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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]].
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| ==Definition==
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| Suppose we observe [[random variable]]s <math>X_1,\ldots,X_n</math>, independent and identically distributed with density ''f''(''X''; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters <math>\theta</math> given the data <math>X_1,\ldots,X_n</math> is
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| :<math>\ell(\theta | X_1,\ldots,X_n) = \sum_{i=1}^n \log f(X_i| \theta) </math>. | |
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| We define the '''observed information matrix''' at <math>\theta^{*}</math> as
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| :<math>\mathcal{J}(\theta^*)
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| = - \left.
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| \nabla \nabla^{\top}
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| \ell(\theta)
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| \right|_{\theta=\theta^*}
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| </math>
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| ::<math>= -
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| \left.
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| \left( \begin{array}{cccc}
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| \tfrac{\partial^2}{\partial \theta_1^2}
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| & \tfrac{\partial^2}{\partial \theta_1 \partial \theta_2}
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| & \cdots
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| & \tfrac{\partial^2}{\partial \theta_1 \partial \theta_n} \\
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| \tfrac{\partial^2}{\partial \theta_2 \partial \theta_1}
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| & \tfrac{\partial^2}{\partial \theta_2^2}
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| & \cdots
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| & \tfrac{\partial^2}{\partial \theta_2 \partial \theta_n} \\
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| \vdots &
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| \vdots &
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| \ddots &
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| \vdots \\
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| \tfrac{\partial^2}{\partial \theta_n \partial \theta_1}
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| & \tfrac{\partial^2}{\partial \theta_n \partial \theta_2}
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| & \cdots
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| & \tfrac{\partial^2}{\partial \theta_n^2} \\
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| \end{array} \right)
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| \ell(\theta)
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| \right|_{\theta = \theta^*}
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| </math>
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| In many instances, the observed information is evaluated at the [[Maximum likelihood|maximum-likelihood estimate]].<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref>
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| ==Fisher information==
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| The [[Fisher information]] <math>\mathcal{I}(\theta)</math> is the [[expected value]] of the observed information given a single observation <math>X</math> distributed according to the hypothetical model with parameter <math>\theta</math>:
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| :<math>\mathcal{I}(\theta) = \mathrm{E}(\mathcal{J}(\theta))</math>.
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| ==Applications==
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| In a notable article, [[Bradley Efron]] and [[David V. Hinkley]] <ref>{{cite journal
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| |last1=Efron |first1=B. |authorlink1=Bradley Efron
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| |last2=Hinkley |first2=D.V. |authorlink2=David V. Hinkley
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| |year=1978
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| |title=Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information
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| |journal=[[Biometrika]]
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| |volume=65 |issue=3 |pages=457–487
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| |doi=10.1093/biomet/65.3.457 |mr=0521817 | jstor = 2335893
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| }}
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| </ref> argued that the observed information should be used in preference to the [[expected information]] when employing [[asymptotic normality|normal approximations]] for the distribution of [[Maximum likelihood|maximum-likelihood estimate]]s.
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| ==See also==
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| * [[Fisher information matrix]]
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| * [[Fisher information metric]]
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| ==References==
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| {{reflist}}
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| [[Category:Information theory]]
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| [[Category:Statistical terminology]]
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| [[Category:Estimation theory]]
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