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I remember reading about this conjecture in Causality (Pearl, 2000).

It states that every dependency between random variables can be explained by (or originates from) a purely causal model. Of course, this causal model might contain additional variables (the confounders).

Can anyone explain what this conjecture is called, who formalised it, and maybe even provide a reference?

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  • $\begingroup$ The philosophy literature on this (and you're in it up to your neck with your question) distinguishes between causal laws and causal mechanisms underlying phenomena (observational correlations). Are you asking about laws or mechanisms vs no underlying structures (anti-realism / skepticism) or about laws (regularities) vs mechanisms (stable decomposable structures)? $\endgroup$ – conjugateprior May 6 '15 at 12:02
  • $\begingroup$ I haven't read Pearl, but doesn't this just come down to "everything is caused by something"? $\endgroup$ – shadowtalker May 6 '15 at 12:29
  • $\begingroup$ @conjugateprior I was not aware about the distinction between causal law and causal mechanism; I guess I meant causal mechanism, though I'm not sure about this. The question is about causal law/mechanism vs. pure correlation. $\endgroup$ – ziggystar May 6 '15 at 12:29
  • $\begingroup$ @ssdecontrol Basically, yes. $\endgroup$ – ziggystar May 6 '15 at 12:29
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This conjecture is called Reichenbach's Principle of Common Cause (RPCC), as it was first made precise by Hans Reichenbach (in 1956; imprecise versions have been around for much longer). The Stanford Encyclopedia of Philosophy has a good discussion and plenty of references.

Tangent: A friend recently asked me this exact question, and in addition, whether there were any counterexamples to the principle. The counterexamples that I'm aware of are: (1) selection bias, (2) logical or part-whole dependence, and (3) temporal trends.

  1. Example of selection bias: college students get admitted if they are EITHER smart OR good at football. This induces a negative correlation between football skills and intelligence within the college population that does not exist in the general population. The selection process, Selection Into College, is a common child of Intelligence and Football Skill rather than a common cause. It induces a dependence because we always implicitly condition on the selected population, and conditioning on a variable in a causal model induces a dependence between its parents.

  2. Example of logical dependence: x and log(x) are correlated. Example of part-whole dependence: my income in the first quarter of the year and the whole year are correlated. Neither of these examples have a well-defined causal model. In an interventionist theory of causation, for a set of variables to have a well-defined causal model, it must be logically possible to intervene on each variable individually without necessarily intervening on others. These could arguably count as cases of causation, if one were to extend the concept of a causal model (in which case they might not be counterexamples to RPCC).

  3. Example of temporal trends: sea levels in Venice and the price of bread in London are both going up, because they are both part of temporal processes that are trending upwards, so they correlate over time. Adjusted for the temporal trend, they don't correlate, reflecting the fact that neither is causally related to the other.

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  • $\begingroup$ Yes, that's what I was looking for. $\endgroup$ – ziggystar Oct 9 '15 at 10:45
  • $\begingroup$ (+1) I would add to the list: 4. Spurious Correlations. $\endgroup$ – usεr11852 May 11 '20 at 0:41
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    $\begingroup$ @usεr11852 Spurious correlation does not contradict RPCC; it's merely an effect on the observational level. $\endgroup$ – ziggystar May 11 '20 at 5:41
  • $\begingroup$ Sorry I should be more careful in my wording. I meant that the RVs only appear dependent in the sense of searching over a large number of tests and failing to account for multiple testing correction. (Failure to control for Type 1 error) $\endgroup$ – usεr11852 May 11 '20 at 8:11

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