Bounding interventional queries from generalized incomplete contingency tables

Abstract: We introduce a method for evaluating interventional queries and Average Treatment Effects (ATEs) in the presence of generalized incomplete contingency tables (GICTs), contingency tables containing a full row of random (sampling) zeros, rendering some conditional probabilities undefined. Rather than discarding such entries or imputing missing values, we model the unknown probabilities as free parameters and derive symbolic expressions for the queries that incorporate them. By extremizing these expressions over all values consistent with basic probability constraints and the support of all variables, we obtain sharp bounds for the query of interest under weak assumptions of small missing frequencies. These bounds provide a formal quantification of the uncertainty induced by the generalized incompleteness of the contingency table and ensure that the true value of the query will always lie within them. The framework applies independently of the missingness mechanism and offers a conservative yet rigorous approach to causal inference under random data gaps.