Further, I’m pretty sure the result is not “X has to cause Y” but “this distribution has measure 0 WRT lebesgue in models where X does not cause Y”
Yes that’s true. Going from “The distributions in which X does not cause Y have measure zero” to “X causes Y” is I think common and seems intuitively valid to me. For example the soundness and completeness of d-separation also only holds but for a set of distributions of measure zero.
I think this could be right, but I also think this attitude is a bit too careless. Conditional independence in the first place has lebesgue measure 0. I have some sympathy for considering something along the lines of “when your posterior concentrates on conditional independence, the causal relationships are the ones that don’t concentrate on a priori measure 0 sets” as a definition of causal direction—maybe this is implied by the finite factored set definition if you supply an additional rule for determining priors, I’m not sure.
Also, this is totally not the Pearlian definition! I made it up.
Yes that’s true. Going from “The distributions in which X does not cause Y have measure zero” to “X causes Y” is I think common and seems intuitively valid to me. For example the soundness and completeness of d-separation also only holds but for a set of distributions of measure zero.
I think this could be right, but I also think this attitude is a bit too careless. Conditional independence in the first place has lebesgue measure 0. I have some sympathy for considering something along the lines of “when your posterior concentrates on conditional independence, the causal relationships are the ones that don’t concentrate on a priori measure 0 sets” as a definition of causal direction—maybe this is implied by the finite factored set definition if you supply an additional rule for determining priors, I’m not sure.
Also, this is totally not the Pearlian definition! I made it up.