Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth

TL;DR

This paper introduces an adaptive nonparametric Bayesian method using a Gaussian random field with an inverse Gamma bandwidth, achieving near-optimal convergence rates across various statistical tasks.

Contribution

It demonstrates that a hierarchical Bayesian approach with a fixed prior can adaptively attain minimax-optimal convergence rates in density estimation, regression, and classification.

Findings

Posterior distribution shrinks at near-minimax rates.

Method is fully adaptive across different data regularities.

Applicable to multiple statistical settings.

Abstract

We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, whatever the regularity level of the data-generating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive.