### 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.

Original language | English |
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Pages (from-to) | 1850-1878 |

Number of pages | 29 |

Journal | Annals of Statistics |

Volume | 42 |

DOIs | |

Publication status | Published - 2014 |

### Keywords

- Approximate posterior, Bayesian inference, Bernstein–von Mises theorem, boundary, nonregular, posterior concentration, SPECT, tomography, total variation distance, variance estimation in mixed models.

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## Profiles

## Professor Peter J Green

- Statistical Science
- School of Mathematics - Professorial Research Fellow
- Cabot Institute for the Environment
- Statistics

Person: Academic , Member

## Cite this

Bochkina, N. A., & Green, P. J. (2014). The Bernstein-von Mises theorem and non-regular models.

*Annals of Statistics*,*42*, 1850-1878. https://doi.org/10.1214/14-AOS1239