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).
Read more about Fisher Information Metric: Definition, Relation To The Kullback–Leibler Divergence, Relation To Ruppeiner Geometry, Change in Entropy, Relation To The Jensen–Shannon Divergence, As Euclidean Metric, As Fubini–Study Metric, Formal Definition
Famous quotes containing the words fisher and/or information:
“For anyone addicted to reading commonplace books ... finding a good new one is much like enduring a familiar recurrence of malaria, with fever, fits of shaking, strange dreams. Unlike a truly paludismic ordeal, however, the symptoms felt while savoring a collection of one man’s pet quotations are voluptuously enjoyable ...”
—M.F.K. Fisher (1908–1992)
“Computers are good at swift, accurate computation and at storing great masses of information. The brain, on the other hand, is not as efficient a number cruncher and its memory is often highly fallible; a basic inexactness is built into its design. The brain’s strong point is its flexibility. It is unsurpassed at making shrewd guesses and at grasping the total meaning of information presented to it.”
—Jeremy Campbell (b. 1931)