## The Annals of Statistics

### The Bernstein–von Mises theorem and nonregular models

#### 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.

#### Article information

Source
Ann. Statist., Volume 42, Number 5 (2014), 1850-1878.

Dates
First available in Project Euclid: 11 September 2014

https://projecteuclid.org/euclid.aos/1410440627

Digital Object Identifier
doi:10.1214/14-AOS1239

Mathematical Reviews number (MathSciNet)
MR3262470

Zentralblatt MATH identifier
1305.62112

#### Citation

Bochkina, Natalia A.; Green, Peter J. The Bernstein–von Mises theorem and nonregular models. Ann. Statist. 42 (2014), no. 5, 1850--1878. doi:10.1214/14-AOS1239. https://projecteuclid.org/euclid.aos/1410440627

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