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Here is more context to the question.

A common example given of the “obviousness” that correlation does not equal causation is that shark attacks correlate with ice cream consumption. The explanation given is that the “actual” cause is that more people go to the beach on days that are hotter - thus the “missing” causal link is that people visit the beach.

But of course - that’s just a correlation too. One could say that it is not that people are on the beach that is the cause, the beach is just a correlation, the cause is that they get in the water.

But that could be a correlation, it is not that they are in the water, it is that they are in the water deeply and swimming, or perhaps it is that the sharks migrate through that time of the year, or perhaps the sun hitting the water reflects at an angle at that time of day that affects the sharks in a particular way and it corresponds to a common time that people eat ice cream.

The rat hole of correlation could go endlessly down different “concepts” and levels of abstraction of causation. Given that the causation at the end of that hole is a correlation, why is causation not just a special case of correlation?

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    Correlation becomes causation when there is a mechanism for it attributable to a natural law. One can be skeptical enough (like Hume) to treat even established laws as "mere correlations", but if so, they are exceptionally stable and strictly adhered to, unlike most other things loosely called "correlations". – Conifold Apr 23 at 21:35
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    This is half of the way to Hume. From a Humean point of view, we cannot ever know the difference between correlation and causation. We can only perceive correlation, and we have noticed that having a mechanism correlates with greater predictive power given the same correlation. That distinction is just a meta-correlation, not a definition. The notion of causation anthropomorphizes nature to fit our feelings of purposefulness. So it is something we can feel, but not something we can know. That does not make it wrong, but we should not think of such decisions as proven facts about reality. – hide_in_plain_sight Apr 24 at 2:13
  • Causation = strong correlation (to be precise) – user34482 Apr 24 at 11:12
  • @Pyrott Not at all. One can see incredibly high correlation without causation (different people’s clapping times in a show); and relatively low correlation although there is causation (molecules’ velocities when there is a change of pressure, making the particles slightly more inclined to move towards the low-pressure area while their motion is otherwise completely random). – Guillermo BCN Apr 24 at 22:44
  • @GuillermoBCN If we want to be rigorous there's no such thing as causation. The first example you mentioned (if I understand it correctly) is a coincidence, not a correlation, otherwise you would have to provide a sound explanation for the connection between the two events. The second example, on the contrary, is a strong correlation, given that there are scientific laws that explain such phenomenon perfectly. But the scientific laws themselves are nothing more than strong correlations of events, they're not absolute, we just see them happening a awful lot of times. – user34482 Apr 25 at 11:55
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Conceptually, the ideas of causation and correlation are as different as apples and oranges. Break it down like this:

  • Causation: the idea that event B is the necessary and inescapable consequence of event A
  • Correlation: the idea that event A and event B occur (or do not occur) together with a given frequency

Causation is an intrinsically temporal relationship; Correlation is an explicitly atemporal relationship. For example, I know that in the game of chess moving a knight will cause a pawn to be captured if it lands on the pawn's square. So say I do some analysis and discover that a pawn being captured correlates with moving a knight with a coefficient of 0.234. What does that correlation tell us about playing the game, which is necessarily a series of causes and effects? The correlation doesn't even (in itself) tell us that it was the knight that captured the pawn, merely that a knight moved and a pawn was captured; We have to import what we know about causation in the game to make that inference.

Now of course causation often (though not always) creates correlation, and people will often use correlations — carefully — as evidence that some causal relationship exists. But that 'carefully' is important, because trying to project from an atemporal observation to a temporal sequence involves a number of suspect inferences.

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This is a fundamental question for the metaphysics of causation. If you hold that causation is just a special form of correlation, this is an example of probabilistic causation. The antithesis of this view is to approach causation as a process.

I don't claim to understand the competing theories in detail but it seems that a central issue here is about facts versus events. If you choose to think in terms of facts (discrete data points), then you will tend to think of causation in terms of correlation. If a certain set of facts are true, then another fact is more likely to be true with some definite probability.

A problem, as you point out in the question, is that the potentially relevant sequence of facts can be infinite. Another way of putting this is that we experience the world to a great extent in terms of events rather than mere facts. There are infinite possibilities of facts occurring in time and space and the precise configuration of these facts matter in complex ways.

The relationships between facts appear to us to be much more than simply statistical, but material. In many kinds of events, free will may also appear to be involved. In order to show that causation is really just a form of correlation, you need to resolve these kinds problems. In other words, you would need to show that what appear to be "events" are actually just a finite and potentially knowable collection of relevant "facts", and that the relationships between these facts can all be expressed as potentially knowable and quantifiable probabilities.

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The difference can be framed in terms of events that are laws of nature and events that are not laws of nature . What then is a law of nature ? A law of nature has the same cause and same event such that C is both necessary and sufficient for E . Correlation then is when C is neither necessary not sufficient for E .

By C I mean Cause and by E I mean the Effect .

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As a different type of answer to this question, one can look at what is needed to make and reason about graphical models of the world, capturing either just correlation or also causation. See, for example, here: https://ftp.cs.ucla.edu/pub/stat_ser/r236-3ed.pdf

If one is interested only in modeling correlation, one needs to go only as far as Section 3 of the above reference. In contrast, to model causation, one needs to model interventions, for example do() in the above reference; this is discussed in the remainder of the above reference. Therefore, one needs fewer mathematical tools to reason about correlation than about causation, indicating that the latter is not just a special case of the former.

See also here: https://plato.stanford.edu/entries/causal-models/

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One answer is because causation can be present without correlation

This has implications for the design of machine intelligence systems that try to derive causality from data.

Consider the number set 1 and -1.

Take the absolute value of each number. (Consider this the "causation" function.)

The result is 1 and 1.

The correlation coefficient between these sets is 0.

It is no longer possible to reliability recreate the original datasets that are created by the causation function "absolute value."

Thus you can have causation without correlation.

As a result, causation is not a subset of correlation - though there may be overlap.

(Credit to John Fowler at the California Institute of Technology)

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