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

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- Solution c

Solution

The joint prior predictive distribution is the Bayesian model evidence, obtained by integrating over the shared parameter:

p (y ∣ α, β) = α β^{α} \int_{m a x (β, M)}^{\infty} θ^{- (α + n + 1)} d θ = \frac{α β ^{α}}{( α + n ) max ( β , M ) ^{α + n}}, y_{j} > 0.

It is zero otherwise. This uniform-Pareto model evidence is a joint density, not the product of separately marginalized observation densities: mixing over the common parameter induces dependence.

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