Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 25J / i

The uniform prior has $s=1$, so the posterior is $\mathrm{Beta}(X+1,n-X+1)$. Its mean and mode areThus the uniform-prior posterior mode is the maximum-likelihood estimator, including $X=0,n$. The mean shrinks toward $1/2$ and equals the maximum-likelihood estimator only when $X=n/2$.