Abstract
The method of Bayesian variable selection via penalized credible regions separates model fitting and variable selection. The idea is to search for the sparsest solution within the joint posterior credible regions. Although the approach was successful, it depended on the use of conjugate normal priors. More recently, improvements in the use of global-local shrinkage priors have been made for high-dimensional Bayesian variable selection. In this paper, we incorporate global-local priors into the credible region selection framework. The Dirichlet–Laplace (DL) prior is adapted to linear regression. Posterior consistency for the normal and DL priors are shown, along with variable selection consistency. We further introduce a new method to tune hyperparameters in prior distributions for linear regression. We propose to choose the hyperparameters to minimize a discrepancy between the induced distribution on R-square and a prespecified target distribution. Prior elicitation on R-square is more natural, particularly when there are a large number of predictor variables in which elicitation on that scale is not feasible. For a normal prior, these hyperparameters are available in closed form to minimize the Kullback–Leibler divergence between the distributions.
Highlights
High dimensional data has become increasingly common in all fields
We show that the consistency of the posterior distribution under a global-local shrinkage prior yields consistency in variable selection under the case of pn → ∞
The focus here is to see if replacing the normal prior with the global-local prior can even further improve the performance of the credible region variable selection approach
Summary
Linear regression is a standard and intuitive way to model dependency in high dimensional data. Consider the linear regression model: Y = Xβ + ε (1). Where X is the n × p high-dimensional set of covariates, Y is the n scalar responses, β = (β1, · · · , βp) is the p-dimensional coefficient vector, and ε is the error term assumed to have E(ε) = 0 and Var(ε) = σ2In. Ordinary least squares is not feasible when the number of predictors p is larger than the sample size n. Bondell & Reich (2012) proposed a penalized regression method based on Bayesian credible regions. The full model is fit using all predictors with a continuous prior. Is used, where σ2 is the error variance term as in (1), and γ is the ratio of prior precision to error precision. The variance, σ2, is often given a diffuse inverse Gamma prior, while γ is the hyperparameter which is either chosen to be fixed or given a Gamma hyperprior.
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