Bernstein - von Mises Theorem for growing parameter dimension

TL;DR

This paper extends classical statistical theorems like Bernstein-von Mises to high-dimensional settings, providing nonasymptotic bounds and conditions for their validity when the parameter dimension grows with the sample size.

Contribution

It offers a nonasymptotic analysis of BvM and related results in high-dimensional models, addressing model misspecification and finite samples.

Findings

BvM holds when p^3/n is small for large dimensions

Fisher expansion valid when p^2/n is small

Results accommodate model misspecification and finite samples

Abstract

This paper revisits the prominent Fisher, Wilks, and Bernstein -- von Mises (BvM) results from different viewpoints. Particular issues to address are: nonasymptotic framework with just one finite sample, possible model misspecification, and a large parameter dimension. In particular, in the case of an i.i.d. sample, the mentioned results can be stated for any smooth parametric family provided that the dimension \(p \) of the parameter space satisfies the condition "\(p^{2}/n \) is small" for the Fisher expansion, while the Wilks and the BvM results require "\(p^{3}/n \) is small".