= 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 is
$$
\boxed{B_{+,-}(y_i)=\frac{q_i/(1-q_i)}{q_0/(1-q_0)}.}
$$
This follows from the <posterior odds> identity when the two models use the original prior conditioned respectively on $\lambda_i>1$ and $\lambda_i\le1$. For the specified prior, $q_0$ is the upper tail at one of $\operatorname{Gamma}(16,16)$; 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 \b[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.
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