The Bernstein-von Mises theorem and nonregular models

TL;DR

This paper investigates the asymptotic behavior of Bayesian posterior distributions in models where the true parameter lies on the boundary, revealing Gaussian and Gamma components and implications for estimation efficiency.

Contribution

It extends the Bernstein-von Mises theorem to nonregular models with boundary parameters, showing new distributional components and efficiency properties.

Findings

Posterior distribution includes Gaussian and Gamma components.

Bayesian inference remains consistent on the boundary.

No bound on estimation efficiency for some boundary parameters.

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.