Consistent High-Dimensional Bayesian Variable Selection via Penalized Credible Regions

Abstract

For high-dimensional data, particularly when the number of predictors greatly exceeds the sample size, selection of relevant predictors for regression is a challenging problem. Methods such as sure screening, forward selection, or penalized regressions are commonly used. Bayesian variable selection methods place prior distributions on the parameters along with a prior over model space, or equivalently, a mixture prior on the parameters having mass at zero. Since exhaustive enumeration is not feasible, posterior model probabilities are often obtained via long Markov chain Monte Carlo (MCMC) runs. The chosen model can depend heavily on various choices for priors and also posterior thresholds. Alternatively, we propose a conjugate prior only on the full model parameters and use sparse solutions within posterior credible regions to perform selection. These posterior credible regions often have closed-form representations, and it is shown that these sparse solutions can be computed via existing algorithms. The approach is shown to outperform common methods in the high-dimensional setting, particularly under correlation. By searching for a sparse solution within a joint credible region, consistent model selection is established. Furthermore, it is shown that, under certain conditions, the use of marginal credible intervals can give consistent selection up to the case where the dimension grows exponentially in the sample size. The proposed approach successfully accomplishes variable selection in the high-dimensional setting, while avoiding pitfalls that plague typical Bayesian variable selection methods.

Keywords:

• Consistency
• Credible region
• LASSO
• Stochastic search.

Acknowledgments

The authors are grateful to the editor, an associate editor, and three anonymous referees for their valuable comments. Bondell’s research was partially supported by an NSF grant DMS 1005612 and NIH grants P01-CA-142538-01 and R01-MH-084022-01. Reich’s research was partially supported by an NIH grant R01-ES-014843-02. The authors thank Gareth James for providing the DASSO code for the Dantzig selector.