Thank you for providing counterexamples! Quite useful. and convinces me that the broadest version of this is false. And apologies for the slow reply—the linked papers took a while to absorb (as you may guess, I’m not a mathematician).
1. I tried for a while, but couldn’t properly follow this paper. From reading the abstract, goung through the figures, your summary, and discussing it with Claude (Opus 4.6), I think the core issue here is the same problem as point 2 - the measures are not independent. My wife’s phrasing here is that “there’s a causal link between position at time t and position at time t+1” (in this case + noise, in the case of point 2 plus velocity times time between measurements).
2. Thank you—slightly embarrassed I didn’t find this one myself. I think the issue here is due to repeated measurement—the moon-earth distance now and at t+1s are obviously connected. I think the missing criteria here is independence—repeatedly measuring the same thing to see evolution over time breaks it. This requires some narrowing of the core thesis—my proposed added text is “I’m talking about correlations that survive proper statistical scrutiny—where the p-value is computed correctly for the data structure at hand. A naive Pearson correlation between two autocorrelated time series, or two monotonic trends, can look impressive while meaning nothing, but that’s not a real correlation any more than a loaded die produces a real test of probability. The well-known tools for handling this (differencing, detrending, cointegration tests, correcting for effective sample size) exist precisely because statisticians already understand that autocorrelation inflates apparent correlation. My claim applies to correlations that remain after you’ve done the statistics right.”
I think the claim “correlation of independent measures implies causation” is still interesting and surprising (probably not to this crowd who point out plenty of more rigorous prior work, but to most biologists at least), though a bit less exciting than my original claim. In particular, it’s less exciting because it doesn’t have a bulletproof checklist for “doing the statistics right”, which may or may not be possible to make.
3. This paper I could understand. I’ve worked through all their examples, and don’t think any are counterexamples to my point. Indeed causation can exist without correlation (as you say, very common in biology and technology), I’ve never claimed otherwise. The paper shows some examples where correlation is high despite the causal path containing intermediates with lower correlation which is intriguing, but I believe every correlation in the paper is explained by a causal path that links the two things correlated. As you say, not necessarily a direct causal connection, but still a “relatively short causal chain linking those things”—an indirect causal connection.
My favorite quote from the paper was “the simple maxim that ‘correlation does not imply causation’ having been superceded by methods such as those set out in [9, 14], and in shorter form in [10]”. Good to see that others (Pearl in addition to Reichenbach) have made something like the point I’m aiming to make here, as well as the restriction from time-series data that you brought to my attention in (2) - “cited limit attention to systems whose causal connections form a directed acyclic graph, together with certain further restrictions, and also do not consider dynamical systems or time series data.”
So far I don’t think cyclic systems cause any correlations between causally-unlinked variables, but the fact that people doing this formally haven’t solved it makes me hesitant to make any claims as an outsider to the field.
Thank you for providing counterexamples! Quite useful. and convinces me that the broadest version of this is false. And apologies for the slow reply—the linked papers took a while to absorb (as you may guess, I’m not a mathematician).
1. I tried for a while, but couldn’t properly follow this paper. From reading the abstract, goung through the figures, your summary, and discussing it with Claude (Opus 4.6), I think the core issue here is the same problem as point 2 - the measures are not independent. My wife’s phrasing here is that “there’s a causal link between position at time t and position at time t+1” (in this case + noise, in the case of point 2 plus velocity times time between measurements).
2. Thank you—slightly embarrassed I didn’t find this one myself. I think the issue here is due to repeated measurement—the moon-earth distance now and at t+1s are obviously connected. I think the missing criteria here is independence—repeatedly measuring the same thing to see evolution over time breaks it. This requires some narrowing of the core thesis—my proposed added text is “I’m talking about correlations that survive proper statistical scrutiny—where the p-value is computed correctly for the data structure at hand. A naive Pearson correlation between two autocorrelated time series, or two monotonic trends, can look impressive while meaning nothing, but that’s not a real correlation any more than a loaded die produces a real test of probability. The well-known tools for handling this (differencing, detrending, cointegration tests, correcting for effective sample size) exist precisely because statisticians already understand that autocorrelation inflates apparent correlation. My claim applies to correlations that remain after you’ve done the statistics right.”
I think the claim “correlation of independent measures implies causation” is still interesting and surprising (probably not to this crowd who point out plenty of more rigorous prior work, but to most biologists at least), though a bit less exciting than my original claim. In particular, it’s less exciting because it doesn’t have a bulletproof checklist for “doing the statistics right”, which may or may not be possible to make.
3. This paper I could understand. I’ve worked through all their examples, and don’t think any are counterexamples to my point. Indeed causation can exist without correlation (as you say, very common in biology and technology), I’ve never claimed otherwise. The paper shows some examples where correlation is high despite the causal path containing intermediates with lower correlation which is intriguing, but I believe every correlation in the paper is explained by a causal path that links the two things correlated. As you say, not necessarily a direct causal connection, but still a “relatively short causal chain linking those things”—an indirect causal connection.
My favorite quote from the paper was “the simple maxim that ‘correlation does not imply causation’ having been superceded by methods such as those set out in [9, 14], and in shorter form in [10]”. Good to see that others (Pearl in addition to Reichenbach) have made something like the point I’m aiming to make here, as well as the restriction from time-series data that you brought to my attention in (2) - “cited limit attention to systems whose causal connections form a directed acyclic graph, together with certain further restrictions, and also do not consider dynamical systems or time series data.”
So far I don’t think cyclic systems cause any correlations between causally-unlinked variables, but the fact that people doing this formally haven’t solved it makes me hesitant to make any claims as an outsider to the field.