The Annals of Statistics

Bernstein–von Mises theorem for linear functionals of the density

Vincent Rivoirard and Judith Rousseau

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In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein–von Mises theorem. The special case of the cumulative distributive function, evaluated at a specific point, is widely considered. In particular, we show that for infinite-dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive, but also the negative, phenomena that can occur in the study of Bernstein–von Mises results.

Article information

Ann. Statist. Volume 40, Number 3 (2012), 1489-1523.

First available in Project Euclid: 5 September 2012

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties

Bayesian nonparametric rates of convergence Bernstein–von Mises adaptive estimation


Rivoirard, Vincent; Rousseau, Judith. Bernstein–von Mises theorem for linear functionals of the density. Ann. Statist. 40 (2012), no. 3, 1489--1523. doi:10.1214/12-AOS1004.

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Supplemental materials

  • Supplementary material: Bernstein–von Mises theorem for linear functionals of the density: Supplementary material. The supplementary material gives the proofs of Proposition 2.1 and of Lemma 2.1.