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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An extended Čencov characterization of the information metric
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by L. L. Campbell PDF
Proc. Amer. Math. Soc. 98 (1986), 135-141 Request permission


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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 98 (1986), 135-141
  • MSC: Primary 62B10; Secondary 53B99
  • DOI:
  • MathSciNet review: 848890