Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 1 / e

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- Solution e

Solution

Use the intended hierarchical Bayesian model: the breed parameters are conditionally independent given

α, β

, and observations are independent given their breed parameters. Put

m_{i} = max (β, M_{i})

. The uniform-Pareto model evidence factorizes over breeds:

p (y_{1}, \dots, y_{I} ∣ α, β) = i = 1 \prod I \frac{α β ^{α}}{( α + n _{i} ) m _{i}^{α + n_{i}}} .

Identical marginal prior distributions alone would not determine this product; conditional independence is the additional assumption.

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