Observed information

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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 many instances, the observed information is evaluated at the maximum-likelihood estimate.[1]

Fisher information[edit]

The Fisher information is the expected value of the observed information given a single observation distributed according to the hypothetical model with parameter :



In a notable article, Bradley Efron and David V. Hinkley [2] 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[edit]


  1. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
  2. ^ 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.