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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 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 ...
By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. [2] [3] It can also be understood to be the infinitesimal form of the relative entropy (i.e., the Kullback–Leibler divergence); specifically, it is the Hessian of
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.
Another popular method is to replace the Hessian with the Fisher information matrix, () = [(^)], giving us the Fisher scoring algorithm. This procedure is standard in the estimation of many methods, such as generalized linear models .
A more general score test can be derived when there is more than one parameter. Suppose that ^ is the maximum likelihood estimate of under the null hypothesis while and are respectively, the score vector and the Fisher information matrix.
The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...