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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.

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I assume that the Laplace prior is on the mean of the Gaussian random variable.

Sadly yes, Gaussian likelihood and Laplace prior do not yield a tractable posterior distribution. You will have to use some MCMC method such as Metropolis-Hastings or slice sampling.

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