Numerical calculation of fisher information

Question

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?

Answer 1

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}