Lets say I have a $2 \times 2$ Fisher information matrix $\mathbf{I}_{\mathbf{a}}$ of the parameter vector $\mathbf{a}$. I want to transform this matrix to find the Fisher information $I_b$ of a scalar parameter $b$, which can be expressed by a function of the parameter vector $\mathbf{a}$, i.e. $b=f(\mathbf{a})$.
According to the Wikipedia article about the Fisher information (at least I think so), we would have: \begin{align} \mathbf{I}_{\mathbf{a}}=\mathbf{J}^\mathrm{T}I_b\mathbf{J}, \end{align} where $\mathbf{J}$ is the Jacobian of our function $f(.)$. Equivalently, I think it should hold that $\mathbf{I}_{\mathbf{a}}^{-1} \mathbf{J}^\mathrm{T}\mathbf{J}$ is the $2\times2$ identity matrix that is scaled by $\frac{1}{I_b}$.
In my coding example however, the matrix is no scaled identity matrix and nothing really adds up so I think there might be something wrong with my approach. I would consequently be very grateful for any kind of help, tutorial or references that you could provide.
According to wikipedia, ${I}_{{b}}=\mathbf{J}^\mathrm{T}\mathbf{I_a}\mathbf{J}$ where $\bf J$ is a 2x1 column matrix. Further, $J_{11} = \dfrac{\partial a_1}{\partial b}$ and $J_{21} = \dfrac{\partial a_2}{\partial b}$