We consider causal models with two observed variables and one latent variables, each variable being discrete, with the goal of characterizing the possible distributions on outcomes that can result from controlling one of the observed variables. We optimize linear functions over the space of all possible interventional distributions, which allows us find properties of the interventional distribution even when we cannot uniquely identify what it is. We show that, under certain mild assumptions about the correlation between controlled variable and the latent variable, the resulting interventional distribution must be close to the observed conditional distribution in a quantitative sense. Specifically, we show that if the observed variables are sufficiently highly correlated, and the latent variable can only take on a small number of distinct values, then the variables will remain causally related after passing to the interventional distribution. Another result, possibly of more general interest, is a bound on the distance between the interventional distribution and the observed conditional distribution in terms of the mutual information between the controlled variable and the latent variable, which shows that the controlled variable and the latent variable must be tightly correlated for the interventional distribution to differ significantly from the observed distribution. We believe that this type of result may make it possible to rigorously consider 'weak' experiments, where the causal variable is not entirely independent from the environment, but only approximately so. More generally, we suggest a connection between the theory of causality to polynomial optimization, which give useful bounds on the space of interventional distributions.

中文翻译：

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因果通道

我们考虑具有两个观察变量和一个潜在变量的因果模型，每个变量都是离散的，目的是表征控制一个观察变量可能导致的结果的可能分布。我们在所有可能的干预分布的空间上优化线性函数，这使我们即使无法唯一地识别它是什么，也可以找到干预分布的属性。我们表明，在关于控制变量和潜在变量之间的相关性的某些温和假设下，所得的干预分布在定量意义上必须接近观察到的条件分布。具体来说，我们表明，如果观察到的变量具有足够高的相关性，并且潜在变量只能采用少量的不同值，那么变量在传递到干预分布后将保持因果相关。另一个可能更普遍感兴趣的结果是，根据受控变量和潜变量之间的互信息，限制了干预分布和观察到的条件分布之间的距离，这表明受控变量和潜变量必须与干预分布紧密相关，以使其与观察到的分布明显不同。我们认为，这种类型的结果可能使我们有可能严格考虑“弱”实验，其中因果变量并非完全独立于环境，而仅与环境无关。更笼统地说，我们建议将因果关系理论与多项式优化之间建立联系，