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

翻译：本文件探讨了在部分遵守后门或前门标准中的调整变量时估计因果影响的问题。对于这些情况,我们通过解决两个非线性优化问题来得出因果关系的界限,并表明界限已经足够。我们使用这种优化方法提出了一个维度削减框架,允许人们用偏差来交换估计力,并利用模拟研究来展示其性能。