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February 2012 The semiparametric Bernstein–von Mises theorem
P. J. Bickel, B. J. K. Kleijn
Ann. Statist. 40(1): 206-237 (February 2012). DOI: 10.1214/11-AOS921


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.


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P. J. Bickel. B. J. K. Kleijn. "The semiparametric Bernstein–von Mises theorem." Ann. Statist. 40 (1) 206 - 237, February 2012.


Published: February 2012
First available in Project Euclid: 29 March 2012

zbMATH: 1246.62081
MathSciNet: MR3013185
Digital Object Identifier: 10.1214/11-AOS921

Primary: 62G86
Secondary: 62F15 , 62G20

Keywords: Asymptotic posterior normality , Bernstein–Von Mises , efficiency , local asymptotic normality , model differentiability , posterior limit distribution , regular estimation , semiparametric statistics

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • February 2012
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