发表机构
University of Electronic Science and Technology of China; Southwest Jiaotong University（电子科技大学; 西南交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于伯努利事件与泊松点过程细化的贝叶斯反问题精确采样框架，通过似然硬币构造实现后验采样，并给出尖锐的计算复杂度分析，适用于椭圆预解问题。
AI 中文摘要
我们为贝叶斯反问题开发了一种精确的直接采样框架，其中有界的前向可观测值可以通过伯努利事件进行查询，而非数值评估。伯恩斯坦-泊松构造将这些事件转化为缩放的高斯似然硬币。用这些硬币对基于先验的泊松点过程进行细化，得到一个后验点过程，其位置在给定基数条件下是独立的后验样本；基数还提供了缩减证据的无偏估计量。主要理论贡献是对实际提前停止的计算工作量进行了尖锐分析。在小噪声区域，工作量由精确拟合集附近的局部先验预测质量决定，揭示了与未知量名义维度不同的预测维度效应。多元扩展可处理相关的高斯观测误差，并给出工厂努力的精度加权分配。对于一类有界的椭圆预解问题，费曼-卡茨表示、泊松杀死和惰性随机级数评估提供了精确的连续伯努利预言机，而无需在后验目标中引入空间离散化或固定参数截断。数值实验展示了参数空间中的后验细化，确认了尖锐的工作制度，并验证了函数值偏微分方程构造。
英文摘要
We develop an exact posterior-sampling framework for Bayesian inverse problems when selected bounded forward observables can be accessed through Bernoulli events. A Bernstein--Poisson construction converts these forward coins into scaled Gaussian likelihood coins, and thinning an inflated prior Poisson point process yields posterior atoms that are iid conditional on their number. For independent Gaussian observations, we derive an exact mean-work identity for the implemented early-stopped factory and sharp small-noise complexity laws governed by local prior-predictive mass near the exact-fit set; a factorial-moment construction extends the likelihood factory to correlated Gaussian errors. The posterior algorithm is model-agnostic once Bernoulli access is available. As one continuum realization, we use Feynman--Kac sampling, Poisson killing, and lazy random-series evaluation for a bounded elliptic resolvent problem with a function-valued coefficient. Numerical experiments validate the forward and likelihood coins, support the predicted work regimes, and demonstrate posterior sampling without deterministic spatial discretization or fixed parameter truncation in the target. A matched-accuracy benchmark against finite-difference prior rejection illustrates how deterministic discretization bias changes the posterior accuracy--cost balance.
