Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 4 / ii / Solution

The posterior mean is a weighted average of the sample proportion and prior mean:Thus it is pulled from 10% toward the prior's 5%, giving about 7.17%. This is an affine shrinkage estimator for a binomial proportion. The prior carries substantial concentration relative to the 90 observations, so the posterior is also more precise under this model: its standard deviation is about $0.01785$, compared with the binomial plug-in standard error $\sqrt{0.1(0.9)/90}=0.03162$. The posterior 95% credible interval $[0.04076,0.11035]$ is narrower and shifted down compared with the simple Wald interval $[0.03802,0.16198]$ based on $9/90$ alone.

The point estimate shrinks toward the historical mean, and the interval is narrower under the informative prior. A credible interval and a frequentist confidence interval have different probability interpretations; the comparison does not make the prior's relevance automatic.