Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 27I / i / Solution

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.