# Fisher information metric

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.

It can be used to calculate the informational difference between measurements.It takes the form:

:$g_\left\{jk\right\}=int frac\left\{partial log p\left(x, heta\right)\right\}\left\{partial heta_j\right\} frac\left\{partial log p\left(x, heta\right)\right\}\left\{partial heta_k\right\} p\left(x, heta\right)dx.$

Substituting $i = -ln\left(p\right)$ from information theory, the formula becomes:

:$g_\left\{jk\right\}=int frac\left\{partial i\left(x, heta\right)\right\}\left\{partial heta_j\right\} frac\left\{partial i\left(x, heta\right)\right\}\left\{partial heta_k\right\} p\left(x, heta\right)dx.$

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:

:$g_\left\{jk\right\}=int frac\left\{partial^2 i\left(x, heta\right)\right\}\left\{partial heta_j partial heta_k\right\} p\left(x, heta\right)dx=mathrm\left\{E\right\}left \left[ frac\left\{partial^2 i\left(x, heta\right)\right\}\left\{partial heta_j partial heta_k\right\} ight\right] .$

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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