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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 ...
The Fisher information metric is particularly simple for the exponential family, which has () = [ () + ] The metric is () = () [()] The metric has a particularly simple form if we are using the natural parameters.
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 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.
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 probability theory and statistics, the generalized extreme value (GEV) distribution [2] is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions.
Using tools from information geometry, the Jeffreys prior can be generalized in pursuit of obtaining priors that encode geometric information of the statistical model, so as to be invariant under a change of the coordinate of parameters. [9] A special case, the so-called Weyl prior, is defined as a volume form on a Weyl manifold. [10]
For such models, there is a natural choice of Riemannian metric, known as the Fisher information metric. In the special case that the statistical model is an exponential family , it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function).