We study asymptotic frequentist coverage and approximately Gaussian properties of Bayes posterior credible sets in nonlinear inverse problems when a Gaussian prior is placed on the parameter of the PDE. The aim is to ensure valid frequentist coverage of Bayes credible intervals when estimating continuous linear functionals of the parameter. Our results show that Bayes credible intervals have conservative coverage under certain smoothness assumptions on the parameter and a compatibility condition between the likelihood and the prior, regardless of whether an efficient limit exists or Bernstein von-Mises (BvM) theorem holds. In the latter case, our results yield a corollary with more relaxed sufficient conditions than previous works. We illustrate the practical utility of the results through the example of estimating the conductivity coefficient of a second order elliptic PDE, where a near-$N^{-1/2}$ contraction rate and conservative coverage results are obtained for linear functionals that were shown not to be estimable efficiently.

翻译：我们研究了当偏微分方程参数采用高斯先验时，非线性反问题中贝叶斯后验可信集的渐近频域覆盖特性及其近似高斯性质。目标在于确保估计参数连续线性泛函时，贝叶斯可信区间具有有效的频域覆盖性。研究结果表明：在参数满足特定光滑性假设、似然函数与先验分布满足相容性条件的前提下，无论是否存在有效极限或伯恩斯坦-冯·米塞斯定理是否成立，贝叶斯可信区间均具有保守覆盖特性。在后一种情形下，我们的研究结果推导出比以往工作条件更宽松的充分条件推论。通过估计二阶椭圆型偏微分方程传导系数的实例，我们展示了该结果的实际应用价值：对于已被证明无法有效估计的线性泛函，我们获得了接近$N^{-1/2}$的收缩率与保守覆盖结果。