当噪声估计掩盖重复贝叶斯反问题中的基失配时

中文摘要 AI 辅助

在贝叶斯反问题中，当真实值含有基外成分时，基受限先验可能失去覆盖。我们证明，估计观测噪声方差可以掩盖这种损失。在线性前向模型下，当基内先验方差主导噪声时，极大似然噪声估计会吸收模型值域补集中的基外能量。残差幅度和观测覆盖检验因此保持在名义水平附近，而场覆盖下降。我们研究共享同一前向算子和同一基的重复问题，该基独立于测试数据固定。在投影到补集后，且条件于拟合的噪声尺度，每个精确检验都是对残差的无尺度方向的检验。我们用John的球度统计量检验其样本谱的形状。在高斯噪声下，其零模型在有限样本量下是精确的，我们推导了其零均值和在比例维度下的功效。在合成问题和预注册的GEBCO地形研究中，该检验检测到交叉验证和观测覆盖检验大多遗漏的结构化基外变异。它无法检测补集中各向同性的高斯基外变异，因为这与噪声尺度的变化无法区分。

英文摘要

Basis-restricted priors in Bayesian inverse problems can lose coverage when the truth has components outside the basis. We show that estimating the observation-noise variance can hide this loss. Under a linear forward model, when the in-span prior variance dominates the noise, the maximum-likelihood noise estimate absorbs the out-of-basis energy in the complement of the model range. Residual-magnitude and observation-coverage checks then stay near nominal while field coverage falls. We study repeated problems sharing one forward operator and one basis, fixed independently of the tested data. After projection onto the complement, and conditionally on the fitted noise scale, every exact test is a test of the scale-free direction of the residuals. We test the shape of their sample spectrum with John's sphericity statistic. Under Gaussian noise its null model is exact at finite sample size, and we derive its null mean and its power at proportional dimension. On synthetic problems and in a preregistered GEBCO topography study, the test detects structured out-of-basis variation that cross-validation and observation-coverage checks largely miss. It cannot detect Gaussian out-of-basis variation that is isotropic in the complement, since that is indistinguishable from a change of noise scale.

发表机构

• University of Cambridge（剑桥大学）

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