Use the shape-rate convention $R_k\sim\operatorname{Gamma}(\alpha_0,\beta_0)$. The factors involving $R_k$ in the <posterior density> are
$$
R_k^{I_k}e^{-s_kR_k}R_k^{\alpha_0-1}e^{-\beta_0R_k}
=R_k^{I_k+\alpha_0-1}e^{-(s_k+\beta_0)R_k}.
$$
By <Poisson-gamma conjugacy>,
$$
R_k\mid I_k,s_k\sim\operatorname{Gamma}(I_k+\alpha_0,s_k+\beta_0),
$$
so its <posterior mean> is $(I_k+\alpha_0)/(s_k+\beta_0)$.

Solution

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

A posterior density is the density of a <posterior distribution> with respect to a chosen reference measure.

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