The Bernstein-von Mises theorem and non-regular models

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 the 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 that depend on the behaviour of the prior distribution on the boundary and have a faster rate of convergence. We also show 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.