Priors learned from legacy reconstructions inherit undetectable overconfidence

Abstract

Where truths are scarce (e.g., seismic and medical imaging), learned priors in ill-posed inverse problems are trained on archives of legacy reconstructions---i.e., an older method's outputs---and their reported uncertainty is taken as data-driven. We show that this prior is, in the population limit, exactly the regularizer that produced its archive of posterior samples, advanced one expectation--maximization step toward the truth. While the step improves the regularizer on the directions the measurements resolve, it leaves the regularizer's assumption on the operator's blind subspace unchanged. An archive of single-best reconstructions collapses the blind interval to zero width. Neither error is detectable in practice, as truths differing only on the blind subspace share the data law, and simulation-based calibration is neutral by construction. We identify from the operator alone which directions the measurements do not inform, and, given a handful of ground-truth models, build intervals there that contain the truth as often as they claim to. We validate these findings on a two-dimensional example with closed-form predictions and in controlled experiments on seismic-imaging and groundwater-flow operators, against priors trained on the truth.