The Bernsteinvon 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 Bernsteinvon 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1211.3434
 Bibcode:
 2012arXiv1211.3434B
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.1214/14AOS1239 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)