Bavesian Adaptive Square-root Lasso

Square-root Lasso is a variant of Lasso.It has scale invariance such that it is no longer necessary to estimate the standard deviation a of the noise.Moreover,it does not rely on Gaussianity or sub-Gaussianity of noise.Combining the advantages of Square-root Lasso and adaptive Lasso,a new regularized method is considered within the Bayesian framework—Bayesian adaptive Square-root Lasso.Using the property that the Laplace distribution can be expressed as a scale mixture of normal distributions,we derive the fully conditional posterior distribution of all unknowns and obtain a Gibbs sampler method for statistics inference.We also introduce latent variable to facilitate sampling and use the Partially Collapsed technique to speed up convergence in our methods.Since the fully Bayesian method cannot produce the sparse solution directly,we adopt the posterior confidence interval method for variable selection.Simulations and real data analyses are used to compare the performance of the Bayesian adaptive Square-root Lasso method with the Bayesian Square-root Lasso method.The results show that the proposed Bayesian adaptive Square-root Lasso method has superior performance.