 Solution

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.