I found that in Bayesian Lasso commonly $\beta \sim N(0,\sigma^2*diag(\tau))$ and $\sigma,\tau \sim \pi(\sigma,\tau)$ is used. Whereas $\pi(\cdot)$ is a product of Laplace distributions.
Is it possible to calculate the conditional density $\tau|\sigma,\beta$? Or do I have to run another Metropolis-Hasting within the sampler in that case?
I have a good draw for $\beta$ and now I want to update $\tau$ accordingly. Here the common Lasso setting does not work.