Exact Bühlmann credibility for gamma claims (${\hat{m}}_{\mathrm{B}}={\hat{m}}_{\mathrm{Bayes}}$)

With the gamma scale inverse-gamma conjugacy model and prior shape $k>2$, the Bühlmann model has $v/a=(k-1)/\alpha$. Its Bühlmann credibility premium equals $\alpha (\lambda +{\sum }_j{X}_j)/(k+n\alpha -1)$, exactly the Bayes estimator under squared error loss of $\alpha \Theta$. Equality holds because the posterior mean is already affine in the observed sample mean.