Edit: also regarding the last sentence, "...because all we have to help us establish causal relationships is correlation". The work of Pearl et al. give us quite a bit more: http://www.michaelnielsen.org/ddi/if-correlation-doesnt-impl...
Yes, correlation suggests causation, i.e. P(causation|correlation) > P(causation) from a Bayes perspective. That doesn't mean you should discount the possibility of ¬causation, merely that its probability is smaller. And "how much smaller" could be very close to 0, so it would still be hasty to say "implies", which linguistically implies "=>".
A better phrasing would be "correlation suggests but does not imply causation". (edit: e.g. as per that xkcd comic, mentioned by other posters. edit2: I mixed up the proof with the OP article. the proof uses "evidence of" which is also good.)
But yes, nice proof nonetheless. I like how causation is basically defined as P(c|a) = 1, showing how most complex philosophical issues are actually irrelevant (for this particular result).
> you can't conclude A causes B
right, this is what I referred to as "=>"
> plus an explanatory theory of the causation, plus evidence
yes, this all works together to build up the "how much smaller". An explanatory theory basically allows you to make predictions and run tests to collect more data to pump into the application of Bayes' theorem as used by that proof, improving your confidence of the difference between P(a|c) and P(a).
Just because you make an observation consistent with your beliefs, does not mean that you can claim all other explanations (complement of your beliefs) are invalid (primarily because you do not know what they are or could be).
> right, this is what I referred to as "=>"
Except that it isn't quite, since you were careful to clarify that your \implies (i.e., '=>' or '⇒') was the \implies of propositional logic, which explicitly disclaims any causal relationship. \implies in that context says precisely and only that the antecedent is false, or the consequent is true. In this sense, 2 + 2 = 4 \implies Barack Obama is currently the president of the US, and 2 + 2 = 5 \implies George Bush is currently the president of the US, even though there is no causal relationship in either case.
The proof only defines "causation" as some event "a" for which "P(c|a) = 1". This is the same property that "=>" has in propositional logic, and there is no implication of philosophical causation here either. But the proof still works, as a consequence of its definitions.
So in other words, the proof says: if causation causes correlation then P(a|c) > P(a) (i.e. correlation is evidence of causation) but we can't say causation is definitely true (P(a) = 1), however you want to define "causes" as long as it has the property that P(c|a) = 1.
Aristotelian logic is a necessary step towards understanding logic, but on its own it's not actually that useful because in reality we just don't have enough things that we can safely approximate as 100% true for it to work reliably.
Incidentally, this also means that the classic lists of "argument fallacies" often contain a few fallacies themselves, as an argument being Aristotelian-fallacious does not mean that it's practically- or probabilistically-fallacious. But I will agree that this is generally a distinction without a difference; I only rarely see someone accuse someone else of committing a fallacy where the accusation is Aristotelian-correct but not probabilistically-correct. But, rolling back around to the original point, most of them are indeed thoughtless recitations of the "correlation does not imply causation" mantra when, probabilistically, the correlation being observed can be reasonably interpreted as evidence.
[1] http://plato.stanford.edu/entries/induction-problem/#BaySub
I do love the work of Pearl, et. al. But even they do not admit Pearson (or any other type) of correlation, but rather the nebulous "statistical dependence". So the proof only works if your statistical dependence tool of choice matches the nature of your causal effect perfectly.
The notion of "Statistical dependence" is not nebulous. X and Y are independent if the joint distribution factorises as
p(X, Y) = p(X) p(Y)
Precisely. They operate purely in probabilistic dependence/independence terminology, from a theoretical point of view.
The "c" in the proof, I assume means "observed correlation". Because we are in fact talking about "[Observed] correlation does not imply [Unobserved] causation", right?
> Statistical dependence can be determined easily if you know the distributions,
Even if you know the distribution, a statistical test will make Type-I/II errors that you would have to take care of.
Actually, I find the text linked above hard to understand, without properly defining 'c.' What's the sample space?
In general, my sentiments are with the xkcd comic strip, but nothing more. Pearl's theories lay a firm foundation for communicating a causal hypothesis and manipulating it algebraically, but the true tests of causal hypothesis are:
- Experimental evidence
- The predictions it makes, in cases where experiments are hard to perform (e.g., in physics, when we make certain causal conjectures about how the universe works).