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I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ \mathbb{I}(\theta)=\mathbb{E}\left[\left. \frac{\partial^2 \text{log }f( X|\theta)}{\partial \theta^2}\right|_{\theta=\theta^*} \right].$$

To calculate this numerically in Matlab is use this formula: $$\mathbb{I}(\theta) = \int_0^1 \frac{\partial^2 \text{log }f( X|\theta)}{\partial \theta^2} \cdot f(X,\theta) \quad dX$$

Am I doing this correct?

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  • $\begingroup$ @Learner Yes ,it is. I corrected my question. $\endgroup$ – Stephen Mar 20 '13 at 12:44
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    $\begingroup$ There is a negative sign missing and $\theta^\star$ should be equal to $\theta$. $\endgroup$ – Learner Mar 20 '13 at 12:52
  • $\begingroup$ @Learner, thanks a lot. $\endgroup$ – Stephen Mar 20 '13 at 12:57
  • $\begingroup$ What's the distribution of X? i.e. what is the functional form of f(x)? Since most Fisher Information problems are defined in terms of the parameters, you may find that you need symbolic methods, not numerical ones. $\endgroup$ – wolfies Apr 26 '13 at 14:38
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The formula i've always used is, \begin{equation} \mathbb{I}_{ij}(\theta) = \int\limits_\mathbb{R} f(x,\theta) \frac{\partial \ln f(x,\theta)}{\partial \theta_{i}} \frac{\partial ln \ f(x,\theta)}{\partial \theta_{j}} dx \end{equation}

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