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

Normal-normal conjugacy givesWith $b={\tau }^2/({\tau }^2+{\sigma }^2)$,

Write $D_s=M_s+{\epsilon }_s$. ThenThe terms are independent and centered, soThe unpooled estimate $D_s$ has mean squared error ${\sigma }^2$, so population shrinkage improves it.

Convolving the population and measurement distributions givesPut $V={\tau }^2+{\sigma }^2$. With a flat prior on $M_0$,

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.

Up to normalization, the joint posterior isA Gibbs sampler cycle consists of:

Each proposal is the exact full conditional and is therefore accepted. If the cycle begins with density $p({\theta }^t\mid D)$, integrating the product of the current posterior and successive conditional kernels over all overwritten coordinates leavesso a full Gibbs sweep preserves the joint posterior.