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.

翻译：本文针对由有限维感兴趣参数刻画的误指定、非独立同分布分层模型，推导了Bernstein-Von Mises定理。我们将该结果应用于包含非线性算子（包括平方积分算子）及PDE约束反问题的分层模型。具体而言，我们考虑了具有参数边界条件的椭圆稳态薛定谔方程，以及带参数势能与边界约束的一般抛物型偏微分方程。通过数值分析合成数据集（涵盖平方积分算子和薛定谔方程），对理论结果进行了补充验证。