Hankel-Christoffel-Nevai 筛选在贝叶斯反问题中的后验相关性
Hankel--Christoffel--Nevai Screening for Bayesian Inverse Problems
查看机构详情
- University of Electronic Science and Technology of China（电子科技大学）
- Southwest Jiaotong University（西南交通大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
提出 Hankel-Christoffel-Nevai 框架，利用似然加权矩矩阵和 Christoffel/Nevai 评分高效筛选贝叶斯反问题中的后验相关候选，并在非线性及 PDE 反问题中验证了计算节省。
中文摘要 AI 辅助
我们引入了一个 Hankel-Christoffel-Nevai 框架，用于在贝叶斯反问题中筛选后验相关候选。一个似然加权矩矩阵记录了贝叶斯更新如何改变先验的几何结构，而 Christoffel 和 Nevai 构造将此信息转换为廉价的关联性评分。Christoffel 比率捕获相对的局部矩质量，而 Nevai 评分提供了似然的稳定多项式局部近似。矩矩阵可以从确定性似然值、有界无偏标记、后验样本或二元无偏似然观测中估计。该构造通过嵌套特征映射扩展到函数值未知量，其中诱导的条件似然具有精确的贝叶斯解释。我们建立了将特征分辨率、多项式局部化和试点估计分开的一致性及误差界。相同的学习几何也可用于重新排序精确随机似然块，在不变后验目标的情况下减少预期工作量。在非线性、基于 PDE 和函数空间反问题上的数值实验展示了有效的后验质量筛选和计算节省。
英文摘要
Likelihood evaluation in Bayesian inverse problems often requires a forward-model solve. We study candidate screening using a likelihood-weighted prior moment matrix and two associated scores: a Christoffel ratio and a Nevai polynomial average. A finite pilot supplies likelihood information that is reused to rank further prior candidates. We connect finite-pilot matrix error, quantitative Legendre localization, and fixed-budget posterior-mass regret. Conditional likelihoods distinguish feature loss from polynomial and sampling errors, and a filtered Gaussian construction permits controlled feature selection and localization. In a nonlinear function-coefficient PDE, equal-budget comparisons separate methods using scalar likelihoods from surrogates using forward outputs. The Christoffel ratio outperforms the tested direct likelihood regressions, but forward-response surrogates capture more posterior mass at small retention budgets. Independent-pilot tests and reference resampling quantify two distinct sources of uncertainty. Controlled Gaussian experiments illustrate finite-filter truncation and the benefit of selecting relevant features. These results identify when moment geometry provides useful screening information, without asserting universal superiority over surrogate models or an exact posterior sampling method.
