note that the resampler itself throws away a ton of information aboutX0 while going from X0 to XT. And that is indeed information which “could have” been relevant, but almost always gets wiped out by noise. That’s the information we’re looking to throw away, for abstraction purposes.
I agree this is true, but why does the Lightcone theorem matter for it?
It is also a theorem that a Gibbs resampler initialized at equilibrium will produce XT distributed according to X, and as you say it’s clear that the resampler throws away a ton of information about X0 in computing it. Why not use that theorem as the basis for identifying the information to throw away? In other words, why not throw away information from X0 while maintaining XT∼X ?
EDIT: Actually, conditioned on X0, it is not the case that XT is distributed according to X.
(Simple counterexample: Take a graphical model where node A can be 0 or 1 with equal probability, and A causes B through a chain of > 2T steps, such that we always have B = A for a true sample from X. In such a setting, for a true sample from X, B should be equally likely to be 0 or 1, but BT∣X0=B0, i.e. it is deterministic.)
Of course, this is a problem for both my proposal and for the Lightcone theorem—in either case you can’t view X0 as a latent that generates X (which seems to be the main motivation, though I’m still not quite sure why that’s the motivation).
Sounds like we need to unpack what “viewing X0 as a latent which generates X” is supposed to mean.
I start with a distribution P[X]. Let’s say X is a bunch of rolls of a biased die, of unknown bias. But I don’t know that’s what X is; I just have the joint distribution of all these die-rolls. What I want to do is look at that distribution and somehow “recover” the underlying latent variable (bias of the die) and factorization, i.e. notice that I can write the distribution as P[X]=∑iP[Xi|Λ]P[Λ], where Λ is the bias in this case. Then when reasoning/updating, we can usually just think about how an individual die-roll interacts with Λ, rather than all the other rolls, which is useful insofar as Λ is much smaller than all the rolls.
Note that P[X|Λ] is not supposed to match P[X]; then the representation would be useless. It’s the marginal ∑iP[Xi|Λ]P[Λ] which is supposed to match P[X].
The lightcone theorem lets us do something similar. Rather all the Xi‘s being independent given Λ, only those Xi’s sufficiently far apart are independent, but the concept is otherwise similar. We express P[X] as ∑X0P[X|X0]P[X0] (or, really, ∑ΛP[X|Λ]P[Λ], where Λ summarizes info in X0 relevant to X, which is hopefully much smaller than all of X).
I’m still not quite sure why the lightcone theorem is a “foundation” for natural abstraction (it looks to me like a nice concrete example on which you could apply techniques) but I think I should just wait for future posts, since I don’t really have any concrete questions at the moment.
I’m still not quite sure why the lightcone theorem is a “foundation” for natural abstraction (it looks to me like a nice concrete example on which you could apply techniques)
My impression is that it being a concrete example is the why. “What is the right framework to use?” and “what is the environment-structure in which natural abstractions can be defined?” are core questions of this research agenda, and this sort of multi-layer locality-including causal model is one potential answer.
The fact that it loops-in the speed of causal influence is also suggestive — it seems fundamental to the structure of our universe, crops up in a lot of places, so the proposition that natural abstractions are somehow downstream of it is interesting.
Okay, that mostly makes sense.
I agree this is true, but why does the Lightcone theorem matter for it?
It is also a theorem that a Gibbs resampler initialized at equilibrium will produce XT distributed according to X, and as you say it’s clear that the resampler throws away a ton of information about X0 in computing it. Why not use that theorem as the basis for identifying the information to throw away? In other words, why not throw away information from X0 while maintaining XT∼X ?
EDIT: Actually, conditioned on X0, it is not the case that XT is distributed according to X.
(Simple counterexample: Take a graphical model where node A can be 0 or 1 with equal probability, and A causes B through a chain of > 2T steps, such that we always have B = A for a true sample from X. In such a setting, for a true sample from X, B should be equally likely to be 0 or 1, but BT∣X0=B0, i.e. it is deterministic.)
Of course, this is a problem for both my proposal and for the Lightcone theorem—in either case you can’t view X0 as a latent that generates X (which seems to be the main motivation, though I’m still not quite sure why that’s the motivation).
Sounds like we need to unpack what “viewing X0 as a latent which generates X” is supposed to mean.
I start with a distribution P[X]. Let’s say X is a bunch of rolls of a biased die, of unknown bias. But I don’t know that’s what X is; I just have the joint distribution of all these die-rolls. What I want to do is look at that distribution and somehow “recover” the underlying latent variable (bias of the die) and factorization, i.e. notice that I can write the distribution as P[X]=∑iP[Xi|Λ]P[Λ], where Λ is the bias in this case. Then when reasoning/updating, we can usually just think about how an individual die-roll interacts with Λ, rather than all the other rolls, which is useful insofar as Λ is much smaller than all the rolls.
Note that P[X|Λ] is not supposed to match P[X]; then the representation would be useless. It’s the marginal ∑iP[Xi|Λ]P[Λ] which is supposed to match P[X].
The lightcone theorem lets us do something similar. Rather all the Xi‘s being independent given Λ, only those Xi’s sufficiently far apart are independent, but the concept is otherwise similar. We express P[X] as ∑X0P[X|X0]P[X0] (or, really, ∑ΛP[X|Λ]P[Λ], where Λ summarizes info in X0 relevant to X, which is hopefully much smaller than all of X).
Okay, I understand how that addresses my edit.
I’m still not quite sure why the lightcone theorem is a “foundation” for natural abstraction (it looks to me like a nice concrete example on which you could apply techniques) but I think I should just wait for future posts, since I don’t really have any concrete questions at the moment.
My impression is that it being a concrete example is the why. “What is the right framework to use?” and “what is the environment-structure in which natural abstractions can be defined?” are core questions of this research agenda, and this sort of multi-layer locality-including causal model is one potential answer.
The fact that it loops-in the speed of causal influence is also suggestive — it seems fundamental to the structure of our universe, crops up in a lot of places, so the proposition that natural abstractions are somehow downstream of it is interesting.