Fisherinformationen

Fisherinformationen, also known as Fisher information, is a concept in statistics and information theory that quantifies the amount of information that an observable random variable carries about an unknown parameter upon which the probability of the variable depends. It is named after the statistician Ronald A. Fisher.

Fisher information is a measure of the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. It is defined as the variance of the score function, which is the derivative of the log-likelihood function with respect to the parameter. The Fisher information matrix is a generalization of this concept for multivariate parameters.

Fisher information is widely used in statistical inference, particularly in the context of maximum likelihood estimation. It provides a lower bound on the variance of any unbiased estimator of the parameter, known as the Cramér-Rao bound. This bound is useful for assessing the efficiency of an estimator.

In Bayesian statistics, Fisher information is used in the context of the Fisher information matrix to construct the Fisher information prior, which is a prior distribution that is proportional to the square root of the determinant of the Fisher information matrix. This prior is often used in empirical Bayes methods.

Fisher information is also used in the context of information theory, where it is used to quantify the amount of information that a random variable carries about a parameter. It is used in the context of the Fisher information inequality, which provides a lower bound on the mutual information between a random variable and a parameter.

In summary, Fisher information is a fundamental concept in statistics and information theory that quantifies the amount of information that an observable random variable carries about an unknown parameter. It is widely used in statistical inference, Bayesian statistics, and information theory.