Sharp sup-norm Bayesian curve estimation

TL;DR

This paper proves that Bayesian methods for sup-norm curve estimation can achieve minimax-optimal rates across various models, enhancing the theoretical understanding of Bayesian nonparametrics.

Contribution

It establishes that the sup-norm posterior concentration approach yields sharp, minimax-optimal rates for a wide range of prior-model pairs beyond conjugate priors.

Findings

Achieves sharp rates for density, regression, and quantile estimation.

Validates the sup-norm posterior concentration approach in non-conjugate settings.

Extends theoretical guarantees for Bayesian curve estimation methods.

Abstract

Sup-norm curve estimation is a fundamental statistical problem and, in principle, a premise for the construction of confidence bands for infinite-dimensional parameters. In a Bayesian framework, the issue of whether the sup-norm-concentration- of-posterior-measure approach proposed by Gin\'e and Nickl (2011), which involves solving a testing problem exploiting concentration properties of kernel and projection-type density estimators around their expectations, can yield minimax-optimal rates is herein settled in the affirmative beyond conjugate-prior settings obtaining sharp rates for common prior-model pairs like random histograms, Dirichlet Gaussian or Laplace mixtures, which can be employed for density, regression or quantile estimation.