Bernstein-von Mises theorem for log-concave posteriors

TL;DR

This paper establishes new Bernstein-von Mises theorems for log-concave posteriors, applicable to both well-specified and misspecified models, using convex analysis without smoothness assumptions.

Contribution

It introduces general Bernstein-von Mises results for log-concave models that do not depend on smoothness, broadening applicability in Bayesian asymptotics.

Findings

Proves Bernstein-von Mises theorems under log-concavity

Applicable to both well-specified and misspecified models

Relies solely on convex analysis techniques

Abstract

We prove new, general versions of Bernstein-von Mises theorem for both well-specified and misspecified models when the log-likelihood is concave in the parameter and the prior distribution is log-concave. Unlike classical versions of Bernstein-von Mises theorem, our versions do not require technical smoothness assumptions, and they solely rely on convex analysis.