I'm intending to implement a Bayesian LASSO inside the Gibbs sampler I use to estimate a multivariate time-series model, but I have a doubt about how to draw this step.
The prior is a Double-Exponential (a.k.a. Laplace distribution) and the likelihood is Gaussian. Unlike Gaussian likelihood with a Gaussian prior, this choice does not seem to conjugate and I can't see a way to use inversion method. How do I draw from this combination? Should I use an approximation like Metropolis-Hastings?
Thanks in advance.