Marginal Scores under Structured Missingness: Coverage and Joint Recovery

Incomplete records constrain marginal scores, while joint predictions depend on relationships that may never be observed together. We characterize when marginal Fisher error controls joint-score error, and at what cost. For Gaussians with a fixed number of observed coordinates per record, we determine the worst recovery constant up to universal factors throughout the budget range. The conditioning cost is cubic at fixed missing fractions and decreases toward one as observations become sufficiently complete. For density-ratio ANOVA models, we prove a global comparison governed by interaction coverage and a matching quadratic gradient cost. A shared background can increase this cost without changing the observations informative about the contrast. Statistical rates weight each interaction's inverse sample count by its gradient energy, and retain the quadratic cost when the background amplitude is unknown. Mixture experiments examine recovery without a prescribed interaction basis. Full-waveform ECG experiments show that identical acquired marginals can support different joint scores and conditional predictions, with acquisition gains depending on the target and cohort.