Gamma scale inverse-gamma conjugacy ($\Theta \mid x\sim \mathrm{InvGamma}(k+n\alpha ,\lambda +{\sum }_j{x}_j)$)

For conditionally independent gamma distribution observations of known shape $\alpha$ and unknown scale $\Theta$, an inverse-gamma distribution prior with shape $k$ and scale $\lambda$ gives an inverse-gamma distribution posterior with shape $k+n\alpha$ and scale $\lambda +{\sum }_j{x}_j$. The posterior mean of the conditional claim expected value $\alpha \Theta$ is $\alpha (\lambda +{\sum }_j{x}_j)/(k+n\alpha -1)$ when the denominator is positive.

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.