Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 1 / c / ii

Table of contents
- Solution ii

Solution

Use the shape-rate convention

R_{k} \sim Gamma (α_{0}, β_{0})

. The factors involving

R_{k}

in the posterior density are

R_{k}^{I_{k}} e^{- s_{k} R_{k}} R_{k}^{α_{0} - 1} e^{- β_{0} R_{k}} = R_{k}^{I_{k} + α_{0} - 1} e^{- (s_{k} + β_{0}) R_{k}} .

By Poisson-gamma conjugacy,

R_{k} ∣ I_{k}, s_{k} \sim Gamma (I_{k} + α_{0}, s_{k} + β_{0}),

so its posterior mean is

(I_{k} + α_{0}) / (s_{k} + β_{0})

.

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