Learning Conditional Distributions from Replicated Aggregates

Learning an individual response distribution from group totals becomes difficult when each measurement carries unknown offsets and the underlying distribution is heavy-tailed. To tackle this, we introduce Pair-Triple Spectral Recovery (PTSR) where pair differences recover the conditional law up to reflection and triple contrasts use spectral phase to resolve the remaining orientation. PTSR provides moment-free recovery while also eliminating the shared nuisance offsets. We provide finite-sample guarantees for reflection selection and connect spectral signal-to-noise to intrinsic reflection information. Across two heavy-tailed benchmarks, PTSR improves conditional-density accuracy and held-out likelihood over winsorized method-of-moments and aggregate-likelihood estimator baselines.