Normal-normal credibility ($\hat{\theta }=Z\overline{X}+(1-Z)\mu ,Z=n{\sigma }_2^2/({\sigma }_1^2+n{\sigma }_2^2)$)

Conditionally independent normal distribution observations of mean $\theta$ and known variance ${\sigma }_1^2$, with normal prior mean $\mu$ and variance ${\sigma }_2^2$, give normal-normal conjugacy with known observation variance. Their posterior mean has the displayed exact credibility estimate form. It minimizes posterior squared-error loss and predicts the conditional mean of the next observation.

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.