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Thus, the Fisher information may be seen as the curvature of the support curve (the graph of the log-likelihood). Near the maximum likelihood estimate, low Fisher information therefore indicates that the maximum appears "blunt", that is, the maximum is shallow and there are many nearby values with a similar log-likelihood. Conversely, high ...
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 ...
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 .
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.
Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables. In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after R.A. Fisher and L.H.C. Tippett who recognised three different forms outlined ...
The infinitesimal form of relative entropy, specifically its Hessian, gives a metric tensor that equals the Fisher information metric; see § Fisher information metric. Fisher information metric on the certain probability distribution let determine the natural gradient for information-geometric optimization algorithms. [17]