IDEAS home Printed from https://ideas.repec.org/a/bla/istatr/v89y2021i2p250-273.html
   My bibliography  Save this article

Information Geometry

Author

Listed:
  • Shun‐ichi Amari

Abstract

Statistical inference is constructed upon a statistical model consisting of a parameterised family of probability distributions, which forms a manifold. It is important to study the geometry of the manifold. It was Professor C. R. Rao who initiated information geometry in his monumental paper published in 1945. It not only included fundamentals of statistical inference such as the Cramér–Rao theorem and Rao–Blackwell theorem but also proposed differential geometry of a manifold of probability distributions. It is a Riemannian manifold where Fisher–Rao information plays the role of the metric tensor. It took decades for the importance of the geometrical structure to be recognised. The present article reviews the structure of the manifold of probability distributions and its applications and shows how the original idea of Professor Rao has been developed and popularised in the wide sense of statistical sciences including AI, signal processing, physical sciences and others.

Suggested Citation

  • Shun‐ichi Amari, 2021. "Information Geometry," International Statistical Review, International Statistical Institute, vol. 89(2), pages 250-273, August.
    • Handle: RePEc:bla:istatr:v:89:y:2021:i:2:p:250-273
    DOI: 10.1111/insr.12464
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/insr.12464
    Download Restriction: no
    File URL: https://libkey.io/10.1111/insr.12464?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Rao C. R. & Sinha Β. K. & Subramanyam K., 1982. "Third Order Efficiency Of The Maximum Likelihood Estimator In The Multinomial Distribution," Statistics & Risk Modeling, De Gruyter, vol. 1(1), pages 1-16, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Mattsson, Lars-Göran & Weibull, Jörgen W., 2023. "An analytically solvable principal-agent model," Games and Economic Behavior, Elsevier, vol. 140(C), pages 33-49.
    2. Li, W. & Rubio, F.J., 2022. "On a prior based on the Wasserstein information matrix," Statistics & Probability Letters, Elsevier, vol. 190(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Sylvia Gottschalk, 2016. "Entropy and credit risk in highly correlated markets," Papers 1604.07042, arXiv.org.
    3. 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.
    4. A. Micchelli, Charles & Noakes, Lyle, 2005. "Rao distances," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 97-115, January.
    5. Kano, Yutaka, 1998. "More Higher-Order Efficiency: Concentration Probability," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 349-366, November.
    6. 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.
    7. 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.
    8. Guoping Zeng, 2013. "Metric Divergence Measures and Information Value in Credit Scoring," Journal of Mathematics, Hindawi, vol. 2013, pages 1-10, October.
    9. 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.
    10. 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).
    11. 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.
    12. 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.
    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. 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.
    15. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:istatr:v:89:y:2021:i:2:p:250-273. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/isiiinl.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.