= Exact Bühlmann–Straub credibility for Poisson-gamma counts
{title2=$\widehat\mu_{\rm cred}=\widehat\mu_{\rm Bayes}$}
With a <Poisson-gamma conjugacy with unequal exposures> model, $m_0=v=\alpha/\beta$ and $a=\alpha/\beta^2$, so the <Bühlmann–Straub credibility factor> is $W/(W+\beta)$. The credibility estimate simplifies to $(\alpha+\sum_jY_j)/(\beta+W)$, exactly the <posterior mean> and hence the <Bayes estimator under squared error loss>. The agreement is exact because that posterior mean is already affine in the exposure-weighted data.
Back to article page
