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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 . 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.

**[1]**J. Aczél and Z. Daróczy,*On measures of information and their characterizations*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Mathematics in Science and Engineering, Vol. 115. MR**0689178****[2]**Shun-ichi Amari,*Differential geometry of curved exponential families—curvatures and information loss*, Ann. Statist.**10**(1982), no. 2, 357–385. MR**653513****[3]**L. L. Campbell,*The relation between information theory and the differential geometry approach to statistics*, Inform. Sci.**35**(1985), no. 3, 199–210. MR**811107**, 10.1016/0020-0255(85)90050-7**[4]**N. N. Čencov,*Statistical decision rules and optimal inference*, Transl. Math. Monographs, vol. 53, Amer. Math. Soc., Providence, R. I., 1981.**[5]**David C. Hamilton, Donald G. Watts, and Douglas M. Bates,*Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions*, Ann. Statist.**10**(1982), no. 2, 386–393. MR**653514****[6]**Noel J. Hicks,*Notes on differential geometry*, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR**0179691****[7]**Roman S. Ingarden, Yoshiharu Sato, Kazuaki Sugawa, and Michiaki Kawaguchi,*Information thermodynamics and differential geometry*, Tensor (N.S.)**33**(1979), no. 3, 347–353. MR**577590****[8]**A. I. Khinchin,*Mathematical foundations of information theory*, Dover Publications, Inc., New York, N. Y., 1957. Translated by R. A. Silverman and M. D. Friedman. MR**0092709****[9]**S. Shahshahani,*A new mathematical framework for the study of linkage and selection*, Mem. Amer. Math. Soc.**17**(1979), no. 211, ix+34. MR**512476**, 10.1090/memo/0211

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
62B10,
53B99

Retrieve articles in all journals with MSC: 62B10, 53B99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0848890-5

Keywords:
Information metric,
Markov mapping

Article copyright:
© Copyright 1986
American Mathematical Society