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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.

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  • $\begingroup$ What is the point of having both $\sigma$ and $\tau$ as hyperparameters? $\endgroup$ – Xi'an Nov 6 '15 at 17:39
  • $\begingroup$ I am not able to run a full gibbs sampling. At the point I have a new $\beta$ i need to update $\tau$ and so on... $\endgroup$ – JohnScott Nov 6 '15 at 17:53
  • $\begingroup$ My point is rather that your prior is $\beta \sim N(0,\sigma^2\tau I)$ so I cannot distinguish $\tau$ from $\sigma$ in the model, all that matters is $\tau\sigma^2$. $\endgroup$ – Xi'an Nov 6 '15 at 17:56
  • $\begingroup$ I see, you are right. But also for $\tau\sigma^2$ I need the conditional. By the way: For instance in the Bayesian Elastic Net there are at least four parameters to set $\rho_{1/2}$ and $r_{1/2}$. Do you know a reference for that? Sorry, for all this odd questions... $\endgroup$ – JohnScott Nov 6 '15 at 18:49
  • $\begingroup$ I think this thread replies to your question. $\endgroup$ – Xi'an Nov 6 '15 at 19:44

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