When Noise Estimation Hides Basis Misspecification in Repeated Bayesian Inverse Problems

2026-09-29

AI Summary

This study addresses the failure of prior coverage in Bayesian inverse problems when the true signal lies outside the basis space, a deficiency often masked by noise variance estimation that absorbs out-of-model energy. To resolve this, we elucidate the mechanism by which noise estimation conceals basis mismatch and propose John’s test of sphericity based on the spectral shape of residuals. Specifically, we construct a scale-invariant residual direction statistic and derive its exact null distribution for finite samples. Experiments on synthetic data and GEBCO bathymetry demonstrate that the proposed method detects structured out-of-basis variations overlooked by cross-validation, substantially enhancing model diagnostic capabilities in Bayesian inverse problems.

Abstract

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.

Problem

• Bayesian inverse problems
• basis misspecification
• noise estimation
• coverage loss
• repeated inverse problems

Innovation

• Bayesian inverse problems
• basis misspecification
• noise estimation
• John's sphericity statistic
• residual spectrum