Observed information

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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

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


  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&ndash, 487. doi:10.1093/biomet/65.3.457. JSTOR 2335893. MR 0521817.
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