Abstract

A causal query will commonly not be identifiable from observed data, in which case no estimator of the query can be contrived without further assumptions or measured variables, regardless of the amount or precision of the measurements of observed variables. However, it may still be possible to derive symbolic bounds on the query in terms of the distribution of observed variables. Bounds, numeric or symbolic, can often be more valuable than a statistical estimator derived under implausible assumptions. Symbolic bounds, however, provide a measure of uncertainty and information loss due to the lack of an identifiable estimand even in the absence of data. We develop and describe a general approach for computation of symbolic bounds and characterize a class of settings in which our method is guaranteed to provide tight valid bounds. This expands the known settings in which tight causal bounds are solutions to linear programs. We also prove that our method can provide valid and possibly informative symbolic bounds that are not guaranteed to be tight in a larger class of problems. We illustrate the use and interpretation of our algorithms in three examples in which we derive novel symbolic bounds. Supplementary materials for this article are available online.

摘要

因果查询通常无法从观察到的数据中识别出来，在这种情况下，无论观察到的变量的测量的数量或精度如何，如果没有进一步的假设或测量变量，就无法设计出查询的估计器。然而，仍然有可能根据观察到的变量的分布得出查询的符号界限。界限，无论是数字的还是符号的，通常比在不合理的假设下得出的统计估计更有价值。然而，即使在没有数据的情况下，由于缺乏可识别的估计值，符号界限也提供了不确定性和信息丢失的衡量标准。我们开发并描述了一种计算符号边界的通用方法，并描述了一类设置，在这些设置中我们的方法保证提供严格的有效边界。这扩展了已知的设置，其中严格的因果界限是线性规划的解决方案。我们还证明，我们的方法可以提供有效且可能提供信息的符号界限，但在较大类问题中不能保证这些符号界限是严格的。我们用三个例子来说明我们的算法的使用和解释，在这些例子中我们推导出新的符号界限。本文的补充材料可在线获取。