Question:
Let $X \sim Bin(n, \pi)$ with $\pi \in (0,1)$ and known $n$. Derive the expression of Jeffreys prior for $\pi$ in the following parametrization:
(a) Original parametrization with parameter $\pi$.
(b) Parametrization with $\phi = logit(\pi)$ as a parameter
(c) Parametrization with $\eta = arcsin(\sqrt{\pi})$ as a parameter
(d) Compute the posteriors for $\pi, \phi$ and $\eta$ under each parametrization considered in parts (a), (b) and (c).
What I have done:
I have no problem calculating the prior for $\pi$, I get
$$f(\pi) \propto \sqrt{\pi^{-1}(1-\pi)^{-1}} \tag1\label1$$ So the posterior for $\pi$ is $$f(\pi|x) \propto L(\pi)\cdot f(\pi) \propto \pi^x(1-\pi)^{n-x}\sqrt{\pi^{-1}(1-\pi)^{-1}} \tag2\label2$$ Where $L(\pi)$ is the likelihood function.
I know that Jeffreys prior is invariant during one-to-one reparametrization and so is the likelihood function. So for both b) and c) I would just replace $\pi$ with $\phi$ or $\eta$ in \eqref{1} and then later do the same for d) in \eqref{2}.
Is this correct? I guess whats confusing me here is that we are to find a parametrization for both $\phi$ and $\eta$ but in my solution above I just used the exact same argument for both making one of the questions redundant.
The only difference I can think about is that for $\eta$ one should probably make sure that this is a one-to-one transformation by noting that $\pi \in (0,1)$ fulfills the requirement.
Thanks in advance.