Per-Input Prior Adequacy: Sources of Information and Conditions of Failure

Tabular foundation models (TFMs) acquire their entire competence from a synthetic prior sampled at meta-training time, and their prediction errors are far from uniform across inputs. Prior evaluation scores a whole dataset with one number, and such a score cannot say where a model is misaligned. Applying those scores per input does not help either. A coverage-style score is uninformative, and a paired improvement ratio is blind to uniform prior gaps because it is a rate statistic. To address this, we propose the reference-averaged log-error ratio (, RA-ratio), a per-input, training-free diagnostic composed of two components, the excess loss of the model over a Bayesian reference and an average of that excess over the reference's prior strength. First, for each input we take the model's log error in excess of that of a closed-form Bayesian reference, the Bayes gap of in-context learning theory. Second, we average that excess over a grid of reference strengths, which removes the reference's strength as a tuning knob rather than selecting it. Third, we validate the diagnostic causally, injecting meta-training tasks from the region it flags and verifying that the error falls there and not in an off-target region. Notably, the RA-ratio attains a mean rank correlation of $\text{abset}\rho =0.76$ with the per-point error on $22$ regression datasets with $3$ seeds, ahead of every reference-free baseline we tried including three conformal variants, and replicates on a second architecture. On a held-out classification family, prior-based per-input diagnostics help where the model's own uncertainty has stopped resolving difficulty, and hurt where it already succeeds.