# 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.

The metric is interesting in several respects. First, it can be understood to be the infinitesimal form of the relative entropy or Kullback–Leibler divergence; specifically, it is the Hessian of the divergence. Alternately, it can be understood as the metric induced by the flat space Euclidean metric, after appropriate changes of variable. When extended to complex projective Hilbert space, it becomes the Fubini–Study metric; when written in terms of mixed states, it is the quantum Bures metric.

Considered purely as a matrix, it is known as the Fisher information matrix. Considered as a measurement technique, where it is used to estimate hidden parameters in terms of observed random variables, it is known as the observed information (this is wrong: it is the expected information; the observed information is not a function of the parameters).

### Other articles related to "information, metric, fisher information metric":

Partition Function (mathematics) - Information Geometry
... reasonable to ask about the structure of this manifold this is the task of information geometry ... This matrix is positive semi-definite, and may be interpreted as a metric tensor, specifically, a Riemannian metric ... Equiping the space of lagrange multipliers with a metric in this way turns it into a Riemannian manifold ...
Fisher Information Metric - Formal Definition
... The Fisher information metric is then an inner product on the tangent space ... This definition of the metric can be seen to be equivalent to the previous, in several steps ...

### Famous quotes containing the words fisher and/or information:

Chicago, Chicago, that toddlin’ town.
—Fred Fisher (1875–1942)

I believe it has been said that one copy of The Times contains more useful information than the whole of the historical works of Thucydides.
Richard Cobden (1804–1865)