Correlation implies Causation (2009)
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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).
I think that you can answer that. Science is not a series of isolated experiments that stand or fall based on their particular data. Instead, all of our judgments of causation depend on a series of nest broad and narrow assumptions about the world. The broadest assumption is perhaps that we have a material world whose substance lacks the ability to intentionally sabotage our experiments and from which we can generate uniformly distributed random samples from. But there are whole range of assumptions below that.
From this view, "extraordinary claims require extraordinary evidence", essentially things are consistent with our existing assumptions still need evidence but not huge amounts. Things that are sudden changes in our whole understanding of the world require much more change. The faster-than-light neutrino experiments, in isolation, were probably a lot more convincing in just their statistics than a lot of experiments that get accepted without comment. But because such other experiments didn't contradict very established positions, their results weren't gone over with a fine-toothed comb. And that's how it should be.
Edit: the thing with a "calculus of casual inference" is that it also would have to include a way of taking into account the range of indirect assumptions that a given casual deduction depends on, so one would something like a knowledge-database.
So while weak(!) correlation does not imply causation, strong correlation pretty much does. It's how the scientific method works. We can't look behind the curtain and take a look at the rule sheet.
>Correlation doesn't imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing 'look over there'.
The universe doesn't guarantee that if P, then Q. At best we can observe that if P at t1, then it is highly likely that Q at t2. We can often simplify that as "if P, then Q", just as we can approximate Einstein's physics with Newtonian physics for low-velocity applications. But at the end of the day, both are only approximations. The clear rules of logic only exist in our head, just as a perfect circle doesn't exist in reality.
If so, whether correlation implies causation is the wrong question to ask. A more important question is what kind of correlations we usually take to imply causation. We're probably looking at correlations that hold exclusively between two sets of events with an extremely high probability, with the right sort of temporal relationship. We could then say that those kinds of correlations simply are what we mean by causation, because there really is nothing else to say.
Once upon a time, most philosophers thought that the mind was some immaterial substance separate from the brain. Now many of us believe that certain functions of the brain are the mind. Perhaps we could apply a similar reductionism to the issue of correlations and causations, too.
You may as well replace the phrase "correlation implies causation" with "correlation implies effectation".
For a careful treatment on correlation and causation, you should read Judea Pearl, one of the great living computer scientists. I highly recommend this casual read: https://www.nyu.edu/classes/shrout/SEM06/pearl.pdf
http://andrewgelman.com/2014/08/04/correlation-even-imply-co...
"That is, correlation in the data you happen to have (even if it happens to be “statistically significant”) does not necessarily imply correlation in the population of interest."
I really wish the author would expand more on this. Maybe due to my shallow knowledge of statistics, I've recently become baffled by the fact that an arbitrary number is used as a confidence interval, to state that something is true or false. And I'd guess most of today's world depends on these confidence intervals. Why is it that we're OK with stating that something is true, if it's true 95% of the time? Or is it the case of good enough, it it ain't broke, don't fix it (until it isn't)?
I guess the process you are talking about involves creating a new model which is able to account for the causal relationship for a larger set of observations, compared to the previous model. In this way, the previous model can be either flushed, or if the conditions under which it fails are known, we can use it for specific cases. Isn't physics prone to the same effect, only to a lesser frequency? Which might be worrisome, because if the opportunity to revise a model comes every 100 years, that might mean that you'll spend your whole life interpreting the world through an inferior model.
It seems like all the author is really saying is that experiments aren't good enough to produce 100% certainty of causation. Not all that shocking. But the author also seems to conflate correlation with uncertainty, and this is probably where the title comes from: increasing certainty from controlled experiments implies causation.
Increased purchases of gifts causes Christmas.
Second, in academic research, we mean 'correlation doesnt imply direct causation'. Because we're talking science (what's significant) not astrology (as above, so below).
For example, the octopus predicts the results of football match correctly most of the time. But as a scientific person, would you say that there is any conceivable causation?
The important word is conceivable.
See for example the irrefutable possibility of https://en.m.wikipedia.org/wiki/Occasionalism
What is the best way of getting summaries of philosophy from as close to the beginning as possible?
The book "Sophie's World" does a great intro here and it a really great weekend read.
http://www.amazon.com/Sophies-World-History-Philosophy-Class...