Same Marginals, Different Query Probabilities: Diagnosing Why Relational Predictors Fail

Low-order marginals do not determine the probability of a Boolean query over uncertain relational data: even- and odd-parity laws on four edges agree on every marginal of up to three edges, yet assign different probabilities to the same query. When a relational predictor errs on such a query, a single error number cannot say why. We propose an information-sufficiency audit that separates three failure sources: query-interface loss, the information discarded by the query representation; finite-context uncertainty, the Bayes risk left by finitely many partially observed worlds; and estimator excess, the risk a predictor adds beyond that Bayes risk. We prove that, under exchangeable dependence on a group of any size, two monotone queries have equal probability under every such law exactly when their union-size signatures agree. The signature predicts which query representations lose information: encoding only the edges a query names imposes an error floor of 0.028 on a four-edge conjunction and a two-pair disjunction, while clause counts and sizes fail on four edges only for a class containing star and path queries, a failure that becomes common on larger groups. On a factorial benchmark we compute Bayes risks from exact posteriors and bracket the excess of each learned predictor. On star and path queries, the network given full incidence does not resolve an improvement beyond the shape-only Bayes risk, so its apparent gain from incidence is explained by excess. With ten times more training data, ensembles are no longer resolved above the query-edge Bayes risk; with thirty times more, the ensemble falls below the query-edge Bayes risk by using edges outside the query, on the benchmark and on independent confirmation tasks, whereas the mean single run is not resolved below that risk.