Bayesian Bridge Regression

J Appl Stat. 2018;45(6):988-1008. doi: 10.1080/02664763.2017.1324565. Epub 2017 May 10.


Classical bridge regression is known to possess many desirable statistical properties such as oracle, sparsity, and unbiasedness. One outstanding disadvantage of bridge regularization, however, is that it lacks a systematic approach to inference, reducing its flexibility in practical applications. In this study, we propose bridge regression from a Bayesian perspective. Unlike classical bridge regression that summarizes inference using a single point estimate, the proposed Bayesian method provides uncertainty estimates of the regression parameters, allowing coherent inference through the posterior distribution. Under a sparsity assumption non the high-dimensional parameter, we provide sufficient conditions for strong posterior consistency of the Bayesian bridge prior. On simulated datasets, we show that the proposed method performs well compared to several competing methods across a wide range of scenarios. Application to two real datasets further revealed that the proposed method performs as well as or better than published methods while offering the advantage of posterior inference.

Keywords: Bayesian Regularization; Bridge Regression; LASSO; MCMC; Scale Mixture of Uniform; Variable Selection.