The famous phrase "correlation does not mean casuation" is not equal to "correlation is indistingushable from casuation", but that's how people seem to often treat it.
Ideally, the paper abstract should be clear about what can or can not be concluded. In the title they say "subsequent" as if suggesting casuality, but in the abstract they say "associated". Confusing for a layman such as me.
My intuition of how to distinguish correlation from casuation:
If cannabis and depression were symmetrically connected, then incease of chances for cannabis use between depressed was equal to the increase of chances for depression between cannabis users.
If cannabis increases chances for depression more that depression increases chances for cannabis, than we conclude cannabis tends to cause depression.
Formula: percentage of depressed between canabis users / percentage of depressed in general population > persentage of cannabis users between depressed / persentage of cannabis users in general population.
Something like that.
(c&d/c)/(d/total) > (c&d/d)/(c/total)
c&d*total/c*d > c&d*total/d*c function takeMeasurement() {
// c (the cause) itself has probability 0.5
// d (the dependent) in absence of c has probability 0.5
// but the presence of c increases it to 0.7
const c = Math.random(1.0) < 0.5;
const d = Math.random(1.0) < (c ? 0.7 : 0.5);
return {c, d};
}
P(c) = 0.5
P(d|!c) = 0.5
P(d|c) = 0.7
P(d) = ?
P(c|!d) = ?
P(c|d) = ?
-------------------------------
P(d) = P(c)*P(d|c) + P(!c)*P(d|!c) =
= P(c)*P(d|c) + (1 - P(c))*P(d|!c) =
= 0.5*0.7 + 0.5*0.5 = 0.35 + 0.25 =
= 0.6
P(c|!d) = P(c) * P(!d|c) / P(!d) =
= P(c) * (1 - P(d|c)) / (1 - P(d)) =
= 0.5 * 0.3 / 0.4 =
= 0.374(9)
P(c|d) = P(c) * P(d|c) / P(d) =
= 0.5 * 0.7 / 0.6 = 0.35 / 0.6 =
= 0.58(3)
-------------------------------
Bayes formula:
P(a|b) = P(a) * P(b|a) / P(b)
function takeMeasurement2() {
const d = Math.random(1.0) < 0.6;
const c = Math.random(1.0) < (d ? 0.58333 : 0.374999);
return {c, d};
}