Regularization plays a critical role in modern statistical research, especially in high-dimensional variable selection problems. Existing Bayesian methods usually assume independence between variables a priori. In this article, we propose a novel Bayesian approach, which explicitly models the dependence structure through a graph Laplacian matrix. We also generalize the graph Laplacian to allow both positively and negatively correlated variables. A prior distribution for the graph Laplacian is then proposed, which allows conjugacy and thereby greatly simplifies the computation. We show that the proposed Bayesian model leads to proper posterior distribution. Connection is made between our method and some existing regularization methods, such as Elastic Net, Lasso, Octagonal Shrinkage and Clustering Algorithm for Regression (OSCAR) and Ridge regression. An efficient Markov Chain Monte Carlo method based on parameter augmentation is developed for posterior computation. Finally, we demonstrate the method through several simulation studies and an application on a real data set involving key performance indicators of electronics companies.
"Bayesian Regularization via Graph Laplacian." Bayesian Anal. 9 (2) 449 - 474, June 2014. https://doi.org/10.1214/14-BA860