The semiparametric Bernstein-von Mises theorem

Abstract

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of Hájek's convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.

Published in at http://dx.doi.org/10.1214/11-AOS921 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Misspecification in infinite-dimensional Bayesian statistics
• A Bernstein-Von Mises Theorem for discrete probability distributions
• Finite sample properties of multiple imputation estimators
• The Bernstein--von Mises theorem for the proportional hazard model
• General frequentist properties of the posterior profile distribution
• Semiparametric posterior limits under local asymptotic exponentiality

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• Adaptive Bernstein-von Mises theorems in Gaussian white noise
• Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems
• Semiparametric Bayesian causal inference
• The Bayesian Analysis of Complex, High-Dimensional Models: Can It Be CODA?
• Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
• Bayesian inference for spectral projectors of the covariance matrix
• Frequentist properties of Bayesian inequality tests
• Monte Carlo inference for semiparametric Bayesian regression
• Bayesian Semi-supervised Inference via a Debiased Modeling Approach