A Bernstein-von Mises theorem for smooth functionals in semiparametric models

Abstract

A Bernstein-von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared $L^2$-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.

Published at http://dx.doi.org/10.1214/15-AOS1336 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Semiparametric Bayesian causal inference
• Bayesian Sparse Linear Regression with Unknown Symmetric Error
• On the Bernstein-von Mises theorem for the Dirichlet process
• Testing Un-Separated Hypotheses by Estimating a Distance
• Bayesian inference for spectral projectors of the covariance matrix
• On frequentist coverage errors of Bayesian credible sets in moderately high dimensions
• Frequentist properties of Bayesian inequality tests
• Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
• Asymptotic Properties for Methods Combining Minimum Hellinger Distance Estimates and Bayesian Nonparametric Density Estimates
• Semi-parametric Bernstein-von Mises in Linear Inverse Problems