Turns out we can deduce causality from correlation, it just requires more than two variables.
This is incorrect without additional assumptions:
Consider the following contrived case. We have a hidden variable H.
H increases B B increases C H decreases C
Depending on the relative strengths of said causations, the correlation of B with C can be arbitrarily close to zero (or even actually zero, although this case is vanishingly unlikely.)
(I believe this is called the Faithfulness assumption, although it’s been too long. The usual handwave is that either we know everything that could potentially affect these variables, or this case is vanishingly improbable to occur, although this latter assumption fails to account for processes that end up optimizing towards zero correlation.)
(The true issue is more “what do you do when no two combinations of variables are independent”, which results in the corner case where you can’t infer much of anything on the causality DAG. Unfortunately, this is generally true, and we handwave by treating low correlations as independent. Unfortunately, P(neutrino hitting A | B) != P(neutrino hitting A | ~B) for pretty much any A and B. (Or vice versa.) The correlation is incredibly tiny, but tiny != 0. (At least assuming A and B are within each others lightcone (neutrinos don’t travel at c, but hopefully you get the gist.). So either we drop true correlations, or allow through spurious ones. Either way this is far more than straight deduction.))
This is incorrect without additional assumptions:
Consider the following contrived case. We have a hidden variable H.
H increases B
B increases C
H decreases C
Depending on the relative strengths of said causations, the correlation of B with C can be arbitrarily close to zero (or even actually zero, although this case is vanishingly unlikely.)
(I believe this is called the Faithfulness assumption, although it’s been too long. The usual handwave is that either we know everything that could potentially affect these variables, or this case is vanishingly improbable to occur, although this latter assumption fails to account for processes that end up optimizing towards zero correlation.)
(The true issue is more “what do you do when no two combinations of variables are independent”, which results in the corner case where you can’t infer much of anything on the causality DAG. Unfortunately, this is generally true, and we handwave by treating low correlations as independent. Unfortunately, P(neutrino hitting A | B) != P(neutrino hitting A | ~B) for pretty much any A and B. (Or vice versa.) The correlation is incredibly tiny, but tiny != 0. (At least assuming A and B are within each others lightcone (neutrinos don’t travel at c, but hopefully you get the gist.). So either we drop true correlations, or allow through spurious ones. Either way this is far more than straight deduction.))