论文标题
高斯工艺先验的非线性反问题中贝叶斯不确定性定量的统计保证
Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors
论文作者
论文摘要
考虑了一类非线性反回归模型中的贝叶斯推断和不确定性定量。提供了回归模型$ \ {\ mathscr g（θ）的分析条件：θ\} $中的θ\和$θ$的高斯工艺先验，因此，对于$θ$的大量线性功能，可以进行半参数效率的推断。证明了一般的半参数伯恩斯坦 - 冯·米斯定理，表明（非高斯）后分布是由以后均值为中心的某些高斯措施近似的。因此，从经常观点的角度来看，基于后验的可靠集是有效的。该理论用在非线性断层扫描问题中出现的两个PDE的应用进行了说明：schrödinger方程的椭圆逆问题和非亚伯X射线变换的反转。部署了新的分析技术，以表明相关的Fisher信息操作员在合适的功能空间之间可逆
Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(θ): θ\in Θ\}$ and on Gaussian process priors for $θ$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $θ$. 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 valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in non-linear tomography problems: an elliptic inverse problem for a Schrödinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces
