Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / d / Solution

{c}

For loss $L(\theta,a)=q(\theta)(\theta-a)^2$ with $q(\theta)>0$, the Bayes estimator is the mean under the posterior reweighted by $q$:
$$
\widehat\theta_B
=\frac{\mathbb E[\theta q(\theta)\mid X]}
{\mathbb E[q(\theta)\mid X]}.
$$


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 Incoming links: Bayes estimator under parameter-weighted squared error