I think a causal perspective is more useful for identifying when correlations are transitive. It is not transitive across colliders (unless you are conditioning on the collider), but it is often transitive across mediators and confounders.
Yes, in practice having a model of what is actually driving the correlations can help you do better than these estimates. A causal model would be helpful for that.
The product estimate for the expected correlation is only useful in a setting where nothing else is known about the relationship between the three variables than the two correlations, but in practice you often have some beliefs about what drives the correlations you observe, and if you’re a good Bayesian you should of course also condition on all of that.
I think a causal perspective is more useful for identifying when correlations are transitive. It is not transitive across colliders (unless you are conditioning on the collider), but it is often transitive across mediators and confounders.
Though also: https://twitter.com/tailcalled/status/1560646503542050817
Yes, in practice having a model of what is actually driving the correlations can help you do better than these estimates. A causal model would be helpful for that.
The product estimate for the expected correlation is only useful in a setting where nothing else is known about the relationship between the three variables than the two correlations, but in practice you often have some beliefs about what drives the correlations you observe, and if you’re a good Bayesian you should of course also condition on all of that.