Priors learned from legacy reconstructions inherit undetectable overconfidence

Abstract

Where truths are scarce (e.g., seismic and medical imaging), a prior for an ill-posed inverse problem is trained on an archive of legacy reconstructions---an older method's outputs---and its uncertainty is treated as data-driven. In the population limit, an archive of posterior samples is the regularizer that produced it, advanced one expectation-maximization step toward the truth. On directions the operator resolves, it improves the assumption; on its blind subspace, the step is the identity, so the assumption survives unchanged however often it is rebuilt. An archive of single-best reconstructions, one per survey, keeps no spread there: the blind interval collapses whatever the penalty was, so error becomes overconfidence. The assumption enters as the archive and leaves as a reported spread, and nothing in deployment tests it. Two truths differing only there share the data law, and no procedure using survey and archive alone can both report a finite blind interval and guarantee coverage over indistinguishable truths. The question requires information the survey does not carry. We provide a resolvability statement that names affected directions from the operator, state how many reference truths are needed to test a prior on them, and use those references to build an interval that contains the truth as claimed, even if the prior is wrong. On synthetic experiments with seismic and groundwater operators, the archive-trained prior's intervals contain the truth less often on the blind subspace than on resolved directions, while the truth-trained control shows no gap of that sign or size. On seismic, a random subspace of the same size gives the same result, so the separation follows the operator. On groundwater, rebuilding the archive using the survey and a handful of boreholes brings the prior's reports toward the truth on directions those boreholes reach, while leaving the rest unchanged.