SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 342-373, January 2020.
We consider the statistical inverse problem of recovering an unknown function $f$ from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of $f$ corresponds to a Tikhonov regularizer $\bar f$ with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein--von Mises theorem for a large collection of linear functionals of $f$, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The result is applied to study three concrete examples that cover both the mildly and severely ill-posed cases: specifically, an elliptic inverse problem, an elliptic boundary value problem, and the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regularizer $\bar f$ is an efficient estimator of $f$, and we derive frequentist guarantees for certain credible balls centered at $\bar{f}$.

中文翻译：

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线性反问题的Bernstein-von Mises定理和不确定性量化

SIAM / ASA不确定性量化期刊，第8卷，第1期，第342-373页，2020年1月。
我们考虑从加性高斯白噪声破坏的线性测量中恢复未知函数$ f $的统计逆问题。我们采用具有标准高斯先验的非参数贝叶斯方法，为此，基于后验的$ f $重构对应于带有可再生内核希尔伯特空间范数罚分的Tikhonov正则化器\\ bar f $。我们证明了$ f $的线性函数的大量集合的半参数Bernstein-von Mises定理，这意味着从频数观点来看，半参数后验估计和不确定性量化是有效且最优的。该结果被用于研究涵盖轻度和重度不适情况的三个具体示例：特别是椭圆反问题，椭圆边值问题和热方程。对于椭圆边界值问题，我们还获得了定理的非参数形式，该定理需要将后验分布收敛到具有最小协方差的先验独立无限维高斯概率测度。结果，得出Tikhonov正则化器$ \ bar f $是$ f $的有效估计量，并且我们得出了某些可靠球的频繁保证，这些可靠球以$ \ bar {f} $为中心。