Turns out the particles → fluid example doesn’t work; it’s not a τ-abstraction (which makes me think the range of applicability of τ-abstraction is considerably narrower than I first thought).
That said, here’s a counterexample which I think works. Variables of the low-level model:
X1...Xn follow an arbitrary structural model
σ is a random permutation
Y1...Yn given by Yi=g(Xσ(i),UYi)
… where U are iid noise terms. So we have some arbitrary structural model, we scramble the variables, and then we compute a function of each. For the high-level model:
X′1...X′n follow the same model as X1...Xn in the low-level model
Z1...Zn given by Zi=g(X′i,UZi)
… so it’s the same as the low-level model, but with the Y variables unscrambled. The mapping between the two is what you’d expect: τ maps X→X′ directly, and uses σ to unscramble Y: Z=Yσ−1. Then the interventions ωτ are similarly simple:
ωτ(Xi←x∗)=(X′i←x∗)
ωτ(Yσ∗−1(i)←y∗,σ←σ∗)=(Zi←y∗)
Note that we can pick any σ∗ we please for the last intervention, but we do need to pick one—we can’t just leave it alone.
I’m pretty sure this checks all the boxes for strong τ-abstraction. But it isn’t a constructive τ-abstraction, since all of the Z’s depend on the same low-level variable σ. In principle, there could still be some other τ which makes the high-level model a constructive abstraction (B&H’s definition only requires that someτ exist between the two models), but I doubt it.
Let me know if you guys spot a hole in this setup, or see an elegant way to confirm that there isn’t some other τ that magically makes it constructive.
Turns out the particles → fluid example doesn’t work; it’s not a τ-abstraction (which makes me think the range of applicability of τ-abstraction is considerably narrower than I first thought).
That said, here’s a counterexample which I think works. Variables of the low-level model:
… where U are iid noise terms. So we have some arbitrary structural model, we scramble the variables, and then we compute a function of each. For the high-level model:
… so it’s the same as the low-level model, but with the Y variables unscrambled. The mapping between the two is what you’d expect: τ maps X→X′ directly, and uses σ to unscramble Y: Z=Yσ−1. Then the interventions ωτ are similarly simple:
Note that we can pick any σ∗ we please for the last intervention, but we do need to pick one—we can’t just leave it alone.
I’m pretty sure this checks all the boxes for strong τ-abstraction. But it isn’t a constructive τ-abstraction, since all of the Z’s depend on the same low-level variable σ. In principle, there could still be some other τ which makes the high-level model a constructive abstraction (B&H’s definition only requires that some τ exist between the two models), but I doubt it.
Let me know if you guys spot a hole in this setup, or see an elegant way to confirm that there isn’t some other τ that magically makes it constructive.