High-Dimensional Bayesian Nonparametrics: Minimax Frequentist Guarantees

Abstract

The increasing availability of high-dimensional datasets presents both opportunities and significant challenges for statistical modeling. In this context, Bayesian nonparametric methods offer immense flexibility for complex data structures by placing priors directly on infinite-dimensional parameter spaces, allowing the model complexity to adapt to the data. However, a critical question in their application, especially in high-dimensional regimes, concerns their frequentist performance characteristics. This paper provides a comprehensive theoretical investigation into high-dimensional Bayesian nonparametric models, focusing on establishing minimax frequentist guarantees. We delve into the conditions under which Bayesian posterior distributions concentrate around the true underlying data-generating mechanism at rates that are minimax optimal in a frequentist sense. By carefully constructing appropriate nonparametric priors and analyzing their interaction with sparsity-inducing mechanisms often essential in high dimensions, we demonstrate that Bayesian nonparametric procedures can achieve optimal rates of convergence, mirroring and sometimes even surpassing the performance of leading frequentist methods. Our work bridges the theoretical gap between the rich expressive power of Bayesian nonparametrics and the robust performance assessment provided by frequentist minimax theory, thereby enhancing confidence in their application to complex, high-dimensional problems. We explore various model classes, including sparse regression and density estimation, illustrating how prior specification plays a pivotal role in achieving these theoretical optimality results.