Fisher_information

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:

$g_{jk} = \int \frac{\partial \log p(x,\theta)}{\partial \theta_j} \frac{\partial \log p(x,\theta)}{\partial \theta_k} p(x,\theta) dx.$

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

$g_{jk} = \int \frac{\partial i(x,\theta)}{\partial \theta_j} \frac{\partial i(x,\theta)}{\partial \theta_k} p(x,\theta) dx.$

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:

$g_{jk} = \int \frac{\partial^2 i(x,\theta)}{\partial \theta_j \partial \theta_k} p(x,\theta) dx = \mathrm{E} \left[ \frac{\partial^2 i(x,\theta)}{\partial \theta_j \partial \theta_k} \right].$

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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References