Correlation is Not Causation
Posted by Jeff Id on May 1, 2011
On the internet, you can meet an amazing set of personalities. Kim øyhus, a physicist, has an unusual website with some interesting commentary and a near megalomania tone to it (think I’m kidding). There are two ‘proofs’ on it, one we’ve been discussing on a different thread and a second which states the following.
Correlation is Evidence of Causation
A proof done with conditional probability.
Definition 1 correlation : c
Definition 2 causation : a
Definition 3 not everything correlates : P(c) < 1
Definition 4 causation give correlation: P(c|a) = 1
P(a|c) : evidence for causation
= P(c|a) P(a) / P(c) : Bayesian inference
= 1 P(a) / P(c) : definition 4
> P(a) : definition 3
Conclusion: P(a|c) > P(a) : Correlation is evidence of causation. Q.e.d.
Which we can all agree with. Correlation is most certainly evidence of causation. Like Kim’s other proof though, this one is also over-interpreted in the conclusion.
Quote from Daniel Dvorkin: The correlation between ignorance of statistics and using “correlation is not causatison” as an argument is close to 1.
So since anyone who puts the effort in can work the math above, and we can all agree that correlation is evidence of causation, why is it that Dvorkin and apparently Kim have such a hard time seeing the other side of the > sign.
Here’s my proof using all of the definitions above and some simple math. This doesn’t rebut Kim’s proof but rather expands on it such that proper conclusions can be drawn.
P(a|c) = P(a ∩ c) / P( c)
Read – the probability of ‘a’ (causation) given ‘c’ (correlation) equals the probability of a and c both occurring divided by the probability of c correlation.
Since by definition above P(c) < 1 – not everything correlates. We can write:
P(a | c) < P (a ∩ c)
In other words, there is a greater probability of causation and correlation existing than causation based correlation alone. So while it is true that correlation is evidence of causation, it is not proof of causation.
Such a simple concept you wonder why you need math to write it down.
My thanks to Kim for the Sunday morning puzzle.