If correlation does not imply causation, what does?

Compulsory xkcd

Right, off I go. If you are a mathematician/statistician/some other kind of badass and haven’t heard of Cox’s theorem: go and learn it now. I would highly recommend reading the first two chapters of “Probability Theory: The Logic of Science” book by E.T. Jaynes rather than wiki though. Not convinced? Stephen Muirhead’s quote: “Such a good book! Something I should have read ages ago”. I am sure that if you happen to know Stephen no more convincing is necessary. Bad luck otherwise.

I can’t praise the book highly enough though. I am surprised that the gist of it isn’t taught at undergraduate level in maths courses. So if you are a mathematician: read the first 2 chapters. If you are a probabilist than read all the relevant parts. If you are a statistician/data scientist: first half is a must. That’s a rule of a thumb I’ve just come up with. You’re welcome.

So what is it that I am trying to get you to read? It’s the fact that from 3 simple postulates one can derive a field known as plausibility reasoning. All mathematicians are introduced into Aristotelian logic yet not all are to plausibility reasoning, which is an extension of it. It’s a field of reasoning which deals with partial information. It is also much more intuitive than Aristotelian logic (which is a special case of it). We are used to saying that if \(A \implies B\) and \(B\) is true nothing can be said about \(A\). But in real life we are tempted to conclude that \(A\) is more likely to occur if we observe \(B\). And this is the everyday definition of the word imply: to make something more likely. I will stick to this definition in the text below and refer to \( \implies \) as deductive implication or \(\text{imply}_d\).

And so we have got this saying: “correlation doesn’t imply causality”. And it’s true in Aristotelian logic when the definition of the word imply is deductive. But we, mere mortals, scientists and statisticians have to resort to the other imply: the imply in plausibility reasoning. Let’s examine what it leads us to.

A new study comes out: “eating asparagus correlates with cancer”. You see a reference to it in the Daily Mail. Kill or cure keeps a long list of things Daily Mail reports that cause or that prevent cancer. Apparently, many things do both. You yawn and ignore it: “correlation doesn’t imply causation”. Feels justified. And it is, but not because of the oft-reapeated adage. It is because there’s an alternative hypothesis to a causal link: Daily Mail being sensational again is more likely. In fact, nothing written on the pages of Daily Mail about cancer can possibly convince you at this stage. The prior probability that Daily Mail is just too incompetent is too high. Such a case where new data can’t possibly convince a person to change his mind is covered in E.T. Jaynes’ book, in Chapter 5 “Queer uses of probability theory”.

On the other hand, let’s suppose you know a thing or two about asparagus and cancer. May be there’s a molecule X in asparagus that has been linked to cancer in a number of credible studies. And here you read it again that X is up to no good. Your alternative hypothesis to the causal link is that X doesn’t cause cancer. Yes, of course it’s Daily Mail, but you follow it up (you wouldn’t normally). And you raise your confidence in the causal hypothesis. Why? Because correlation is evidence for causation and counterevidence against lack of causation. So the posterior odds \( \frac{\mathbb{P(\text{causality} | \text{correlation})}} {\mathbb{P(\neg\text{causality} | \text{correlation})}} \) have increased. End result is different this time: you are more convinced in the causality.

The fact that same new knowledge can result in different beliefs is covered by E.T. Jaynes in Chapter 6: Elementary parameter estimation. It highlights the importance of prior knowledge.

In the second case the adage about causality doesn’t seem to apply. The fact is whether a correlation implies a causal effect is highly dependant on your alternative hypothesis. In many many cases, a causal link is the most plausible explanation. Which can only be shown via correlations. And so, in general, the adage is wrong. But here’s the worrying bit: we insist on spreading the adage further without explaining the above subtleties.

My preliminary fix would be amending the saying like so: “correlation doesn’t imply causation when there’s an alternative explanation”. The fix is needed because I dislike fully general counter arguments. They empower people with an ability to reject inconvenient truths: “Oh look, they are trying to prove me wrong! Nice try, but correlation doesn’t imply causation anyway.” The emphasis on the alternative is crucial, I tried explaining it above, but consult Chapter 4: Elementary Hypothesis Testing if it wasn’t clear: it has all the missing details, including actual calculations.

How could my fix help? The rejecter of the claim is now obliged to propose that there’s something hidden going on. It could very well be that she is sceptical of the source or that she believes that an alternative explanation will soon be found. Such rejections are more useful (and could easily be valid): they indicate that there is no productive resolution to the disagreement at the moment. Providing more and more evidence at this very stage would be an exercise in futility. However, one must bear “absence of evidence is evidence of absence” in mind. As time goes on and there’s no sensible alternative on the horizon, one must accept that the probability of the causal link is increasing…

By the way, the alternative text on the xkcd comic which I used as a tagline is the following: “Correlation doesn’t \(\text{imply}_d\) causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing ‘look over there’.”

I hope this has been interesting and you are curious about E. T. Jaynes’s book now. Bye for now!