Misspecified Bernstein-Von Mises theorem for hierarchical models

TL;DR

This paper establishes a Bernstein-von Mises theorem for misspecified hierarchical models, including PDE-based inverse problems, and validates the theory with numerical experiments on synthetic data.

Contribution

It extends Bernstein-von Mises results to complex hierarchical models with non-linear operators and PDE constraints, addressing misspecification issues.

Findings

Theoretical proof of Bernstein-von Mises in misspecified hierarchical models.

Numerical validation on synthetic data for PDE-based inverse problems.

Insights into the asymptotic behavior of Bayesian posterior in complex models.

Abstract

We derive a Bernstein von-Mises theorem in the context of misspecified, non-i.i.d., hierarchical models parametrized by a finite-dimensional parameter of interest. We apply our results to hierarchical models containing non-linear operators, including the squared integral operator, and PDE-constrained inverse problems. More specifically, we consider the elliptic, time-independent Schr\"odinger equation with parametric boundary condition and general parabolic PDEs with parametric potential and boundary constraints. Our theoretical results are complemented with numerical analysis on synthetic data sets, considering both the square integral operator and the Schr\"odinger equation.