Correlation implies Causation (2009)(web.archive.org)
web.archive.org
Correlation implies Causation (2009)
http://web.archive.org/web/20091113124223/http://dailymeh.tumblr.com/post/238289205
14 comments
I think the point of "correlation does not imply causation", refers to the literal prepositional logic sense of "=>".
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).
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).
Actually, as I see it, the main point of "correlation does not imply causation" is mostly "correlation does not imply a particular causal relationship". While coincidence is possible, as well, its mostly about the fact that you can't conclude A causes B from a correlation between A and B alone, because the correlation may be due to the fact that B causes A, or the fact that A and B are both caused by C. That's why you need the correlation, plus an explanatory theory of the causation, plus evidence to reject alternative causal relationships, to have a decently strong basis for concluding a particular causal relationship.
I think, what I said is the same as what you said, just with some more maths.
> 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).
> 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).
Belief in "correlation implies causation" admits the Law of Excluded Middle fallacy.
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).
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).
> > you can't conclude A causes B
> 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.
> 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.
"is" in language can often mean "is a subset of", I was using the term that way. Whilst you are right that "=>" disclaims any philosophical relationship, the proof covers all definitions of "cause" that one might reasonably come up with. So "you can't conclude A => B" implies (with probability 1) that "you can't conclude A causes B":
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.
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.
That's not entirely true, at least, using the normal definition of "causes" that is of interest in correlation vs. causation discussions, which certainly includes "causes" which are contributors to the occurrence of an effect but do not alone guarantee it (e.g., smoking causes cancer, but it is not true that smoking implies cancer in the propositional logic sense.)
There are additional possible causal stories, as well. It could be that A and B are completely independent but both causes of C, and your observations of A and B are gated by C. Most people aren't aware of this possibile explanation.
Shorter: Correlation does not Aristotelian-imply causation. Correlation probabilistically-implies causation.
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.
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.
[deleted]
I'm no expert on this subject but the proof in the first article seems a bit dodgy. Bayesian inference is a type of induction [1]. So using it to prove induction merely begs the question?
[1] http://plato.stanford.edu/entries/induction-problem/#BaySub
[1] http://plato.stanford.edu/entries/induction-problem/#BaySub
The author doesn't really apply Bayesian inference. He applies Bayes' rule which is a mathematical property.
P(c|a) is not 1. If we assume correlation is Pearson, then the "shape" of the causation has to be a linear effect for P(c|a) = 1. A U-shaped effect will probably give you a Pearson 0, etc.
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.
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.
> But even they do not admit Pearson (or any other type) of correlation, but rather the nebulous "statistical dependence".
The notion of "Statistical dependence" is not nebulous. X and Y are independent if the joint distribution factorises as
Precisely. They operate purely in probabilistic dependence/independence terminology, from a theoretical point of view.
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.
I understand, but from an inference point of view, it is nebulous because the distributions are unknown. Statistical dependence can be determined easily if you know the distributions, but not if you only have samples.
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?
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?
I partly agree with you, but I wouldn't go to the extent of using "easily" in this phrase --
> 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).
> 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).
The actual title is "If correlation doesn’t imply causation, then what does?" which I think is a much more useful title and question.
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.
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.
This argument is absurd. It takes the obvious truth that causation will certainly lead to correlation and blindly flips it around and claims it's somehow profound. Just bad reasoning from beginning to end. The correct reasoning - causation leads to correlation - is the basis of all science. When scientific theories are evaluated, the razor is a check that the theory correctly predicts observed results. You can't cook up a theory that fits a set of already known data and claim you have formulated a causal relationship unless you can then predict the NEXT results over a substantial set of trials.
I think the author is trying to say that coorelation is the starting point in the hunt for causation in many investigations.
If so, the author says it poorly. Yes, you wouldn't look for a causal link without correlation, that would be insane, but the existence of correlation tells you only that there might be causation, nothing more.
but there is no other way to actually show causation. You can't 'look at' causation. You can look at a million chess games, you'll never be able to see the rules themselves. All you see is correlation between how the pieces move in every game and from this you can try to figure the rules out, if any exist at all.
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.
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.
https://xkcd.com/552/
>Correlation doesn't imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing 'look over there'.
>Correlation doesn't imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing 'look over there'.
I would go one step further and argue that "causation" or "causality" is just a label that we attach to a special kind of correlation. The very concept of causality, it seems to me, is an attempt to impose human logical structures on a messy world that is fundamentally probabilistic.
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.
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.
First, correlation is symmetric between effects and their causes, while causation is not.
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
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
This article seems to miss the role of theories in physical sciences. When we talk about 'cause' we understand that to mean that some chain of events, governed by the rules of physics, lead to the result. Yes, those rules of physics were arrived at largely as a result of observation of correlations, but no one is going to propose military coups leading orange harvests as a fundamental physical law on the basis of observed correlation.
Correlation doesn't even imply correlation!
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."
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."
Does not necessarily is not the same thing as does not.
> They had done the chemical reaction that blew up the lab 175 times before without incident; then, suddenly, something went wrong and the lab went boom and real, actual people died.
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 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)?
We are OK with it because almost always we don't know a better way. Every time we use statistics and have certain finite sample size, these confidence intervals will emerge.
The only place we are ok with this is in the social sciences where rigorous causation can never be proven. No physics theory is considered 'valid' because of a confidence threshold. Pharmacology and epidemiology flirt with this too, because the underlying causal mechanisms can be fiendishly difficult to determine, but we see 'new evidence' stories almost every day that flush whatever last week's 'wisdom' was.
>we see 'new evidence' stories almost every day that flush whatever last week's 'wisdom' was.
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.
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.
The distinction is obvious: correlation does not include an ordering, but causation does. You can observe that two things both happened, and that is correlation. You can observe that one thing happened, and then another thing happened after, and stipulate causation. You can increase your certainty by a controlled experiment.
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.
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.
> You can observe that one thing happened, and then another thing happened after, and stipulate causation.
Increased purchases of gifts causes Christmas.
Increased purchases of gifts causes Christmas.
Correlation has a formula, it detects the linear relation between two variables. So the quadratic relation is actually having zero correlation.
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.
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.
Just not sound, philosophically. I don't understand why the author wants to make the claim.
See for example the irrefutable possibility of https://en.m.wikipedia.org/wiki/Occasionalism
See for example the irrefutable possibility of https://en.m.wikipedia.org/wiki/Occasionalism
I'd say that God has been pretty consistent so far (or giving the illusion that it is, if you prefer).
This is quoting Hume... I always find it hard to get to understand these figures, because I feel I need to know who influences them, but then I need to know who influences them... and so on.
What is the best way of getting summaries of philosophy from as close to the beginning as possible?
What is the best way of getting summaries of philosophy from as close to the beginning as possible?
Study philosophy from it's base principles. Start with pre-socratic philosophers and keep reading until you get to modern day.
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...
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...
Will do - thanks!
Not sure about "close to the beginning" but Will Durant's "The Story of Philsophy" is a great survey.
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...