- Fisher information metric
In

information geometry , the**Fisher information metric**is a particularRiemannian metric which can be defined on a smooth statistical manifold, i.e., asmooth manifold whose points areprobability measures defined on a commonprobability space .It can be used to calculate the informational difference between measurements.It takes the form:

:$g\_\{jk\}=int\; frac\{partial\; log\; p(x,\; heta)\}\{partial\; heta\_j\}\; frac\{partial\; log\; p(x,\; heta)\}\{partial\; heta\_k\}\; p(x,\; heta)dx.$

Substituting $i\; =\; -ln(p)$ from

information theory , the formula becomes::$g\_\{jk\}=int\; frac\{partial\; i(x,\; heta)\}\{partial\; heta\_j\}\; frac\{partial\; i(x,\; heta)\}\{partial\; heta\_k\}\; p(x,\; heta)dx.$

Which can be thought of intuitively as: "The distance between two points on a statistical differential manifold is the amount of information between them, i.e. the informational difference between them."

An equivalent form of the above equation is:

:$g\_\{jk\}=int\; frac\{partial^2\; i(x,\; heta)\}\{partial\; heta\_j\; partial\; heta\_k\}\; p(x,\; heta)dx=mathrm\{E\}left\; [\; frac\{partial^2\; i(x,\; heta)\}\{partial\; heta\_j\; partial\; heta\_k\}\; ight]\; .$

**ee also***

Cramér-Rao bound

*Fisher information

*Bures metric **References*** Shun'ichi Amari - "Differential-geometrical methods in statistics", Lecture notes in statistics, Springer-Verlag, Berlin, 1985.

* Shun'ichi Amari, Hiroshi Nagaoka - "Methods of information geometry", Translations of mathematical monographs; v. 191, American Mathematical Society, 2000.

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