SYMPLECTIC STRUCTURES ON STATISTICAL MANIFOLDS

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 respe...

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Bibliographic Details
Published inJournal of the Australian Mathematical Society (2001) Vol. 90; no. 3; pp. 371 - 384
Main Author NODA, TOMONORI
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2011
Subjects
Statistical models
moment map
symplectic manifold
statistical model
primary 53D20; secondary 60D05
statistical manifold
dually flat space
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Summary: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.
ISSN:1446-7887
1446-8107
DOI:10.1017/S1446788711001285