Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 18H / c

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.