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This paper addresses the problem of estimating causal effects when adjustmentvariables in the back-door or front-door criterion are partially observed. Forsuch scenarios, we derive bounds on the causal effects by solving twonon-linear optimization problems, and demonstrate that the bounds aresufficient. Using this optimization method, we propose a framework fordimensionality reduction that allows one to trade bias for estimation power,and demonstrate its performance using simulation studies.

Bounds on Causal Effects and Application to High Dimensional Data