Observed Information - Definition

Definition

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

\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.

Read more about this topic:  Observed Information

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