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Historically, information geometry can be traced back to the work of
C. R. Rao, who was the first to treat the
Fisher matrix as a
Riemannian metric.[2][3] The modern theory is largely due to
Shun'ichi Amari, whose work has been greatly influential on the development of the field.[4]
Classically, information geometry considered a parametrized
statistical model as a
Riemannian manifold. For such models, there is a natural choice of Riemannian metric, known as the
Fisher information metric. In the special case that the statistical model is an
exponential family, it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function). In this case, the manifold naturally inherits two flat
affine connections, as well as a canonical
Bregman divergence. Historically, much of the work was devoted to studying the associated geometry of these examples. In the modern setting, information geometry applies to a much wider context, including non-exponential families,
nonparametric statistics, and even abstract statistical manifolds not induced from a known statistical model. The results combine techniques from
information theory,
affine differential geometry,
convex analysis and many other fields.
The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book, Methods of Information Geometry,[5] and the more recent book by Nihat Ay and others.[6] A gentle introduction is given in the survey by Frank Nielsen.[7] In 2018, the journal Information Geometry was released, which is devoted to the field.
Contributors
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The history of information geometry is associated with the discoveries of at least the following people, and many others.
This section may need to be rewritten to comply with Wikipedia's
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As an interdisciplinary field, information geometry has been used in various applications.
^Rao, C. R. (1945). "Information and Accuracy Attainable in the Estimation of Statistical Parameters". Bulletin of the Calcutta Mathematical Society. 37: 81–91. Reprinted in Breakthroughs in Statistics. Springer. 1992. pp. 235–247.
doi:
10.1007/978-1-4612-0919-5_16.
S2CID117034671.
^Nielsen, F. (2013). "Cramér-Rao Lower Bound and Information Geometry". In Bhatia, R.; Rajan, C. S. (eds.). Connected at Infinity II: On the Work of Indian Mathematicians. Texts and Readings in Mathematics. Vol. Special Volume of Texts and Readings in Mathematics (TRIM). Hindustan Book Agency. pp. 18–37.
arXiv:1301.3578.
doi:
10.1007/978-93-86279-56-9_2.
ISBN978-93-80250-51-9.
S2CID16759683.
^Amari, Shun'ichi; Nagaoka, Hiroshi (2000). Methods of Information Geometry. Translations of Mathematical Monographs. Vol. 191. American Mathematical Society.
ISBN0-8218-0531-2.
^Ay, Nihat;
Jost, Jürgen; Lê, Hông Vân; Schwachhöfer, Lorenz (2017). Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 64. Springer.
ISBN978-3-319-56477-7.
^Amari, Shun'ichi (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Berlin: Springer-Verlag.
ISBN0-387-96056-2.
^Murray, M.; Rice, J. (1993). Differential Geometry and Statistics. Monographs on Statistics and Applied Probability. Vol. 48.
Chapman and Hall.
ISBN0-412-39860-5.
^Marriott, Paul; Salmon, Mark, eds. (2000). Applications of Differential Geometry to Econometrics. Cambridge University Press.
ISBN0-521-65116-6.