Bayesian-Frequentist Convergence for High-Dimensional Nonparametric Inference

Abstract

This paper provides a comprehensive analysis of Bayesian-Frequentist convergence in the context of high-dimensional nonparametric inference. In an era characterized by increasingly complex and large datasets, both Bayesian and Frequentist methodologies face significant challenges, particularly concerning the curse of dimensionality and computational intractability. We explore the theoretical underpinnings that facilitate the asymptotic equivalence of these two distinct statistical paradigms, focusing on posterior contraction rates, minimax optimality, and the Bernstein-von Mises theorem. Special attention is given to how specific prior constructions and regularization techniques, such as shrinkage priors and Reproducing Kernel Hilbert Spaces, enable robust inference and ensure desirable convergence properties in high-dimensional settings. We synthesize recent advancements in the literature, highlighting conditions under which Bayesian credible sets achieve valid frequentist coverage and posterior distributions attain optimal frequentist rates. The discussion extends to practical implications, including computational considerations and the interpretation of uncertainty quantification in complex models, demonstrating how a deeper understanding of this convergence can enhance the reliability and applicability of nonparametric methods across various scientific disciplines.