Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 28K / b / iii / Solution

For every estimator, its worst-case risk dominates its average risk under the uniform prior. Since the preceding Bayes estimator minimizes that average,The estimator $X/n$ has risk exactly $1/n$ throughout the interior, so it attains this lower bound. Under either endpoint convention discussed above its supremum over $[0,1]$ is also $1/n$. ThusThis constant-risk Bayes argument works also for $n=1$, when the only observed counts are endpoints and the posterior integrability argument still selects their corresponding decisions.