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Approximate Gibbs sampler for Bayesian Huberized lasso

The Bayesian lasso is well-known as a Bayesian alternative for Lasso. Although the advantage of the Bayesian lasso is capable of full probabilistic uncertain quantification for parameters, the corresponding posterior distribution can be sensitive to outliers. To overcome such problem, robust Bayesian regression models have been proposed in recent years. In this paper, we consider the robust and efficient estimation for the Bayesian Huberized lasso regression in fully Bayesian perspective. A new posterior computation algorithm for the Bayesian Huberized lasso regression is proposed. The proposed approximate Gibbs sampler is based on the approximation of full conditional distribution and it is possible to estimate a tuning parameter for robustness of the pseudo-Huber loss function. Some theoretical properties of the posterior distribution are also derived. We illustrate performance of the proposed method through simulation studies and real data examples.