The semiparametric Bernstein-von Mises theorem

TL;DR

This paper extends the Bernstein-von Mises theorem to semiparametric models, showing under certain conditions that the posterior distribution for the parameter of interest is asymptotically normal, enabling efficient Bayesian inference.

Contribution

It establishes a semiparametric Bernstein-von Mises theorem under interpretable conditions, broadening the scope of Bayesian asymptotic normality results.

Findings

Posterior distribution is asymptotically normal in semiparametric models.

Bayesian estimators achieve efficiency similar to frequentist methods.

Conditions include differentiability, metric entropy, and prior support in Kullback-Leibler neighborhoods.

Abstract

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of H\'{a}jek's convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.