Summary. This paper considers the problem of estimating the average con-trolled direct effect (ACDE) of a treatment on an outcome, in the presence of unmeasured confounders between an intermediate variable and the outcome. Such confounders render the direct effect unidentifiable even in cases where the total effect is unconfounded (hence identifiable). Kaufman et al. (2005) applied a linear programming software to find the minimum and maximum possible values of the ACDE for specific numerical data. In this paper, we apply the symbolic Balke-Pearl (1997) linear programming method to derive closed-form formulas for the upper and lower bounds on the ACDE under various assumptions of monotonicity. These universal bounds enable clini-cal experimenters to assess the direct effect of treatment from observed data with minimum computational effort, and they further shed light on the sign of the direct effect and the accuracy of the assessments.

We use the implicitization procedure to generate polynomial equality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network with hidden variables. We show how we may reduce the complexity of the implicitization problem and make the problem tractable in certain causal Bayesian networks. We also show some preliminary results on the algebraic structure of polynomial constraints. The results have applications in distinguishing between causal models and in testing causal models with combined observational and experimental data. 1

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