Bayesian Lasso Prior. I The solution of 3 can be interpreted as the posterior mode of in the above Bayesian model. You can include a Laplace prior in a Bayesian model and then the posterior is proportional to the lassos penalized likelihood.
With this prior independent Laplace ie. We provide insight into the reasons for this sub-optimality and propose a new class of Dirichlet-Laplace DL priors which are optimal and lead to efficient posterior computation. The Lasso estimate for linear regression parameters can be interpreted as a Bayesian posterior mode estimate when the priors on the regression parameters are indepen-dent double-exponential Laplace distributions.
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The benefit was submitting the prior information on top of the quality generalized linear regression model. It is the maximum a posteriori MAP estimate of the coefficient vector when the prior distribution of its coordinates are independent mean-zero Gaussians with the same variance and the likelihood of the data is. Class of Gaussian scale mixture priors including the Bayesian Lasso 21 and other commonly used choices such as ridge regression are sub-optimal. Bayesian approaches to lasso have also been developed Park and Casella 2008 Hans 2009 Hans 2010 but have not yet been sucessfully extended to high dimensional data.
