Observed information

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 variables $X_1,\ldots,X_n$, independent and identically distributed with density f(X; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters θ given the data $X_1,\ldots,X_n$ is

$\ell(\theta | X_1,\ldots,X_n) = \sum_{i=1}^n \log f(X_i| \theta)$.

We define the observed information matrix at θ ^* as

$\mathcal{J}(\theta^*) = - \left. \nabla \nabla^{\top} \ell(\theta) \right|_{\theta=\theta^*}$

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

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

Fisher information

The Fisher information $\mathcal{I}(\theta)$ is the expected value of the observed information:

$\mathcal{I}(\theta) = \mathrm{E}(\mathcal{J}(\theta))$.

Applications

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.

References

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. MR0521817.