Definition
Given a statistical manifold, with coordinates given by, one writes for the probability distribution. Here, is a specific value drawn from a collection of (discrete or continuous) random variables X. The probability is normalized, in that
The Fisher information metric then takes the form:
The integral is performed over all values x of all random variables X. Again, the variable is understood as a coordinate on the Riemann manifold. The labels j and k index the local coordinate axes on the manifold.
When the probability is derived from the Gibbs measure, as it would be for any Markovian process, then can also be understood to be a Lagrange multiplier; Lagrange multipliers are used to enforce constraints, such as holding the expectation value of some quantity constant. If there are n constraints holding n different expectation values constant, then the manifold is n-dimensional. In this case, the metric can be explicitly derived from the partition function; a derivation and discussion is presented there.
Substituting from information theory, an equivalent form of the above definition is:
Read more about this topic: Fisher Information Metric
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)