PACT: Learning Couplings under Empirical Marginal Constraints

Observations from different sources often provide rich information about individual distributions but limited evidence of how outcomes correspond. Learning reliable joint models from such partially paired data is a common challenge in conditional prediction and risk assessment. We introduce Pair-Aligned Conditional Transport (PACT), which combines relation learning with marginal calibration. We show that empirical marginal constraints during training not only shape model outputs but also systematically alter relation learning: adding independent endpoint observations can even reverse the direction of the leading learning bias. Building on this finding, we establish a theoretical link between objective bias and parameter displacement and derive a correction for first-order expected objective bias. Controlled experiments validate the predictions of the analysis, while applications demonstrate PACT's value for conditional prediction and cross-policy risk modeling. This work provides a statistical basis for understanding how dispersed observations jointly influence the learning of joint distributions.