= Bühlmann–Straub credibility estimate
{c}
{title2=$\widehat\mu=Z\overline X_w+(1-Z)m_0$}
For total exposure $W=\sum_jm_j$ and weighted average $\overline X_w=\sum_jm_jX_j/W$, the best affine <mean squared error> estimate of the conditional mean is $Z\overline X_w+(1-Z)m_0$, where $Z=Wa/(Wa+v)$. If the coefficients sum to $B$, their error is $a(1-B)^2+v\sum_jb_j^2/m_j$. <Cauchy-Schwarz inequality> minimizes the latter term at $b_j=B m_j/W$, and a one-dimensional quadratic minimization then gives $B=Z$.
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