= Gamma prior for an inverse-gamma scale
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{title2=$\Theta\mid x\sim\operatorname{Gamma}(\alpha+nk,\lambda+\sum_i x_i^{-1})$}
For <inverse-gamma distribution> observations of known shape $k$ and scale $\Theta$, their <likelihood function> in $\Theta$ is proportional to $\Theta^{nk}\exp(-\Theta\sum_i x_i^{-1})$. A shape-$\alpha$, rate-$\lambda$ <gamma distribution> <conjugate prior> therefore gives the displayed <Bayesian posterior>. The <Bayes estimator under squared error loss> of the conditional claim <expected value> $\Theta/(k-1)$ is $(\alpha+nk)/[(k-1)(\lambda+\sum_i x_i^{-1})]$. It depends on reciprocals, rather than the <sample mean> of the claims.
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