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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])
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Abstract

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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

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