Misspecified Bernstein–von Mises theorem for hierarchical models

Abstract

We derive a Bernstein–von Mises theorem in the context of misspecified, non-i.i.d., hierarchical models parametrised 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 Schrödinger equation with parametric boundary condition and general parabolic PDEs with parametric potential and boundary constraints. Our theoretical results are com-plemented with a numerical analysis of synthetic data sets, considering both the square integral operator and the Schrödinger equation.