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We develop a Bayesian variable selection method, called SVEN, based on ahierarchical Gaussian linear model with priors placed on the regressioncoefficients as well as on the model space. Sparsity is achieved by usingdegenerate spike priors on inactive variables, whereas Gaussian slab priors areplaced on the coefficients for the important predictors making the posteriorprobability of a model available in explicit form (up to a normalizingconstant). The strong model selection consistency is shown to be attained whenthe number of predictors grows nearly exponentially with the sample size andeven when the norm of mean effects solely due to the unimportant variablesdiverge, which is a novel attractive feature. An appealing byproduct of SVEN isthe construction of novel model weight adjusted prediction intervals. Embeddinga unique model based screening and using fast Cholesky updates, SVEN produces ahighly scalable computational framework to explore gigantic model spaces,rapidly identify the regions of high posterior probabilities and make fastinference and prediction. A temperature schedule guided by our model selectionconsistency derivations is used to further mitigate multimodal posteriordistributions. The performance of SVEN is demonstrated through a number ofsimulation experiments and a real data example from a genome wide associationstudy with over half a million markers.

Model Based Screening Embedded Bayesian Variable Selection for Ultra-high Dimensional Settings