Bernstein-von Mises theorem for log-concave posteriors

TL;DR

This paper develops Bernstein-von Mises type results for Bayesian posteriors under log-concavity, relying solely on convex analysis rather than classical smoothness or domination assumptions. It treats well-specified, nearly misspecified, and misspecified models, showing that the rescaled posterior converges to a Gaussian on the tangent cone in the first two cases, while in the misspecified case the limit concentrates on a second-order tangent set determined by the active constraints of the prior support. The analysis covers broad convex constraint sets for the prior support and introduces geometric objects like the tangent and normal cones to describe the asymptotic shapes. These results extend nonparametric and high-dimensional Bayesian asymptotics to non-smooth settings and provide a principled framework for constrained Bayesian inference under misspecification, with potential extensions to twice-epi-differentiable priors.

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.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:main_case1']}
• Lemma 1
• Theorem 3
• Remark 1
• proof
• Theorem 4
• proof
• Remark 2