= Normal-normal conjugacy with unequal exposures
{title2=$\Theta\mid X\sim N((v\mu+a\sum_jm_jX_j)/(v+aW),av/(v+aW))$}
For independent normal observations conditional on $\Theta$, with mean $\Theta$ and variances $v/m_j$, and normal prior mean $\mu$ and variance $a$, write $W=\sum_jm_j$. Multiplying likelihood and prior adds precisions $W/v+1/a$. Completing the square gives the displayed <posterior distribution>. Its <posterior mean> is $Z\overline X_w+(1-Z)\mu$, where $Z=aW/(v+aW)$ and $\overline X_w=\sum_jm_jX_j/W$. Thus the <Bayes estimator under squared error loss> is exactly the <Bühlmann–Straub credibility estimate>, not merely an approximation to it. Its posterior variance is $av/(v+aW)$.
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