Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors
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DOI：
10.1214/21-aos2082
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发表时间：
2020-07
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影响因子：
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通讯作者：
F. Monard;Richard Nickl;G. Paternain
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文献类型：
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作者：
F. Monard;Richard Nickl;G. Paternain
研究了一类非线性逆回归模型的贝叶斯推断和不确定性量化问题。分析条件的回归模型$\{\mathscr G（\theta）：\theta \in \Theta\}$和高斯过程先验的$\theta$提供这样的半参数有效的推理是可能的一个大类的线性泛函的$\theta$。一个一般的半参数Bernstein-von Mises定理证明，表明（非高斯）后验分布近似的高斯措施集中在后验均值。作为一个结果后基于可信集被证明是有效的，从频率论的角度来看是最佳的。该理论被证明涵盖了两个典型的应用程序中出现的非线性层析成像问题的偏微分方程：第一个涉及的椭圆形的Schr\“odinger方程的反问题，第二个非阿贝尔$X$-射线变换的反演。新的偏微分方程技术的发展表明，相关的Fisher信息算子之间的适当的函数空间是可逆的。
Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ and on Gaussian process priors for $\theta$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $\theta$. A general semi-parametric Bernstein-von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence posterior-based credible sets are shown to be valid and optimal from a frequentist point of view. The theory is demonstrated to cover two prototypical applications with PDEs that arise in non-linear tomography problems: the first concerns an elliptic inverse problem for the Schr\"odinger equation, and the second the inversion of non-Abelian $X$-ray transforms. New PDE techniques are developed to show that the relevant Fisher information operators are invertible between suitable function spaces.