I think different people mean different things with “causation”.
On the one hand, we have things where A makes B vastly more likely. No lawyer tries to argue that while their client shot the victim in the head (A) and the victim died (B), it could still be the case that the cause of death was old age and their client was simply unlucky. This is the strictest useful definition of causation.
Things get more complicated when A is just one of many factors contributing to B. Nine (or so) of ten lung carcinoma are “caused” by smoking, we say. But for the individual smoker cancer patient, we can only give a probability that their smoking was the cause.
On the far side, on priors I find it likely that the genes which determine eye colors in humans might also influence the chance that they get depression due to a long causal chain in the human body. Perhaps blue eyed people have an extra 10−6 or 10−9 chance to get depression compared to green eyed people after correcting for all the confounders, or perhaps it is the other way round. So some eye colors are possibly what one might call a risk factor for depression. This would be the loosest (debatably) useful definition of causation.
For such very weak “causations”, a failure to find a significant correlation does not imply that there is no “causation”. Instead, the best we can do is say something like “The likelihood of the observed evidence in any universe where eye color increases the depression risk by more than a factor of 2.3×10−3 (or whatever) is less than one in 3.5 millions.” That is, we provide bounds instead of just saying there is no correlation.
I think different people mean different things with “causation”.
On the one hand, we have things where A makes B vastly more likely. No lawyer tries to argue that while their client shot the victim in the head (A) and the victim died (B), it could still be the case that the cause of death was old age and their client was simply unlucky. This is the strictest useful definition of causation.
Things get more complicated when A is just one of many factors contributing to B. Nine (or so) of ten lung carcinoma are “caused” by smoking, we say. But for the individual smoker cancer patient, we can only give a probability that their smoking was the cause.
On the far side, on priors I find it likely that the genes which determine eye colors in humans might also influence the chance that they get depression due to a long causal chain in the human body. Perhaps blue eyed people have an extra 10−6 or 10−9 chance to get depression compared to green eyed people after correcting for all the confounders, or perhaps it is the other way round. So some eye colors are possibly what one might call a risk factor for depression. This would be the loosest (debatably) useful definition of causation.
For such very weak “causations”, a failure to find a significant correlation does not imply that there is no “causation”. Instead, the best we can do is say something like “The likelihood of the observed evidence in any universe where eye color increases the depression risk by more than a factor of 2.3×10−3 (or whatever) is less than one in 3.5 millions.” That is, we provide bounds instead of just saying there is no correlation.