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

Treat the diagnosis counts as the observed incidence series and define their total infectiousness byA conditionally independent Poisson observation model for the renewal process isInitial infections before the observation window can be included in the definition of $s_t$.

Use the shape-rate convention $R_k\sim \mathrm{Gamma}({\alpha }_0,{\beta }_0)$. The factors involving $R_k$ in the posterior density areBy Poisson-gamma conjugacy,so its posterior mean is $(I_k+{\alpha }_0)/(s_k+{\beta }_0)$.