Constraint-Based Framework for Bounding Probabilities of Causation under Partial Structural Assumptions and Auxiliary Statistical Information

Probabilities of causation (PoCs) are fundamental to individual level explanation and decision making, yet they are inherently counterfactual and not point identifiable from data in general. Existing bounds either disregard available auxiliary information, require complete structural assumptions and auxiliary statistical information, or rely on restrictive binary settings, limiting their practical use. In real world applications, structural assumptions and auxiliary statistical information are often incomplete but still informative. This paper proposes a unified constraint based framework for bounding PoCs using partial structural assumptions and partial auxiliary statistical information. We show how partial structural assumptions and partial auxiliary statistical information can be systematically incorporated as constraints in an optimization formulation, even when such information is available only for subsets of variables or covariate values. The resulting framework yields tighter and formally valid bounds without full identifiability and naturally accommodates multiple sources of incomplete structural and statistical information within a single optimization program. This approach extends the applicability of PoCs to realistic settings where both structural assumptions and auxiliary statistical information are incomplete.