= Gamma rate gamma conjugacy
{title2=$\Theta\mid x\sim\operatorname{Gamma}(A+nr,B+\sum_i x_i)$}
Conditionally independent <gamma distributions> with known shape $r$ and unknown rate $\Theta$ have likelihood proportional to $\Theta^{nr}e^{-\Theta\sum_i x_i}$. A <gamma distribution> prior of shape $A$ and rate $B$ therefore updates to shape $A+nr$ and rate $B+\sum_i x_i$. When $A+nr>1$, the <posterior mean> of the conditional claim mean $r/\Theta$ is $r(B+\sum_i x_i)/(A+nr-1)$. For $A=rk+1$ and $B=k\mu$, this is a <credibility estimate> with weight $n/(n+k)$. It is the reciprocal-rate version of <gamma scale inverse-gamma conjugacy>.
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