A measure of the amount of information an observable random variable carries about an unknown parameter, used to quantify sensitivity.Fisher information is a concept from statistics and information theory that quantifies how much information an observable random variable provides about an unknown parameter. It measures the sensitivity of the probability distribution to changes in that parameter. If a small change in the parameter causes a large shift in the likelihood of observed data, the Fisher information is high; if the data barely changes, it is low.
Introduced by Ronald Fisher in the 1920s as part of his foundational work on maximum likelihood estimation.
Formally, it is the variance of the score, which is the derivative of the log-likelihood with respect to the parameter. For a given probability distribution, the Fisher information is a single number (or a matrix for multiple parameters) that sets a lower bound on the variance of any unbiased estimator, known as the Cramér–Rao bound. This makes it a fundamental tool for assessing the precision of statistical estimates.
It is used in experimental design to optimize measurements, in quantum metrology to achieve precision beyond classical limits, and in machine learning for parameter estimation.
In quantum mechanics, the Fisher information is related to the speed of evolution of a quantum state, linking statistics to the Heisenberg uncertainty principle.
