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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 distributions. It can be used to calculate the distance between probability distributions. [2] The metric is interesting in several aspects.
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 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 maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision.
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 Bayesian statistics, the Jeffreys prior is a non-informative prior distribution for a parameter space.Named after Sir Harold Jeffreys, [1] its density function is proportional to the square root of the determinant of the Fisher information matrix:
where is the Fisher information matrix of the model at point θ. Generally, the variance measures the degree of dispersion of a random variable around its mean. Thus estimators with small variances are more concentrated, they estimate the parameters more precisely.