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

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.

References in corpus (3)

• Convergence rates of posterior distributions for noniid observations
• A Bernstein-Von Mises Theorem for discrete probability distributions
• The Bernstein--von Mises theorem for the proportional hazard model

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• Frequentist Consistency of Variational Bayes
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• Posterior consistency in linear models under shrinkage priors
• A Bernstein-von Mises theorem for smooth functionals in semiparametric models
• Adaptive Bernstein-von Mises theorems in Gaussian white noise
• Quasi-Bayesian analysis of nonparametric instrumental variables models
• Nearly optimal Bayesian Shrinkage for High Dimensional Regression
• Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems
• Necessary and Sufficient Conditions for High-Dimensional Posterior Consistency under $g$-Priors
• On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems
• Semiparametric posterior limits under local asymptotic exponentiality
• Optimal Bayesian estimation in random covariate design with a rescaled Gaussian process prior
• Optimal Bayesian estimation in stochastic block models
• Dirichlet-Laplace priors for optimal shrinkage
• Large ball probability, Gaussian comparison and anti-concentration
• Accuracy of Gaussian approximation in nonparametric Bernstein -- von Mises Theorem
• Bayesian estimation in differential equation models
• On frequentist coverage errors of Bayesian credible sets in moderately high dimensions
• Approximating posteriors with high-dimensional nuisance parameters via integrated rotated Gaussian approximation
• Bayesian factorizations of big sparse tensors
• The semiparametric Bernstein-von Mises theorem for models with symmetric error
• Nonasymptotic Laplace approximation under model misspecification
• Functional Horseshoe Priors for Subspace Shrinkage