Fisher Matrix

$$[I(\Theta ){]}_{ij}=-E\left[\frac{{\partial }^2\log p(x;\Theta )}{\partial {\theta }_i\partial {\theta }_j}\right]$$

Where $[I(\Theta ){]}_{ij}$ denotes the $i^{\prime }th$ row and $j^{\prime }th$ column.

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Example 7

page=13 Let $x=A+w$ where

• $A$ is an unknown DC level
• $w$ is Gaussian noise with unknown variance $\xi ={\sigma }^2$.
• $x=x[0],x[1],\dots ,x[N-1]$ are our observations

Then our parameter vector of interest is

$$\Theta =\left[\begin{array}{c}A\\ \xi \end{array}\right]\in {\mathbb{R}}^2$$

This gives us the following Fisher Matrix

$$I(\Theta )=\left[\begin{array}{cc}-E\left[\frac{{\partial }^2}{\partial A^2}\log p(x;\Theta )\right]& -E\left[\frac{{\partial }^2}{\partial A\partial \xi }\log p(x;\Theta )\right]\\ -E\left[\frac{{\partial }^2}{\partial \xi \partial A}\log p(x;\Theta )\right]& -E\left[\frac{{\partial }^2}{\partial {\xi }^2}\log p(x;\Theta )\right]\end{array}\right]$$

See further calculations at page=14.