I am going through the "Fitting a Line" example from here.

$f_1 = ax_1 + b$ and $f_2 = ax_2 + b$ are the models used to observed two data points in $R^2$. If $\sigma_1$ and $\sigma_2$ is the uncertainity in measuring $x_1$ and $x_2$, how do you calculate the Fisher information matrix?

I can't get to the solution they got to on Page 4.

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    $\begingroup$ There isn't enough information provided in your post to answer the question: your models must specify the possible distributions of all random quantities. $\endgroup$ – whuber Apr 14 '18 at 17:33
  • $\begingroup$ I agree. I am just trying to assume a Gaussian model with variance $\sigma_i$ while measuring $x_i$ and see if I reach the solution proposed in the paper. If that were true, how do I go about it? $\endgroup$ – pushkar Apr 14 '18 at 17:59

I got it. What I got wrong in my understanding was that we are trying to estimate $a$ and $b$, not the $ x_i$s.

In that case,

\begin{equation} \frac{\partial{f_i}}{\partial{a}} = x_i, \frac{\partial{f_i}}{\partial{b}} = 1. \end{equation}

Extending this to find the Fisher Information Matrix gives,

\begin{equation} F = \begin{bmatrix} \frac{x_1^2}{\sigma_1^2} + \frac{x_2^2}{\sigma_2^2} & \frac{x_1}{\sigma_1^2} + \frac{x_2}{\sigma_2^2} \\ \frac{x_1}{\sigma_1^2} + \frac{x_2}{\sigma_2^2} & \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} \end{bmatrix} \end{equation}


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