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On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.
On the volume elements on a manifold
J. Moser · 1965
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Statistical decision rules and optimal inference
N. N. Čencov · 1982
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The analysis of linear partial differential operators. I
L. Hörmander · 1983
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Differential-geometrical methods in statistics
S.-I. Amari · 1985
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Die Fisher-Information und symplektische Strukturen
T. Friedrich · 1991
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Natural operations in differential geometry
I. Kolář, P. W. Michor, and J. Slovák · 1993
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The convenient setting of global analysis
A. Kriegl and P. W. Michor · 1997
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Topics in differential geometry
P. W. Michor · 2008
Cited alongside, same era.
Sobolev metrics on the manifold of all Riemannian metrics
M. Bauer, P. Harms, and P. W. Michor · 2013
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Geometry of diffeomorphism groups, complete integrability and geometric statistics
B. Khesin, J. Lenells, G. Misiołek, and S. C. Preston · 2013
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Information geometry and sufficient statistics
N. Ay, J. Jost, H. V. Le, and L. Schwachhöfer · 2014
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Generalized Hunter–Saxton equations, optimal information transport, and factorisation of diffeomorphisms
K. Modin · 2014
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