In
information geometry
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to p ...
, Chentsov's theorem states that the
Fisher information metric In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, ''i.e.'', a smooth manifold whose points are probability measures defined on a common probability space. ...
is, up to rescaling, the unique
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
on a
statistical manifold that is invariant under
sufficient statistics
In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the pa ...
.
See also
*
Fisher information
In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that model ...
*
Sufficient statistic
In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the pa ...
*
Information geometry
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to p ...
References
* N. N. Čencov (1981), ''Statistical Decision Rules and Optimal Inference'', Translations of mathematical monographs; v. 53, American Mathematical Society, http://www.ams.org/books/mmono/053/
* Shun'ichi Amari, Hiroshi Nagaoka (2000) ''Methods of information geometry'', Translations of mathematical monographs; v. 191, American Mathematical Society, http://www.ams.org/books/mmono/191/ (Theorem 2.6)
*
{{DEFAULTSORT:Chentsov's theorem
Differential geometry
Information geometry
Statistical distance