Robust Frequentist Coverage for Adaptive High-Dimensional Bayesian Nonparametrics

Abstract

Bayesian nonparametrics (BNP) offers a flexible framework for statistical inference, yet its application to high-dimensional data faces significant challenges from the "curse of dimensionality" and computational complexity. Crucially, while Bayesian credible sets provide a natural measure of uncertainty, their frequentist coverage often lacks robustness in adaptive and high-dimensional settings. This paper addresses the critical problem of achieving robust frequentist coverage for adaptive high-dimensional Bayesian nonparametric procedures. We propose a novel framework that integrates carefully designed adaptive hierarchical priors, scalable computational strategies, and post-hoc calibration methods. Our methodology allows BNP models to adapt to unknown sparsity and smoothness in high-dimensional data while ensuring credible regions possess reliable frequentist coverage probabilities. Through theoretical analysis and extensive empirical investigation, demonstrating coverage rates consistently near nominal levels (e.g., 92-97% for 95% credible intervals in simulations), we show that our approach yields statistically efficient procedures with robust frequentist guarantees, effectively bridging Bayesian flexibility and frequentist desiderata in modern high-dimensional data analysis.