Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 1 / b / Solution

For the $m$ independent Bernoulli trials, the log likelihood function, apart from its constant, is $\ell (\theta )=x\log \theta +(m-x)\log (1-\theta )$. Its negative expected second derivative, the Fisher information, isFor $m>0$, the Jeffreys prior is proportional to its square root. Since $\mathrm{B}(1/2,1/2)=\pi$, its normalized form isThe Jeffreys prior is invariant under smooth one-to-one reparameterization as a measure. If $\varphi =h(\theta )$, then $I_{\varphi }(\varphi )=I_{\theta }(\theta )(d\theta /d\varphi {)}^2$, so $\sqrt{I_{\varphi }}d\varphi =\sqrt{I_{\theta }}|d\theta |$. Thus recomputing it in the new coordinate gives precisely the transformed prior, rather than a new prior. In several dimensions the corresponding density is $\sqrt{\det I(\theta )}$, with the same Jacobian transformation property.