Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 2 / c / Solution

A constant improper prior would give the formal posterior density kernel $N^{-1}{1}_{\{N\ge y\}}$. However,There is no normalized posterior distribution under this prior. The failure is not simply that the prior is improper: some improper priors give proper posteriors. Here posterior propriety itself fails after this single observation, so probabilities, quantiles and a posterior mean cannot be assigned to that kernel.