Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation
Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation
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统计反问题的伯恩斯坦-冯·米塞斯定理 I：薛定谔方程
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Richard Nickl
中科院分区：
文献类型：
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作者：
Richard Nickl
本文研究了偏微分方程u - fu =0\mathcal O ~\text{s. t.}中确定未知势f>0的反问题。} u = g \text { on } \partial \mathcal O，$$其中$\mathcal O$是$\mathbb R^d$中的有界$C^\infty$-域，$g>0$是给定的规定边界值的函数。数据由被加性高斯噪声破坏的解u组成。一个非参数贝叶斯函数f$前设计和伯恩斯坦-冯米塞斯定理证明，这需要的后验分布的意见是近似在一个合适的函数空间由无限维高斯测量，具有“最小”的协方差结构在信息理论意义上。因此，后验分布在小噪声限制下对$f$执行有效和最优的频率统计推断。
The inverse problem of determining the unknown potential $f>0$ in the partial differential equation $$\frac{\Delta}{2} u - fu =0 \text{ on } \mathcal O ~~\text{s.t. } u = g \text { on } \partial \mathcal O,$$ where $\mathcal O$ is a bounded $C^\infty$-domain in $\mathbb R^d$ and $g>0$ is a given function prescribing boundary values, is considered. The data consist of the solution $u$ corrupted by additive Gaussian noise. A nonparametric Bayesian prior for the function $f$ is devised and a Bernstein - von Mises theorem is proved which entails that the posterior distribution given the observations is approximated in a suitable function space by an infinite-dimensional Gaussian measure that has a `minimal' covariance structure in an information-theoretic sense. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on $f$ in the small noise limit.
$X$ 射线变换的高效非参数贝叶斯推理
DOI：
10.1214/18-aos1708
发表时间：
2019
期刊：
The Annals of Statistics
影响因子：
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作者：
Monard, François;Nickl, Richard;Paternain, Gabriel P.
通讯作者：
Paternain, Gabriel P.