09 July 2025
Anyone familiar with the empirical sciences knows the mantra: correlation does not imply cause. Whenever we observe two quantities go up and down in unison, we are cautious to conclude that one causes the other and ask for further evidence. But what about the other direction: does cause imply correlation? The question is of course not trivial. If true, the statement would allow one to rule out causal relationships were no correlation is observed.
My tendency is, and has been, to answer affirmatively. If one thinks about archetypical examples of cause and effect (e.g. a billiard ball hitting another, eating and feeling satiated), one observes an almost perfect correlation (e.g. the cue ball hitting the colored ball an the latter moving tend to always happen together, the more I eat the more satiated I am and the more I am satiated the more I have eaten). Nevertheless, the issue is more subtle than one may anticipate. Lets think about it.
In order to show this, it is enough to find an instance where a causal relationship exists but no correlation is observed. The reader may be able to come up with different ones, but let me describe one that I like and that sparked the writing of this post.
Imagine a world with two types of people $A$ and $B$, and two types of food $a$ and $b$. If a person eats the incorrect food (e.g. a person of type $A$ eats food $b$) they die, otherwise nothing happens. Every one regardless of their type has a certain chance $p$ of dying of natural causes. Finally, people know what food to eat to be safe (i.e. no one eats the incorrect food).
If you were to collect data from this world you would obtain something along these lines:
| person type | food type | dead |
|---|---|---|
| $A$ | $a$ | False |
| $B$ | $b$ | False |
| $A$ | $a$ | False |
| $B$ | $b$ | False |
| $A$ | $a$ | True |
| ... | ... | ... |
And if you were to compute the correlation between food type and dead you would see a number very close to 0. In fact, in the described scenario the two variables are perfectly uncorrelated. Since:
$$P(\texttt{dead}|\texttt{food type}) = p = P(\texttt{dead}),$$
the two variables are independent and thus uncorrelated.
This is a clear example where there is a causal relationship between two variables, yet no correlation is observed. As such, this should be enough proof for the fact that cause does not imply correlation.
Hume is a guy that thought a lot about causation, and thus it is worth listening to him. As an 18th philosopher he was not thinking in terms of data or correlations. Rather he wanted to answer the question how does the human mind recognize causality? He pointed out that there is no particular stimulus from which we can recognize that something is causing something else. There is a stimulus from which we recognize red (i.e. 🟥) or straightness (i.e. |), but there is no counterpart for causality. So, from what stimuli do we conclude causality? Hume answers: from constant conjuction. And what is that? Simple, if A always follows B, then A and B are constantly conjoined. And this implies that A and B are correlated. So if we asked Hume, he would say that cause does imply correlation.
But what would he answer to the irrefutable argument from the previous section? It would be something along the lines of:
You tell us to assume that the incorrect food type causes death, but how would you know this? What would you have had to observe to conclude that such statement is true? Certainly, you must have seen multiple people of type $A$ eating $b$, or vice versa, and dying shortly after.
Hume (hypothetically).
And this would be a pretty good answer.
All things considered, I guess the conclusion is the following. If you are a scientist or a mathematician, cause does not imply correlation. But, if you are a philosopher or someone interested in epistemology, cause does imply correlation.