Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 3 / d / Solution

Let $S={\sum }_s(D_s-\bar{D}{)}^2$. Integrating out $M_0$ givesNear zero, integrability requires $k>-1$; at infinity it requiresFor integer $k$ the posterior is proper exactly whenThe choice $k=0$ is allowed for $N>3$ and yields a proper posterior, but it is an improper flat prior on a scale parameter and is not invariant under reparameterization, so sensitivity to more principled scale priors should be checked.