Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2 / a / Solution

For a regular scalar statistical model, the one-observation Fisher information isUnder the usual differentiation and interchange-of-integral conditions it also equals $-{\mathbb{E}}_{\theta }[{\partial }_{\theta }^2\log p(Y\mid \theta )]$. The Jeffreys prior isThis prior measure is invariant under smooth one-to-one reparameterization, but may be improper. For independent identically distributed observations the information multiplier $n$ changes only the prior's proportionality constant.