Description: Fisher Information is a statistical measure that quantifies the amount of information that an observable random variable provides about an unknown parameter. It is based on estimation theory and is used to assess the precision of estimators in statistical models. In more technical terms, it is defined as the inverse of the expected Fisher information matrix, implying that greater information leads to lower variance of the estimator. This measure is fundamental in information theory and has applications in various fields, including statistics, economics, and biology. Fisher Information is used to compare different estimators and to determine the efficiency of an estimator in relation to the Cramér-Rao lower bound, which establishes a lower limit for the variance of unbiased estimators. In summary, Fisher Information is crucial for understanding how data observations can influence parameter estimation and data-driven decision-making.
History: Fisher Information was introduced by British statistician Ronald A. Fisher in the 1920s. Fisher, regarded as one of the fathers of modern statistics, developed this measure as part of his work in estimation theory and experimental design. His contribution was fundamental in establishing the foundations of statistical inference, and Fisher Information became a central concept in his research. Over the years, the theory has evolved and been integrated into various fields of study, solidifying its role as an essential tool in statistics.
Uses: Fisher Information is used in various statistical applications, including parameter estimation, model comparison, and evaluating the efficiency of estimators. It is particularly useful in the context of regression models and analysis of variance, where it helps determine the amount of information that can be obtained from observed data. Additionally, it is applied in experimental design, where the goal is to maximize Fisher Information to achieve more accurate estimates.
Examples: A practical example of Fisher Information can be found in estimating the mean of a normal population. If a sample of data is available, Fisher Information can be calculated to assess how much information about the mean is obtained from that sample. Another example is in regression analysis, where it is used to compare different models and determine which provides the best estimation of parameters.
