A Calibrated Bayesian Approach to High-Dimensional Nonparametric Inference

Abstract

High-dimensional nonparametric inference presents a significant challenge in modern statistics, driven by the increasing complexity and scale of data in fields ranging from genomics to image analysis. While Bayesian nonparametric (BNP) methods offer a flexible framework for modeling complex data structures without restrictive parametric assumptions, their performance in high dimensions often faces issues of computational intractability, prior sensitivity, and, crucially, a lack of calibration in their uncertainty quantification. Standard Bayesian credible intervals, while intuitively appealing, do not always possess the desired frequentist coverage properties, especially in complex, high-dimensional settings where model misspecification or prior choices can have profound effects. This paper proposes a calibrated Bayesian approach designed to address these limitations. We develop a methodology that combines the inherent flexibility and regularization capabilities of Bayesian nonparametric models with post-hoc calibration techniques, specifically focusing on semiparametric posterior corrections. By adapting and extending these correction methods, we aim to achieve robust uncertainty quantification that exhibits desirable frequentist properties, such as accurate coverage probabilities for low-dimensional functionals of interest, even when the underlying full posterior might be difficult to calibrate directly. Our approach leverages the rich posterior distribution provided by BNP methods and subsequently refines it to ensure better frequentist validity for key inferential targets. We outline the theoretical framework, detailing how to construct these calibrated posterior distributions and discuss their computational implementation. Through this work, we seek to bridge the gap between Bayesian flexibility and frequentist rigor, offering a powerful tool for reliable inference in complex, high-dimensional nonparametric problems.