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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.

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.