Prior laundering: learned priors with inherited, undetectable overconfidence

Ali Siahkoohi, Sina Alemohammad

Learned generative priors are increasingly used for ill-posed Bayesian inverse problems, their posterior uncertainty treated as earned from data. But training one requires truths, scarce in seismic and medical imaging, so the recourse is an archive of legacy reconstructions---prior laundering. Where the measurements are uninformative the posterior reverts to the prior, so the uncertainty reported there is the archive's, not the data's, and nothing in deployment reveals it: truths differing only on those directions induce identical data laws, and self-consistency checks such as simulation-based calibration pass whatever the prior believes. That belief has an exact source: averaging the legacy posterior over the measurements yields the old regularizer advanced a single expectation--maximization step---improved where the data resolve, frozen where they cannot. It is overconfident wherever the inherited belief is tighter than the truth. A single-best archive is worse, collapsing the blind credible interval to zero width. Deployed, a diffusion prior fit to the archive under-covers the operator's blind subspace, unlike a truth-trained control, and a normalizing flow does the same on a nonlinear groundwater operator. We recommend reporting which directions the measurements resolve, separating confidence the data support from belief inherited through the pipeline.