Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 2 / a / Solution

For a regular one-parameter sampling distribution, the Fisher information and Jeffreys prior areUnder the usual differentiation and integrability conditions, $I(\theta )=-{\mathbb{E}}_{\theta }[{\partial }_{\theta }^2\log p_Y(Y\mid \theta )]$. This prior distribution transforms as a density under smooth one-to-one reparameterizations, so the rule is coordinate invariant. Its integral need not be finite; posterior propriety must still be established if it is an improper prior.