Calibrated Bayesian Nonparametrics for High-Dimensional Minimax Inference

Abstract

High-dimensional data analysis presents significant challenges for classical and Bayesian inference alike, particularly when seeking robust uncertainty quantification and optimal statistical performance. This paper introduces a novel framework for Calibrated Bayesian Nonparametrics specifically designed for high-dimensional minimax inference. We address the critical need for Bayesian methods that not only exhibit strong theoretical guarantees in terms of contraction rates matching minimax lower bounds but also provide reliable uncertainty estimates through well-calibrated credible sets. Traditional Bayesian nonparametrics, while offering flexibility, often struggle with computational scalability and theoretical calibration in high dimensions without careful prior specification. Our methodology leverages adaptive prior constructions, such as sparsity-inducing priors or hierarchical Gaussian process priors over function spaces, coupled with a rigorous theoretical analysis of posterior contraction rates. We demonstrate that, under specific regularity conditions and appropriate prior choices, our proposed Bayesian nonparametric models achieve optimal minimax rates of convergence for various high-dimensional estimation problems, including regression, density estimation, and graphical model inference. Furthermore, we establish the frequentist validity of Bayesian credible sets, ensuring that the uncertainty statements are well-calibrated and provide accurate coverage probability for the true unknown parameter. This work bridges the gap between the flexibility of Bayesian nonparametrics, the optimality guarantees of minimax theory, and the crucial requirement of calibrated uncertainty quantification in complex, high-dimensional settings, offering a powerful and theoretically sound approach for modern statistical inference.