\(\)
posterior \(\propto\) \(\pi(\beta_0)\left( \prod_{i=1}^{p} \rho(\tilde{\lambda}_i)\right)^{-1}\exp\left(-\frac{1}{2}(\tilde{\lambda}^\top \tilde{\lambda} + \beta^\top \Sigma(\tilde{\lambda})^{-1}\beta) + \ell(\beta_0,\beta,\tilde{\lambda}) \right) \)
\(\propto \pi(\beta_0) \left( \prod_{i=1}^{p} \rho(\tilde{\lambda}_i)^{-1} \exp\left( -\frac{\beta_i^2}{2\rho^2(\tilde{\lambda}_i)} \right) \right) \left( \prod_{i=1}^{p} \exp\left( - \tilde{\lambda}_i^2 \right) \right) \exp \left( \ell(\beta_0,\beta,\tilde{\lambda}) \right)\)
If \(\phi_\theta(t)\) is the joint CF of the posterior,  \(\varphi_\theta(t_i,0)\) is the CF of the marginal distribution for \(\theta_i\).