发表机构
California Institute of Technology（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究贝叶斯反问题中先验到后验映射在 Wasserstein 度量下的稳定性，通过似然正则性和先验类条件建立一致、Hölder 与 Lipschitz 稳定性，并数值验证于 Darcy 反问题。
AI 中文摘要
贝叶斯反问题中的先验通常通过离散化、超参数估计或生成建模来近似。因此，理解先验近似误差如何传播到后验及后续预测非常重要。在本工作中，我们研究先验到后验映射的稳定性，其中先验和后验的扰动均在同一 Wasserstein 度量 $W_p$（$p\geq1$）下测量。我们确定了似然正则性和可容许先验类的条件，这些条件确保了一致、Hölder 和 Lipschitz 稳定性。对于在有界集上一致连续的有界似然，当先验类具有一致可积的 $p$ 阶矩和共同的正证据下界时，一致稳定性成立。在似然具有全局 Hölder 正则性和先验高阶矩一致有界的条件下，耦合论证导出了具有锐利指数的 Hölder 估计。对于 $p>1$，Lipschitz 似然不一定给出 Lipschitz 稳定性，即使先验具有有界支撑。我们通过插值论证，在一致 Poincaré 界和具有在先验下一致有界本质振荡的全局 Lipschitz 负对数似然条件下，建立了 Lipschitz 稳定性。对于可分 Hilbert 空间上的高斯先验和具有加性高斯噪声的有界 Lipschitz 正演模型，这些估计给出了后验 $W_2$ 界，即使对于相互奇异的先验扰动也成立。针对两层 Darcy 反问题的数值实验验证了预测的 Hölder 和 Lipschitz 速率，以及由此产生的对 Lipschitz 感兴趣量的后验均值和标准差误差的控制。
英文摘要
Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior map where both prior and posterior perturbations are measured in the same Wasserstein metric $W_p$, $p\geq1$. We identify verifiable conditions on likelihood regularity and admissible prior classes that ensure uniform, Hölder, and Lipschitz stability. For bounded likelihoods that are uniformly continuous on bounded sets, uniform stability holds over prior classes with uniformly integrable $p$-th moments and a common positive evidence lower bound. With global Hölder regularity of the likelihood and uniform bounds on higher prior moments, a coupling argument leads to a Hölder estimate with a sharp exponent. For $p>1$, a Lipschitz likelihood need not give Lipschitz stability, even for priors with bounded support. We establish Lipschitz stability through an interpolation argument under uniform Poincaré bounds and a globally Lipschitz potential with uniformly bounded essential oscillation under the priors. For Gaussian priors with additive Gaussian noise and bounded Lipschitz forward models, these estimates give posterior $W_2$ bounds even for mutually singular prior perturbations. Numerical experiments for a Darcy inverse problem illustrate the predicted Hölder and Lipschitz rates and the resulting control of errors in the posterior mean and standard deviation of a Lipschitz quantity of interest.
Comments27 pages, 2 figures. Compare to v1: the proof of Theorem 5.2 is simplified, Example 4.4 is updated, related work is updated, and minor language edits are included
