Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T08:43:00.051Z Has data issue: false hasContentIssue false
  • English
  • Français

SYMPLECTIC STRUCTURES ON STATISTICAL MANIFOLDS

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  15 June 2011

TOMONORI NODA
TOMONORI NODA*
Affiliation:
Department of Mathematics, Osaka Dental University, 8-1, Kuzuhahanazono-cho, Hirakata-shi, Osaka 573-1121, Japan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini–Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases.

Keywords

symplectic manifoldmoment mapstatistical manifoldstatistical modeldually flat space

MSC classification

Secondary: 53D20: Momentum maps; symplectic reduction
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 371 - 384
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Amari, S. and Nagaoka, H., Methods of Information Geometry, Translations of Mathematical Monographs, 191 (American Mathematical Society, Providence, RI, and Oxford University Press, Oxford, 2000), translated from the 1993 Japanese original by Daishi Harada.Google Scholar
[2]Barndorff-Nielsen, O. E. and Jupp, P. E., ‘Statistics, yokes and symplectic geometry’, Ann. Fac. Sci. Toulouse Math. 6 (1997), 389–427.CrossRefGoogle Scholar
[3]Cena, A. and Pistone, G., ‘Exponential statistical manifold’, Ann. Inst. Statist. Math. 59 (2007), 27–56.CrossRefGoogle Scholar
[4]Eguchi, S., ‘Geometry of minimum contrast’, Hiroshima Math. J. 22 (1992), 631–647.CrossRefGoogle Scholar
[5]Friedrich, T., ‘Die Fisher-Information und symplektische Strukturen’, Math. Nachr. 153 (1991), 273–296.CrossRefGoogle Scholar
[6]Kirillov, A. A. and Yur’ev, D. V., ‘Kähler geometry of the infinite-dimensional homogeneous space M=Diff+(S 1)/Rot(S 1)’, Funktsional. Anal. i Prilozhen. 20 (1986), 79–80 (in Russian).Google Scholar
[7]Lauritzen, S., ‘Statistical manifolds’, in: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 10 (Inst. Math. Statist., Hayward, CA, 1987), pp. 163–216.CrossRefGoogle Scholar
[8]Matsuzoe, H., ‘Geometry of contrast functions and conformal geometry’, Hiroshima Math. J. 29 (1999), 175–191.CrossRefGoogle Scholar
[9]Nakamura, Y., ‘Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions’, Japan J. Indust. Appl. Math. 10 (1993), 179–189.CrossRefGoogle Scholar
[10]Nakamura, Y., ‘Gradient system associated with probability distributions’, Japan J. Indust. Appl. Math. 11 (1994), 21–30.CrossRefGoogle Scholar
[11]Shishido, Y., ‘Strong symplectic structures on spaces of probability measures with positive density function’, Proc. Japan Acad. 81 (2005), 134–136.Google Scholar
[12]Vân, L. H., ‘Monotone invariants and embeddings of statistical manifolds’, in: Advances in Deterministic and Stochastic Analysis (World Scientific, Hackensack, NJ, 2007), pp. 231–254.CrossRefGoogle Scholar
[13]Weinstein, A., ‘Symplectic manifolds and their Lagrangian submanifolds’, Adv. Math. 6 (1971), 329–346.CrossRefGoogle Scholar