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

arXiv: https://arxiv.org/abs/1912.01189 · HTML: https://mlsanity.com/paper/1912.01189/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2019.

## Built on by (most cited first, 4 of 4)

- [A Performance-Driven Benchmark for Feature Selection in Tabular Deep Learning](https://mlsanity.com/paper/2311.05877/) (2023)
- [Towards a Unified Framework for Uncertainty-aware Nonlinear Variable Selection with Theoretical Guarantees](https://mlsanity.com/paper/2204.07293/) (2022)
- [Efficient Model Compression for Bayesian Neural Networks](https://mlsanity.com/paper/2411.00273/) (2024)
- [Uncertainty of Network Topology with Applications to Out-of-Distribution Detection](https://mlsanity.com/paper/2511.18813/) (2025)

## Abstract (arXiv)

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.