The inverse problem of determining the unknown potential $f > 0$ in the partial differential equation $$\frac{\Delta}{2} u - fu =0 \: \mathrm{on} \mathcal O, \:\: u = g \mathrm{on} \partial \mathcal O,$$ where $\mathcal O$ is a bounded $C^\infty$-domain in $\mathbb R^d$ and $g > 0$ is a given source function, 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 by an infinite-dimensional Gaussian measure that has a 'minimal' covariance structure in an information-theoretic sense. The function space in which this approximation holds true is shown to carry the finest topology permitted for such a result to be possible. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on $f$ in the small noise limit.

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统计逆问题的伯恩斯坦–冯·米塞斯定理I：薛定er方程

在偏微分方程$$ \ frac {\ Delta} {2} u-fu = 0 \：\ mathrm {on} \ mathcal O，\：\：u中确定未知势$ f> 0 $的反问题= g \ mathrm {on} \ partial \ mathcal O，$$其中$ \ mathcal O $是$ \ mathbb R ^ d $中的有界$ C ^ \ infty $域，并且$ g> 0 $是给定源功能，被认为。数据由加性高斯噪声破坏的解$ u $组成。设计了函数$ f $的非参数贝叶斯先验，并证明了伯恩斯坦–冯·米塞斯定理，该定理表明，给定观测值的后验分布是由信息中具有“最小”协方差结构的无穷维高斯测度近似的-理论意义。显示了近似值成立的函数空间，它具有允许出现这种结果的最佳拓扑。结果，后验分布在小噪声范围内对$ f $进行了有效和最佳的频度统计推断。