What reparametrization of vector of parameters $\theta$ makes the Jeffreys prior $$\sqrt{\det I(\theta)}$$ correspond to the uniform prior?
A change of parametrization from $\theta$ to $\eta$ changes the Fisher information as follows (I think):
\begin{align} I(\eta) &= (\nabla_{\eta}{\theta}) I(\theta \mid X) {(\nabla_{\eta}{\theta})}^T \end{align}
E.g., for a Bernoulli distribution parametrized by probability $p\in[0,1]$, the Jeffreys prior is proportional to $\sqrt{\frac1{p(1-p)}}$. I believe the transformation in this case of
$$\eta = \int \sqrt{\frac1{p(1-p)}}\, dp = 2\arcsin(\sqrt{p})$$
has a uniform Jeffreys prior?
How do I generalize this to vector-valued values?
