also, it appears that the two diagrams in the Frankenstein Rule section differ in their d-separation of (x_1 \indep x_4 | x_5) (which doesn’t hold in the the left), so these are not actually equivalent (we can’t have an underlying distribution satisfy both of these diagrams)
Oh, we can totally have an underlying distribution satisfy both of these diagrams. The key is that, while the right diagram asserts (x_1 \indep x_4 | x_5), the left does not say that x_1 can’t be independent of x_4 given x_5. Remember the interpretation: an underlying distribution satisfies a DAG if-and-only-if the distribution factors over that DAG. We neither assume nor prove minimality; the DAG does not need to be minimal.
So, for instance, a distribution in which all five variables are unconditionally independent would satisfy every diagram over those variables.
You are right that the two diagrams are not equivalent (i.e. there exist distributions which satisfy either one but not the other), and we’re not claiming they’re equivalent. We’re just saying “assume that some distribution satisfies both of these two diagrams; what other diagrams must the distribution then satisfy?”.
also, it appears that the two diagrams in the Frankenstein Rule section differ in their d-separation of (x_1 \indep x_4 | x_5) (which doesn’t hold in the the left), so these are not actually equivalent (we can’t have an underlying distribution satisfy both of these diagrams)
Oh, we can totally have an underlying distribution satisfy both of these diagrams. The key is that, while the right diagram asserts (x_1 \indep x_4 | x_5), the left does not say that x_1 can’t be independent of x_4 given x_5. Remember the interpretation: an underlying distribution satisfies a DAG if-and-only-if the distribution factors over that DAG. We neither assume nor prove minimality; the DAG does not need to be minimal.
So, for instance, a distribution in which all five variables are unconditionally independent would satisfy every diagram over those variables.
You are right that the two diagrams are not equivalent (i.e. there exist distributions which satisfy either one but not the other), and we’re not claiming they’re equivalent. We’re just saying “assume that some distribution satisfies both of these two diagrams; what other diagrams must the distribution then satisfy?”.
Ah that’s right. Thanks that example is quite clarifying!