José María, 2016. "Evolution of the global inequality in greenhouse gases emissions using multidimensional generalized entropy measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 146-157.
    2. Berkane, Maia & Oden, Kevin & Bentler, Peter M., 1997. "Geodesic Estimation in Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 35-46, October.
    3. Anderson, William & Farazmand, Mohammad, 2024. "Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    4. A. Micchelli, Charles & Noakes, Lyle, 2005. "Rao distances," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 97-115, January.
    5. Ghosh, Indranil, 2023. "On the issue of convergence of certain divergence measures related to finding most nearly compatible probability distribution under the discrete set-up," Statistics & Probability Letters, Elsevier, vol. 203(C).
    6. García, Gloria & M. Oller, Josep, 2001. "Minimum Riemannian risk equivariant estimator for the univariate normal model," Statistics & Probability Letters, Elsevier, vol. 52(1), pages 109-113, March.
    7. Lovric, Miroslav & Min-Oo, Maung & Ruh, Ernst A., 2000. "Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 36-48, July.
    8. Sylvia Gottschalk, 2016. "Entropy and credit risk in highly correlated markets," Papers 1604.07042, arXiv.org.
    9. Abhik Ghosh & Ayanendranath Basu, 2017. "The minimum S-divergence estimator under continuous models: the Basu–Lindsay approach," Statistical Papers, Springer, vol. 58(2), pages 341-372, June.
    10. Mengütürk, Levent Ali, 2018. "Gaussian random bridges and a geometric model for information equilibrium," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 465-483.
    11. Shun‐ichi Amari, 2021. "Information Geometry," International Statistical Review, International Statistical Institute, vol. 89(2), pages 250-273, August.
    12. Asok K. Nanda & Shovan Chowdhury, 2021. "Shannon's Entropy and Its Generalisations Towards Statistical Inference in Last Seven Decades," International Statistical Review, International Statistical Institute, vol. 89(1), pages 167-185, April.
    13. Gottschalk, Sylvia, 2017. "Entropy measure of credit risk in highly correlated markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 478(C), pages 11-19.
    14. Guoping Zeng, 2013. "Metric Divergence Measures and Information Value in Credit Scoring," Journal of Mathematics, Hindawi, vol. 2013, pages 1-10, October.
    15. José M. Oller & Albert Satorra & Adolf Tobeña, 2019. "Unveiling pathways for the fissure among secessionists and unionists in Catalonia: identities, family language, and media influence," Palgrave Communications, Palgrave Macmillan, vol. 5(1), pages 1-13, December.
    16. Luigi Augugliaro & Angelo M. Mineo & Ernst C. Wit, 2013. "Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 471-498, June.

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