Causal Channels
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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.
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