Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 18H (Statistics)

LetThus $d_t$ patients recover during month $t$, while $r_t-d_t$ survive that month without recovery. The likelihood function is

The independent $\mathrm{Beta}(T,1)$ prior densities are proportional to $q_t^{T-1}$. Beta-binomial conjugacy therefore leaves the coordinates posteriorly independent, with

The probability of no recovery through month $M$ is the product of the conditional survival probabilities, soUnder quadratic loss, the Bayes estimator is the posterior mean. Posterior independence and the mean of a Beta distribution giveConsequently

The posterior expected loss at decision $\delta$ isAt continuity points its derivative isThus the risk is minimized at a posterior one-third quantile:More generally, any $\delta$ satisfyingis optimal. The quantile lies below the posterior median, reflecting the smaller penalty assigned to underestimation.