Semiparametric Bernstein-von Mises Theorem: Second Order Studies

TL;DR

This paper investigates second order properties of Bayesian estimators in semiparametric models, revealing an interference phenomenon and proposing dependent priors that enhance efficiency and adaptivity, supported by theoretical results and simulations.

Contribution

It introduces a second order semiparametric Bernstein-von Mises theorem, highlighting the impact of nuisance function estimation on parametric inference and proposing new dependent priors for improved adaptivity.

Findings

More accurate nuisance estimation improves parametric inference accuracy.

Dependent priors enable adaptive and efficient Bayesian inference.

Independent priors may fail to achieve optimal rates without strict assumptions.

Abstract

The major goal of this paper is to study the second order frequentist properties of the marginal posterior distribution of the parametric component in semiparametric Bayesian models, in particular, a second order semiparametric Bernstein-von Mises (BvM) Theorem. Our first contribution is to discover an interesting interference phenomenon between Bayesian estimation and frequentist inferential accuracy: more accurate Bayesian estimation on the nuisance function leads to higher frequentist inferential accuracy on the parametric component. As the second contribution, we propose a new class of dependent priors under which Bayesian inference procedures for the parametric component are not only efficient but also adaptive (w.r.t. the smoothness of nonparametric component) up to the second order frequentist validity. However, commonly used independent priors may even fail to produce a desirable root-n contraction rate for the parametric component in this adaptive case unless some stringent assumption is imposed. Three important classes of semiparametric models are examined, and extensive simulations are also provided.