Let $X \sim Bin(n, \pi)$ with $\pi \in (0,1)$ and known $n$.
Derive Jeffreys' prior for $\pi$ in the following parametrizations:
a) Original parametrization $\pi$
b) Parametrization with $\phi = \text{logit}(\pi)$
c) Compute the posteriors for $\pi$ and $\phi$.
What I have done:
Finding Jeffreys' prior ($f(\pi) \propto \sqrt{J(\pi)}$) for $\pi$ and finding the prior is fairly straightforward I got
$$f(\pi) \propto \left(\frac{1}{\pi(1-\pi)} \right)^{1/2}$$ Which gives to following posterior
$$f(\pi|x) \propto L(\pi)\cdot f(\pi) \propto \pi^x(1-\pi)^{n-x}\cdot \left(\frac{1}{\pi(1-\pi)} \right)^{1/2} = \pi^{x-1/2}(1-\pi)^{n-x-1/2}$$
I'm fairly sure this is correct what I'm uncertain about is $\phi$. I know that the Jeffreys prior is invariant under one-to-one reparametrizations and so is the likelihood function. Does this mean that for $\phi = \text{logit}(\pi)$ i could simply take
$$f(\phi) \propto \left(\frac{1}{\pi(1-\phi)} \right)^{1/2} \qquad L(\phi) = \binom{n}{x}\phi^x(1-\phi)^{n-x}$$
i.e just replace $\pi$ with $\phi$ and not needing to work with the $\text{logit}(\cdot)$ function at all.