Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 2 / h / Solution

The posterior probability $q_i=\mathbb{P}({\lambda }_i>1\mid y_i)$ answers a natural directional question: how likely is this area's risk to exceed the reference risk? It accounts for uncertainty rather than declaring an increase from the point estimate alone. However, it does not distinguish a negligible increase from a clinically important one, and it combines prior beliefs with evidence from the data. With an informative prior distribution, a large probability may already have been present before observation, or a genuinely surprising count may be strongly moderated by the prior.

To measure the evidence supplied by the observations, compare posterior and prior odds. If $q_0=\mathbb{P}({\lambda }_i>1)$ under the chosen proper prior distribution, the Bayes factor for increased versus nonincreased risk isThis follows from the posterior odds identity when the two models use the original prior conditioned respectively on ${\lambda }_i>1$ and ${\lambda }_i\le 1$. For the specified prior, $q_0$ is the upper tail at one of $\mathrm{Gamma}(16,16)$; centering its mean at one does not make this tail probability exactly $1/2$. To address seriousness itself, choose a substantive excess $c>0$ and report the posterior probability of ${\lambda }_i>1+c$, together with an interval for the size of the excess. An evidence ratio and a meaningful risk threshold answer different, complementary questions.