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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

An extended Čencov characterization of the information metric


Author: L. L. Campbell
Journal: Proc. Amer. Math. Soc. 98 (1986), 135-141
MSC: Primary 62B10; Secondary 53B99
MathSciNet review: 848890
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Abstract: Čencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In Čencov's theorem, the underlying differentiable manifold is the probability simplex $ \Sigma _1^n{x_i} = 1, x_i > 0$. For some purposes of using geometry to obtain insights about probability, it is more convenient to regard the simplex as a hypersurface in the positive cone. In the present paper Čencov's result is extended to the positive cone. The proof uses standard techniques of differential geometry but does not use the language of category theory.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0848890-5
PII: S 0002-9939(1986)0848890-5
Keywords: Information metric, Markov mapping
Article copyright: © Copyright 1986 American Mathematical Society