In mathematics, in information geometry, the Fisher information metric is a metric tensor for a statistical differential manifold.
It can be used to calculate the informational difference between measurements.
It takes the form:
Substituting i = - ln(p) from information theory, the formula becomes:
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_{ij} = -\int \frac{\partial^2 i(x,\theta)}{\partial \theta_i \partial \theta_j} p(x,\theta) dx = -\mathrm{E} \left[ \frac{\partial^2 i(x,\theta)}{\partial \theta_i \partial \theta_j} \right].](a/../upload/math/8836de5cdd2f35d9a0502b7174692253.png)
See also
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, Transactions of mathematical monographs; v. 191, American Mathematical Society, 2000.