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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 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 ...
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 distributions. It can be used to calculate the distance between probability distributions. [2] The metric is interesting in several aspects.
The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions . As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids .
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.
In information theory, the principle of minimum Fisher information (MFI) is a variational principle which, when applied with the proper constraints needed to reproduce empirically known expectation values, determines the best probability distribution that characterizes the system. (See also Fisher information.)
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.
The log-likelihood function being plotted is used in the computation of the score (the gradient of the log-likelihood) and Fisher information (the curvature of the log-likelihood). Thus, the graph has a direct interpretation in the context of maximum likelihood estimation and likelihood-ratio tests .