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.
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.