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October 2014 The Bernstein–von Mises theorem and nonregular models
Natalia A. Bochkina, Peter J. Green
Ann. Statist. 42(5): 1850-1878 (October 2014). DOI: 10.1214/14-AOS1239

Abstract

We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the “true” solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein–von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography.

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Natalia A. Bochkina. Peter J. Green. "The Bernstein–von Mises theorem and nonregular models." Ann. Statist. 42 (5) 1850 - 1878, October 2014. https://doi.org/10.1214/14-AOS1239

Information

Published: October 2014
First available in Project Euclid: 11 September 2014

zbMATH: 1305.62112
MathSciNet: MR3262470
Digital Object Identifier: 10.1214/14-AOS1239

Subjects:
Primary: 62F12 , 62F15

Keywords: Approximate posterior , Bayesian inference , Bernstein–von Mises theorem , Boundary , nonregular , posterior concentration , SPECT , tomography , total variation distance , variance estimation in mixed models

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 5 • October 2014
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