So if we can approximate the CF reasonably well, we can just ifft the marginal CF to get the posterior.
We have
\(\varphi_\theta = \left( \mathcal{F}(\pi(\beta_0)) \prod_{i = 1}^{p} \mathcal{F}(\pi(\beta_i,\tilde{\lambda}_i)) \right) * \mathcal{F}(\mathcal{L}(\beta_0,\beta,\tilde{\lambda}))\)
        \(= \left( \mathcal{F}(\pi(\beta_0)) \prod_{i = 1}^{p} \mathcal{F}(\pi(\beta_i,\tilde{\lambda}_i)) \right) * \mathcal{F}(\mathcal{L}(X\beta_+,\tilde{\lambda};y))\)
For a GLM, the likelihood is from the exponential family, and can be written,
\(\mathcal{L}(\beta_0,\beta,\tilde{\lambda};X,y) = \left( \prod_{i=1}^{n} h(y_i) \right) g(\eta)^n \exp \left( \eta^\top \sum_{i=1}^{n} T(y_i)\right)\)
with \(\eta = \eta(\theta)\) a function of all of the parameters. For canonical GLMs, \(\eta(\theta) = \theta\).
We also have: