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The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized and explored by the statistician Sir Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
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 log-likelihood is also particularly useful for exponential families of distributions, which include many of the common parametric probability distributions. The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a ...
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 quantum Fisher information equals four times the variance for pure states [| ,] = (). For mixed states, when the probabilities are parameter independent, i.e., when () =, the quantum Fisher information is convex:
The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log- Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution ).
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.)