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2013-09-28T21:35:55Z

64.134.175.70: update Dow Divisor

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 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

:<math>\ell(\theta | X_1,\ldots,X_n) = \sum_{i=1}^n \log f(X_i| \theta) </math>.

We define the '''observed information matrix''' at <math>\theta^{*}</math> as

:<math>\mathcal{J}(\theta^*)
= - \left.
\nabla \nabla^{\top}
\ell(\theta)
\right|_{\theta=\theta^*}
</math>

::<math>= -
\left.
\left( \begin{array}{cccc}
\tfrac{\partial^2}{\partial \theta_1^2}
& \tfrac{\partial^2}{\partial \theta_1 \partial \theta_2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_1 \partial \theta_n} \\
\tfrac{\partial^2}{\partial \theta_2 \partial \theta_1}
& \tfrac{\partial^2}{\partial \theta_2^2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_2 \partial \theta_n} \\
\vdots &
\vdots &
\ddots &
\vdots \\
\tfrac{\partial^2}{\partial \theta_n \partial \theta_1}
& \tfrac{\partial^2}{\partial \theta_n \partial \theta_2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_n^2} \\
\end{array} \right)
\ell(\theta)
\right|_{\theta = \theta^*}
</math>

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>

==Fisher information==
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>:

:<math>\mathcal{I}(\theta) = \mathrm{E}(\mathcal{J}(\theta))</math>.

==Applications==
In a notable article, [[Bradley Efron]] and [[David V. Hinkley]] <ref>{{cite journal
|last1=Efron |first1=B. |authorlink1=Bradley Efron
|last2=Hinkley |first2=D.V. |authorlink2=David V. Hinkley
|year=1978
|title=Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information
|journal=[[Biometrika]]
|volume=65 |issue=3 |pages=457–487
|doi=10.1093/biomet/65.3.457 |mr=0521817 | jstor = 2335893
}}
</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.

==See also==
* [[Fisher information matrix]]
* [[Fisher information metric]]

==References==
{{reflist}}

[[Category:Information theory]]
[[Category:Statistical terminology]]
[[Category:Estimation theory]]

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