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

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- Solution d

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 generating

λ_{i}^{(s)} \sim Gamma (y_{i} + 1/2, E_{i}), i = 1, \dots, I .

Rank the risks within that joint draw, taking rank one to mean highest risk:

r_{i}^{(s)} = 1 + \sum_{k \neq = i} 1 {λ_{k}^{(s)} > λ_{i}^{(s)}} .

The empirical posterior rank distribution and highest-risk probability are

P (r_{i} = k ∣ y) = \frac{1}{S} s = 1 \sum S 1 {r_{i}^{(s)} = k}, P (i highest ∣ y) = \frac{1}{S} s = 1 \sum S 1 {r_{i}^{(s)} = 1} .

Independent 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.

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