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

For $n\ge 1$, the binomial log-likelihood is $X\log \theta +(n-X)\log (1-\theta )$ plus a constant. Its maximizer, including endpoints, is ${\hat{\theta }}_{\mathrm{MLE}}=X/n$. For $0<\theta <1$, the Fisher information isThis follows either from the expected negative second derivative or the variance of $(X-n\theta )/(\theta (1-\theta ))$. At $0,1$ the usual regular score formula does not apply; the interior information diverges as an endpoint is approached.

For a symmetric Beta distribution prior $\mathrm{Beta}(s,s)$, conjugacy gives $\theta \mid X\sim \mathrm{Beta}(X+s,n-X+s)$, with posterior mean $(X+s)/(n+2s)$. All three priors below are instances of this calculation.