On Information Geometrical Structures

Information geometry is a modern differential geometric approach to statistics, in particular theory of information. The main motivation for this expository survey article is the lack of compact material that mainly address to mathematical audience because of the interdisciplinary content. Information geometry simply described as applying the techniques of differential geometry to statistical models, represented as manifolds of probability distributions. This can be done either done by putting the concept of divergences on the center or the Fisher metric. This paper is motivated from the latter approach.

___

  • A. P. Dawid, Invited discussion of Defining the statistical probılem (with applications to second-order efficiency, Ann. Statist., 3, 1231-1234, 1975.
  • B. Efron, Defining the Curvature of a Statistical Problem (with applications to second order efficiency), Ann. Statist., 3, 1189ı1242, 1975.
  • H. Jeffreys, An invariant form for the prior probability in estima- tion problems, Proceedings of Royal Society of London, Series A, Mathematical and Physical Sciences, 186, 453-461, 1946.
  • J. Hofbauer, K. Sigmund, Evolutionaty games and population dynamics, Cambridge Univ Press, 1998.
  • J.Jost, Information geometry, Lecture Notes.
  • J. Jost, F. M. Şimsir, Affine harmonic maps, Analysis, 29, 185ı197, 2009.
  • J. Jost, F. M. Simsir, Nondivergence harmonic maps, Harmonic maps and differential geometry, Contemp. Math., 542, 231-238, 2011.
  • T. Kurose, Dual connections and affine geometry, Mathematische Zeitschrift, 203, pp. 115-121, 1990.
  • R. E. Kass, P.W. Vos, Geometrical Foundations of Asymptotic Inference, Wiley Series in Probability and Statistics, New York, 1997.
  • S.L. Lauritzen, Statistical manifolds, in: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics Lecture Notes, 10, pp. 163-218, Berkeley 1987.
  • L. Malago, G. Pistone, Information geometry of the Gaussian Disıtribution in view of stochastic optimization, in: FOGA’15 Proıceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII, 150-162, Wales 2015.
  • M. K. Murray, J. W. Rice, Differential Geometry and Statistics, Chapman Hall,1993.
  • M. Noguchi, Geometry of Statistical Manifolds, Differantial Ge- ometry and its Applications 2, pp. 197-222, 1992.
  • C. R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bulletin of the Calcutta Mathematical Society , 37, 81-91, 1945.
  • Y. Sato, K. Sugawa and M. Kawaguchi, The geometrical structure of parameter space of the two dimensional normal distribution, Division of Information Engineering, Hokkaido Univ., Sapporo, Japan, 1977.
  • C. E. Shannon, A mathematical theory of communication, Bell. Syst. Tech. J., 27, 379-423 and 623-656, 1948.
  • H. Shima, The Geometry of Hessian Structures, World Scientific, 2007.
  • H. Shima, Hessian manifolds of constant Hessian sectional curva- ture, J. Math. Soc. Japan, 47(4), 735-753, 1994.
  • L. T. Skovgaard, Riemannian geometry of the multivariate norımal model, Scandinavian Journal of Statistics, 11(4), pp. 211-223, 1984.