High-Dimensional Nonparametric Inference: Reconciling Bayesian Beliefs with Frequentist Truths

Abstract

This paper explores the intricate relationship and potential for reconciliation between Bayesian and Frequentist paradigms in the challenging domain of high-dimensional nonparametric inference. In contexts where the number of parameters far exceeds the sample size, both methodologies offer distinct advantages and face unique hurdles. Bayesian methods provide a coherent framework for incorporating prior knowledge and quantifying uncertainty, often yielding automatically adaptive procedures through appropriate prior specification. Conversely, Frequentist approaches offer robust guarantees in terms of coverage and minimax optimality, without reliance on subjective priors. We investigate theoretical frameworks, such as adaptive shrinkage priors and empirical Bayes methods from the Bayesian perspective, and penalized likelihood and minimax rate-optimal estimators from the Frequentist viewpoint, demonstrating how under high-dimensional settings, their performance can converge or complement each other. We specifically analyze conditions under which Bayesian credible sets achieve valid frequentist coverage and when frequentist confidence intervals can be interpreted through a Bayesian lens. The aim is to bridge the conceptual and practical divides, offering insights into the construction of inference procedures that are simultaneously robust, adaptive, and provide interpretable uncertainty quantification in complex, high-dimensional data environments. This work contributes to a deeper understanding of statistical foundations and practical methodology, particularly for fields grappling with large-scale data and intricate models.