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The Fisher information matrix plays a role in an inequality like the isoperimetric inequality. [29] Of all probability distributions with a given entropy, the one whose Fisher information matrix has the smallest trace is the Gaussian distribution. This is like how, of all bounded sets with a given volume, the sphere has the smallest surface area.
In information geometry, the Fisher information metric [1] 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. [clarification needed]
In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log-likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information.
In econometrics, the information matrix test is used to determine whether a regression model is misspecified.The test was developed by Halbert White, [1] who observed that in a correctly specified model and under standard regularity assumptions, the Fisher information matrix can be expressed in either of two ways: as the outer product of the gradient, or as a function of the Hessian matrix of ...
The formula also holds without taking the real part ... Quantum Fisher information matrix is a part of a wider family of quantum statistical distances. [12]
The family of all normal distributions can be thought of as a 2-dimensional parametric space parametrized by the expected value μ and the variance σ 2 ≥ 0. Equipped with the Riemannian metric given by the Fisher information matrix, it is a statistical manifold with a geometry modeled on hyperbolic space.
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It states that the precision of any unbiased estimator is at most the Fisher information; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance. An unbiased estimator that achieves this bound is said to be (fully) efficient.