Normal-normal credibility with unequal exposures ($Z=W{\sigma }^2/(v+W{\sigma }^2),W={\sum }_j{m}_j$)

For conditionally independent annual averages $X_j\mid \theta \sim N(\theta ,v/m_j)$ and prior $\theta \sim N(\mu ,{\sigma }^2)$, the posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, with ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Completing the quadratic likelihood gives posterior precision ${\sigma }^{-2}+W/v$. This is an exact Bayesian credibility version of the Bühlmann–Straub credibility estimate. Increasing the between-risk prior variance increases the experience weight; increasing process variance decreases it. The next total pure premium is $m_{n+1}$ times the estimated per-life mean.