Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 43 / 4 / b / Solution

Write $A=\alpha +\beta$ and keep $p_0=\alpha /A$ fixed. Then $\alpha =p_{0}A$, $\beta =(1-p_0)A$, while the fixed observed years and exposures leave $M$ unchanged. The credibility factor isAs prior concentration $A$ increases without bound,The beta prior variance is $p_0(1-p_0)/(A+1)$, which tends to zero while its mean stays fixed. Increasing $A$ therefore makes the prior increasingly concentrated and reduces the influence of the same finite data. This explains the limiting Bayesian credibility estimate, rather than merely treating $\alpha ,\beta$ as two larger arbitrary parameters.