Chemistry - Fisher information metric

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:

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

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

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

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:

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

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.

Categories: Differential geometry

Fisher information metric

See also

References