Minimax Optimal High-Dimensional Nonparametric Inference: A Bayesian-Frequentist Confluence

Abstract

High-dimensional nonparametric inference presents significant challenges due to the "curse of dimensionality," where the sample size required for accurate estimation grows exponentially with dimension. This paper explores the fascinating confluence between Bayesian and Frequentist paradigms in achieving minimax optimal rates in such challenging settings. We delve into theoretical frameworks where appropriately constructed Bayesian priors, particularly those exhibiting sparsity or smoothness adaptivity, lead to posterior distributions that contract at rates equivalent to the Frequentist minimax lower bounds. This optimality implies that Bayesian estimators, such as posterior means or medians, inherently possess strong Frequentist guarantees. We review the foundational concepts of minimax theory, Bayesian posterior consistency, and the conditions under which these two perspectives align, demonstrating that careful prior specification can bridge the theoretical gap and yield robust, theoretically sound inference methods in high-dimensional spaces. The discussion highlights the practical implications of this confluence, offering insights into developing adaptive and efficient statistical procedures.