Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 27I (Principles of Statistics)

Complete the square in the independent normal prior and likelihood:Thus, conditional on the full observations, the hospital parameters remain independent andIf $j$ is the observed minimum index and $X_j=x$, its conditional distribution is exactly this formula with $X_j=x$. No extra truncation is introduced because the selection index is already a function of the conditioned data. Its posterior mean is shrunk toward zero, so the worst measured performance is not simply identified with the worst true performance.

Assume $\alpha ,\beta ,\kappa >0$, with the given gamma using shape and rate. PutThe joint posterior density before normalization isComplete each square and integrate over all ${\theta }_i$. The Gaussian integral contributes ${\tau }^{-n/2}$, givingThis hierarchical form completely specifies the normal-gamma distribution of the posterior. For comparison, $\tau \mid \theta ,X$ has shape $\alpha +n$ and rate $\beta +\frac{1}{2}{\sum }_i[{\theta }_i^2+\kappa (X_i-{\theta }_i{)}^2]$; confusing this conditional shape with the marginal shape would lose the Gaussian-integration factor.

Marginally $\theta$ has a multivariate Student distribution with $2A=2\alpha +n$ degrees of freedom and scale matrix $B/[A(1+\kappa )]I_n$. The coordinates are conditionally independent given $\tau$, but are generally dependent after integrating their shared precision. This is conjugate shrinkage with uncertainty in the overall noise scale.