I want to express the variance of a posterior. The posterior is expressed as a partial composition with a Gamma and inverse-Gamma:
$\Pi(\alpha,t)=Gamma(t,r(t,\alpha))$ with $r(t,\alpha)\sim invGamma(t,\alpha)$.
How can I express the variance of $\Pi$ depending on $t$ which is an integer and $\alpha>0$?
My tentative answer is: I have a random variable $X\sim \mbox{Gamma}(t,Y)$ where $Y$ is itself a random variable $Y\sim \mbox{invGamma}(t,\alpha)$. The variance of $X$ write $Var(X) = E\left[ Var(X\vert Y)\right] + Var \left[ E(x\vert Y)\right]$, this yields: \begin{equation} Var(X) = t E\left( \frac{1}{Y^2} \right) + t^2 Var\left(\frac{1}{Y} \right) \end{equation} $\frac{1}{Y}\sim \mbox{Gamma}(t,\alpha^{-1})$ and hence $\frac{1}{Y^2}$ follows a generalized gamma distribution with parameter $p=\frac{1}{2}$, $d=\frac{t}{2}$ and $a=\alpha^{-2}$. Hence
\begin{equation} Var(X) = \alpha^{-2} t (t+2)(t+1)+ t^3 \alpha^2 \end{equation}
I think that this doesn't make sense, it cannot be that with more observations my posterior is more dispersed (less precise). Can anyone help me spot the mistake?