Minimax Adaptive Bayesian Nonparametrics for High-Dimensional Models

Abstract

High-dimensional data analysis presents significant challenges to traditional statistical methods, primarily due to the "curse of dimensionality." Bayesian nonparametrics offers a powerful framework for flexible modeling without restrictive parametric assumptions, but ensuring optimal performance in high-dimensional settings requires careful consideration of adaptivity and minimax optimality. This paper explores the theoretical foundations and practical implications of minimax adaptive Bayesian nonparametrics for high-dimensional models. We delve into how carefully constructed prior distributions, particularly sparsity-inducing priors and hierarchical models, enable Bayesian procedures to achieve optimal rates of convergence that match the minimax lower bounds, even when the underlying sparsity level or smoothness parameters are unknown. The concept of adaptivity is central, allowing these methods to perform optimally across a wide range of underlying data generating processes without requiring prior knowledge of these specific characteristics. We survey recent advancements in this field, focusing on applications in high-dimensional regression, density estimation, and variable selection. By integrating insights from frequentist minimax theory with Bayesian inference, we demonstrate how Bayesian approaches can provide not only uncertainty quantification but also strong theoretical guarantees regarding their asymptotic performance. The paper highlights the critical role of prior specification, posterior contraction rates, and computational strategies in developing truly adaptive and minimax optimal Bayesian nonparametric methods for the complex landscape of high-dimensional data.