  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.
