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We present a class of stochastic interventions for causal inference with discrete treatments defined through the marginals of the solution to a cost-penalized Kullback–Leibler projection problem. This formulation seeks a stochastic plan that minimizes the expected cost of reallocating treatments while penalizing the divergence from the independent product of the organic propensity scores and a prespecified target distribution. This problem arises as a limiting case of a relaxed optimal transport problem without marginal constraints. The resulting source marginal yields a generalization of incremental propensity score interventions (IPIs), accommodating arbitrary target distributions and cost structures. Like classical IPIs, these interventions are governed by a tunable parameter and do not require the positivity assumption for identification. We derive the influence functions for the corresponding expected outcomes and develop semiparametric estimators that remain robust under model misspecification, as demonstrated in a simulation study. We illustrate the practical utility of these methods by analyzing policies for ADHD treatment and their impact on children’s academic achievement.
