无力收敛的后验收敛：粗糙贝叶斯逆问题的分辨率稳定采样
Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems
- School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
- School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对粗糙贝叶斯逆问题，提出分辨率稳定采样方法，揭示力收敛与后验收敛的不匹配，通过理论分析与数值实验验证其在达西流等场景的有效性。
AI中文摘要：
即使基于梯度的采样器使用的精确灵敏度不收敛，贝叶斯后验也能在模型细化下收敛。本文研究离散尺度不变粗糙势的这种不匹配及其对马尔可夫链哈密顿量提议（Metropolized Hamiltonian proposals）的影响。对于魏尔斯特拉斯截断（Weierstrass truncations），相邻经典力增量在粗糙区域呈几何增长，而固定加性商一致收敛，固定乘性商在远离原点处局部收敛。匹配的杰克逊商（Jackson quotient）不仅因收敛性，还因膨胀协方差及在固有尺度处的精确有限闭包而具有特殊性。一致负对数似然近似被证明可隐含显式的总变差、赫尔利格及有界兴趣量收敛，适用于贝叶斯逆问题的高斯前向映射准则。在算法层面，可测的“冲击-漂移-冲击”映射为三角剪切，因此精确的马尔可夫校正无需提议场的可微性。局部一致场收敛产生固定长度提议、接受函数及常见紧集上的有界利普希茨核的收敛，而首个经典哈密顿蒙特卡洛（HMC）半步无固定步长细化极限。数值实验验证了预测的力细化率，表明后验稳定可与精确梯度HMC的严重分辨率相关重调共存，并在二维达西流（Darcy flow）中展示无灵敏度收敛的后验收敛。计算用于评估细化一致性，而非声称提议场间的通用效率排序。
英文摘要：
Bayesian targets may converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this probability--sensitivity mismatch and its consequences for Metropolized Hamiltonian proposals. A vanishing-amplitude wiggly-energy model first gives the basic analytic obstruction: the potential perturbation tends to zero while its classical derivative is of order $r_\varepsilon/\varepsilon$. We then show that the same scaling arises naturally in a periodic elliptic inverse problem, where homogenization makes the forward map and Gaussian likelihood converge while differentiation with respect to a microscopic scale parameter retains an $O(1)$ oscillatory contribution. This provides a PDE origin for the single-scale wiggly mechanism. The main construction concerns a more demanding nested Weierstrass hierarchy, interpreted as an analytically tractable prototype for repeated corrector contributions across geometrically separated scales. There all previously resolved scales persist, adjacent classical-force increments grow geometrically like $(ab)^N$, and the limiting rough component may fail to possess a classical derivative. In this self-similar setting the matched Jackson quotient is structurally adapted to the refinement through dilation covariance and exact finite closure. Uniform negative-log-likelihood approximation yields explicit total-variation, Hellinger, and bounded quantity-of-interest bounds. Measurable kick--drift--kick maps remain exact after Metropolis correction, local field convergence propagates to fixed-length proposals and kernels, and the first classical HMC half-kick can have no fixed-step refinement limit. Numerical experiments on scale-structured inverse problems test the resulting resolution-stability mechanism across one- and two-dimensional inverse problems.
