Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

TL;DR

This paper establishes Bernstein-von Mises theorems for Gaussian regression models with an increasing number of regressors, demonstrating asymptotic normality and adaptivity for Bayesian estimators in high-dimensional settings.

Contribution

It extends Bernstein-von Mises theorems to nonparametric and semiparametric Gaussian regression models with growing regressors, including applications to Sobolev and Hölder classes.

Findings

Asymptotic normality of the posterior in high-dimensional Gaussian regression.

Minimax convergence rates achieved for functionals in Sobolev and Hölder classes.

Bayesian estimators exhibit adaptivity in the considered models.

Abstract

This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises Theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and $C^{\alpha}$ classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.