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. Definition Suppose we observe random variables , independent and identically distributed with density f(X; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters given the data is . We define the observed information matrix at as In many instances, the observed information is evaluated at the maximum-likelihood estimate. Alternative definition Andrew Gelman, David Dunson and Donald Rubin define observed information instead in terms of the parameters' posterior probability, : Fisher information The Fisher information is the expected value of the observed information given a single observation distributed according to the hypothetical model with parameter : . Applications In a notable article, Bradley Efron and David V. Hinkley argued that the observed information should be used in preference to the expected information when employing normal approximations for the distribution of maximum-likelihood estimates. See also Fisher information matrix Fisher information metric References Information theory Estimation theory