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

Take the exposure $E_i$ as known and positive. The Poisson distribution log-likelihood, up to a parameter-independent constant, is ${\ell }_i({\lambda }_i)=y_i\log {\lambda }_i-E_i{\lambda }_i$. Its first two derivatives areSince $\mathbb{E}(Y_i)=E_i{\lambda }_i$, the Fisher information isThe Jeffreys prior is proportional to its square root. Removing the constant $\sqrt{E_i}$ givesThis is an improper prior: its integral diverges at infinity.

Multiplying the Poisson distribution likelihood by the Jeffreys prior gives the posterior density kernelThus, in shape-rate convention,It is a proper posterior distribution even when $y_i=0$, provided $E_i>0$. The posterior mean and posterior variance are $(y_i+1/2)/E_i$ and $(y_i+1/2)/E_i^2$. This is Poisson-gamma conjugacy with unequal exposures, with the zero-rate prior interpreted as an improper kernel.

A proper gamma approximation to a Poisson Jeffreys prior uses shape $1/2$ and a small positive rate $\epsilon$:Its density is proportional to ${\lambda }_i^{-1/2}{e}^{-\epsilon {\lambda }_i}$, so where $\epsilon {\lambda }_i$ is small its kernel approximates the Jeffreys prior. After observing the data, Poisson-gamma conjugacy gives $\mathrm{Gamma}(y_i+1/2,E_i+\epsilon )$, which converges to the preceding posterior distribution as $\epsilon \downarrow 0$. This is an approximation of an improper kernel on the relevant parameter range, not a weak limit to a normalized Jeffreys probability distribution. The shape must be $1/2$; a small-shape gamma prior would approximate a different power of ${\lambda }_i$.

Under independent area counts and independent area prior distributions, the joint posterior distribution factors into the gamma posteriors. For each simulation $s=1,\dots ,S$, draw one joint vector by independently generatingRank the risks within that joint draw, taking rank one to mean highest risk:The empirical posterior rank distribution and highest-risk probability areIndependent gamma draws make this direct Monte Carlo integration, so Markov chain Monte Carlo is unnecessary here. Continuous posteriors give ties probability zero, and the highest-risk probabilities sum to one. The same ranking calculation can be applied to dependent joint draws from a richer hierarchical Bayesian model; ranking separate posterior means would discard the uncertainty that the program is intended to measure.

For a shape-rate gamma distribution $\mathrm{Gamma}(a,b)$, the mean and variance conditions requireThe first equation gives $a=b$, and the second then gives $a=b=16$. HenceThis proper prior distribution is centered on the national reference risk, while allowing genuine between-area variation.

The new prior density contributes the factor ${\lambda }_i^{15}{e}^{-16{\lambda }_i}$. Multiplication by the Poisson distribution likelihood givesIts posterior mean displays the resulting shrinkage:The observed risk ratio $y_i/E_i$ is pulled towards one, most strongly for small exposures. The posterior variance is $(y_i+16)/(E_i+16{)}^2$.

The informative gamma distribution prior distribution discourages extreme risks caused by small denominators. Its influence is strongest where the exposure is small, while large-exposure areas are predominantly determined by their likelihood. Consequently an area previously appearing near the top because of a noisy high raw risk ratio may have its highest-risk probability reduced, and the posterior rank distributions of poorly observed areas tend to overlap more rather than suggesting a precise league table.

There is no universal claim that every rank distribution becomes wider or narrower: unequal exposures produce unequal shrinkage and can change the ordering, while the new prior also changes the uncertainty in the risks. In the limiting case of negligible information for every area, all area posteriors approach the same independent continuous prior distribution. Symmetry then gives each area probability $1/I$ of occupying each rank. This illustrates why modest differences between weakly observed areas should not yield confident rankings.

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.