= 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 $a,v,m_j>0$ and <independent random variables>[independent] normal observations conditional on $\Theta$, with <mean> $\Theta$ and <variances> $v/m_j$, and normal <prior distribution>[prior] <mean> $\mu$ and <variance> $a$, write $W=\sum_jm_j$. Multiplying <likelihood function>[likelihood] and <prior distribution>[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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