The task is to prove that reparametrization of multivariable Fisher information with diffeomorphism $\eta: \Theta \rightarrow H$ is $$ I(\eta)=\bigg(\frac{\partial \theta^T}{\partial \eta}\bigg)I(\theta) \bigg(\frac{\partial \theta}{\partial \eta^T}\bigg), $$ for given $I(\theta)$.
My solution:
We know that multivariable Fisher information is $$ I(\theta)=\mathbb{E}\bigg\{\frac{\partial}{\partial\theta}\log{(f_\theta(x))}\bigg[\frac{\partial}{\partial \theta}\log{(f_\theta(x))}\bigg]^T\bigg\}. $$
We can apply chain rule $$ \frac{\partial}{\partial \eta}= \frac{\partial \theta}{\partial \eta}\frac{\partial}{\partial \theta} \Rightarrow \frac{\partial}{\partial \theta} = \bigg(\frac{\partial \theta}{\partial \eta}\bigg)^{-1}\frac{\partial}{\partial \eta} %\theta'(\eta)\frac{\partial}{\partial\theta} $$
Applying it to the first equation and changing $\theta \mapsto \eta$ we obtain $$ I(\eta)=\mathbb{E}\bigg\{\frac{\partial\theta}{\partial\eta}\frac{\partial}{\partial\theta}\log{(f_\eta(x))}\bigg[\frac{\partial\theta}{\partial\eta}\frac{\partial}{\partial \theta}\log{(f_\eta(x))}\bigg]^T\bigg\}. $$ Now it's somewhere close to solution, what I would like to do is to take this $\big(\frac{\partial\theta}{\partial\eta}\big)$s out of the expected value and tell that it is done. But I don't think it is correct.
Can anyone help?
Here are the proof steps. The first 3 lines are a Binomial reparametrization example. You can ignore them, and they will not impact the proof below. However, it will help you to understand the proof when you feel confuse.