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$ \phi _{\alpha _{pi}}$ for ARD prior


$\displaystyle \phi_{\alpha_{pi}}\vert.$ $\displaystyle \sim$ $\displaystyle Ga\left(\frac{1}{2}+\tilde{a}_{\alpha},
\frac{\alpha_{pi}^2}{2}+\tilde{b}_{\alpha}\right)$ (43)