= Normal-normal credibility
{title2=$\widehat\theta=Z\overline X+(1-Z)\mu,\quad Z=n\sigma_2^2/(\sigma_1^2+n\sigma_2^2)$}
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.
Back to article page
