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This work develops rigorous theoretical basis for the fact that deep Bayesianneural network (BNN) is an effective tool for high-dimensional variableselection with rigorous uncertainty quantification. We develop new Bayesiannon-parametric theorems to show that a properly configured deep BNN (1) learnsthe variable importance effectively in high dimensions, and its learning ratecan sometimes "break" the curse of dimensionality. (2) BNN's uncertaintyquantification for variable importance is rigorous, in the sense that its 95%credible intervals for variable importance indeed covers the truth 95% of thetime (i.e., the Bernstein-von Mises (BvM) phenomenon). The theoretical resultssuggest a simple variable selection algorithm based on the BNN's credibleintervals. Extensive simulation confirms the theoretical findings and showsthat the proposed algorithm outperforms existing classic andneural-network-based variable selection methods, particularly in highdimensions.

Variable Selection with Rigorous Uncertainty Quantification using Deep  Bayesian Neural Networks: Posterior Concentration and Bernstein-von Mises  Phenomenon