Variable Selection with Rigorous Uncertainty Quantification using Deep Bayesian Neural Networks: Posterior Concentration and Bernstein-von Mises Phenomenon
This provides a theoretical foundation for using deep BNNs in variable selection with reliable uncertainty estimates, addressing a key challenge in high-dimensional statistics, though it is incremental in building on existing Bayesian and neural network approaches.
The paper tackles high-dimensional variable selection with rigorous uncertainty quantification using deep Bayesian neural networks, showing that properly configured BNNs can effectively learn variable importance and achieve correct coverage of 95% credible intervals, with simulations confirming outperformance over existing methods in high dimensions.
This work develops rigorous theoretical basis for the fact that deep Bayesian neural network (BNN) is an effective tool for high-dimensional variable selection with rigorous uncertainty quantification. We develop new Bayesian non-parametric theorems to show that a properly configured deep BNN (1) learns the variable importance effectively in high dimensions, and its learning rate can sometimes "break" the curse of dimensionality. (2) BNN's uncertainty quantification for variable importance is rigorous, in the sense that its 95% credible intervals for variable importance indeed covers the truth 95% of the time (i.e., the Bernstein-von Mises (BvM) phenomenon). The theoretical results suggest a simple variable selection algorithm based on the BNN's credible intervals. Extensive simulation confirms the theoretical findings and shows that the proposed algorithm outperforms existing classic and neural-network-based variable selection methods, particularly in high dimensions.