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.
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 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.
- Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
- Efron, B.; Hinkley, D.V. (1978). "Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information". Biometrika 65 (3): 457–487. doi:10.1093/biomet/65.3.457. JSTOR 2335893. MR 0521817.