Elastic net Zou and Hastie (2005) is a flexible regularization and variable selection method that uses a mixture of $L_1$ and $L_2$ penalties. It is particularly useful when there are much more predictors than the sample size. This paper proposes a Bayesian method to solve the elastic net model using a Gibbs sampler. While the marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian elastic net, the Bayesian elastic net has two major advantages. Firstly, as a Bayesian method, the distributional results on the estimates are straightforward, making the statistical inference easier. Secondly, it chooses the two penalty parameters simultaneously, avoiding the "double shrinkage problem" in the elastic net method. Real data examples and simulation studies show that the Bayesian elastic net behaves comparably in prediction accuracy but performs better in variable selection.
Qing Li. Nan Lin. "The Bayesian elastic net." Bayesian Anal. 5 (1) 151 - 170, March 2010. https://doi.org/10.1214/10-BA506