Hierarchical view of laplace prior - technical question

Hello everyone, here's a problem that has been bothering me for quite some time now.

When doing regression one can adopt a laplace prior on the weights W of the regression in order to encourage sparsity. However, directly working with the laplacian prior makes things awkward as MAP estimation of the weights W is non-linear.

A common solution to this problem (e.g. Figueirido: Adaptive sparseness for supervised learning) is to adopt a hierarchical view of the laplacian prior:

• Each weight Wi (i is the index) has a zero-mean Gaussian prior p(wi|Ti) = N(wi|0,Ti) where Ti is the variance of the i-th weight.
• Each variance Ti has an exponential prior p(Ti|k) = -k/2 exp(-k/2 * Ti)

Now we can integrate out the variance Ti and obtain a laplacian prior on Wi:

• p(Wi|k) = S p(Wi|Ti) p(Ti|k) dTi = sqrt(k)/2 exp(-sqrt(k) abs(Wi))

The S above is meant to be integration from zero to infinity.

Map estimation can now be performed using an EM algorithm.

My question: how is this integral worked out? I've tried to follow references, up to (very challenging for me) papers that introduced this representation based on gaussian scale mixtures. Could somebody help?

Thanks in advance! N.