Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 19H / i

Under squared-error loss, the posterior expected loss decomposes asIt is uniquely minimized by the posterior mean. Integrating the Gamma posterior gives $\mathbb{E}[\theta \mid x]=b^2{\int }_0^{\infty }{\theta }^2{e}^{-b\theta }d\theta =2/b$. Hence the Bayes estimator is