nit: I found your graphical quantifiers and implications a bit confusing to read. It’s the middle condition that looks weird.
If I’ve understood, I think you want (given a set of Xi, there exists Ω with all of):
∀i.[X¯i→Xi→Ω]∀Γ.(∀i.[X¯i→Xi→Γ])⟹[X→Ω→Γ]X(→Ω)→Ω
(the third line is just my crude latex rendition of your third diagram)
Is that order of quantifiers and precedence correct?
In words, you want an Ω:
which is a redund over the Xi
where any Γ which is a redund over the Xi is screened from X by Ω
which is approximately deterministic on X
An unverified intuition (I’ll unlikely work on this further): would the joint distribution of all candidate Γs work as Ω? It screens off the X from any given Γ, right (90% confidence)? And it is itself a redund over the X I think (again 90% confidence ish)? I don’t really grok what it means to be approximately deterministic but it feels like this should be? Overall confidence 30% ish, deflating for outside and definedness reasons.
Yup, that’s exactly the right idea, and indeed constructing Ω by just taking a tuple of all the redund Γ‘s (or a sufficient statistic for that tuple) is a natural starting point which works straightforwardly in the exact case. In the approximate case, the construction needs to be modified—for instance, regarding the third condition, that tuple of Γ’s (or a sufficient statistic for it) will have much more entropy conditional on X than any individual Γ.
nit: I found your graphical quantifiers and implications a bit confusing to read. It’s the middle condition that looks weird.
If I’ve understood, I think you want (given a set of Xi, there exists Ω with all of):
∀i.[X¯i→Xi→Ω]∀Γ.(∀i.[X¯i→Xi→Γ])⟹[X→Ω→Γ]X(→Ω)→Ω
(the third line is just my crude latex rendition of your third diagram)
Is that order of quantifiers and precedence correct?
In words, you want an Ω:
which is a redund over the Xi
where any Γ which is a redund over the Xi is screened from X by Ω
which is approximately deterministic on X
An unverified intuition (I’ll unlikely work on this further): would the joint distribution of all candidate Γs work as Ω? It screens off the X from any given Γ, right (90% confidence)? And it is itself a redund over the X I think (again 90% confidence ish)? I don’t really grok what it means to be approximately deterministic but it feels like this should be? Overall confidence 30% ish, deflating for outside and definedness reasons.
Yup, that’s exactly the right idea, and indeed constructing Ω by just taking a tuple of all the redund Γ‘s (or a sufficient statistic for that tuple) is a natural starting point which works straightforwardly in the exact case. In the approximate case, the construction needs to be modified—for instance, regarding the third condition, that tuple of Γ’s (or a sufficient statistic for it) will have much more entropy conditional on X than any individual Γ.