I'm not sure what motivation for worst-case reasoning you're thinking about here. Maybe just that there are many disjunctive ways things can go wrong other than bad capability evals and the AI will optimize against us?

This getting very meta, but I think my Real Answer is that there's an analogue of You Are Not Measuring What You Think You Are Measuring for plans. Like, the system just does not work any of the ways we're picturing it at all, so plans will just generally not at all do what we imagine they're going to do.

(Of course the plan could still in-principle have a high chance of "working", depending on the problem, insofar as the goal turns out to be easy to achieve, i.e. most plans work by default. But even in that case, the planner doesn't have counterfactual impact; just picking some random plan would have been about as likely to work.)

The general solution which You Are Not Measuring What You Think You Are Measuring suggested was "measure tons of stuff", so that hopefully you can figure out what you're actually measuring. The analogy of that technique for plans would be: plan for tons of different scenarios, failure modes, and/or goals. Find plans (or subplans) which generalize to tons of different cases, and there might be some hope that it generalizes to the real world. The plan can maybe be robust enough to work even though the system does not work at all the ways we imagine.

But if the plan doesn't even generalize to all the low-but-not-astronomically-low-probability possibilities we've thought of, then, man, it sure does seem a lot less likely to generalize to the real system. Like, that pretty strongly suggests that the plan will work only insofar as the system operates basically the way we imagined.

And for this exact failure mode, I think that improvements upon various relatively straight forward capability evals are likely to quite compelling as the most leveraged current interventions, but I'm not confident.

Personally, my take on basically-all capabilities evals which at all resemble the evals developed to date is You Are Not Measuring What You Think You Are Measuring; I expect them to mostly just not measure whatever turns out to matter in practice.

Yes, there is a story for a canonical factorization of $\Lambda$, it's just separate from the story in this post.

Sounds like we need to unpack what "viewing $X^0$ as a latent which generates $X$" is supposed to mean.

I start with a distribution $P\left[X\right]$. 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 $P\left[X\right]={\sum }_{i}P\left[X_i|\Lambda \right]P\left[\Lambda \right]$, where $\Lambda$ is the bias in this case. Then when reasoning/updating, we can usually just think about how an individual die-roll interacts with $\Lambda$, rather than all the other rolls, which is useful insofar as $\Lambda$ is much smaller than all the rolls.

Note that $P\left[X|\Lambda \right]$ is not supposed to match $P\left[X\right]$; then the representation would be useless. It's the marginal ${\sum }_{i}P\left[X_i|\Lambda \right]P\left[\Lambda \right]$ which is supposed to match $P\left[X\right]$.

The lightcone theorem lets us do something similar. Rather all the $X_i$'s being independent given $\Lambda$, only those $X_i$'s sufficiently far apart are independent, but the concept is otherwise similar. We express $P\left[X\right]$ as ${\sum }_{X^0}P\left[X|X^0\right]P\left[X^0\right]$ (or, really, ${\sum }_{\Lambda }P\left[X|\Lambda \right]P\left[\Lambda \right]$, where $\Lambda$ summarizes info in $X^0$ relevant to $X$, which is hopefully much smaller than all of $X$).

$\Lambda$ is conceptually just the whole bag of abstractions (at a certain scale), unfactored.

Yeah, that probably works.

If you have sets of variables that start with no mutual information (conditioning on $X^0$), and they are so far away that nothing other than $X^0$ could have affected both of them (distance of at least $2T$), then they continue to have no mutual information (independent).

Yup, that's basically it. And I agree that it's pretty obvious once you see it - the key is to notice that distance $2T$ implies that nothing other than $X^0$ could have affected both of them. But man, when I didn't know that was what I should look for? Much less obvious.

I don't understand why the distribution of $X^0$ must be the same as the distribution of $X$. It seems like it should hold for arbitrary $X^0$.

It does, but then $X^T$ doesn't have the same distribution as the original graphical model (unless we're running the sampler long enough to equilibrate). So we can't view $X^0$ as a latent generating that distribution.

But this theorem is only telling you that you can throw away information that could never possibly have been relevant.

Not quite - note that the resampler itself throws away a ton of information about $X^0$ while going from $X^0$ to $X^T$. 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.

So the reason this is interesting (for the thing you're pointing to) is not that it lets us ignore information from far-away parts of $X^T$ which could not possibly have been relevant given $X^0$, but rather that we want to further throw away information from $X^0$ itself (while still maintaining conditional independence at a distance).

Ah, no, I suppose that part is supposed to be handled by whatever approximation process we define for $\Lambda$? That is, the "correct" definition of the "most minimal approximate summary" would implicitly constrain the possible choices of boundaries for which $\Lambda$ is equivalent to $X_0$?

Almost. The hope/expectation is that different choices yield approximately the same $\Lambda$, though still probably modulo some conditions (like e.g. sufficiently large $T$).

What's the $n/2$ here? Is it meant to be $m/2$?

System size, i.e. number of variables.

The new question is: what is the upper bound on bits of optimization gained from a bit of observation? What's the best-case asymptotic scaling? The counterexample suggests it's roughly exponential, i.e. one bit of observation can double the number of bits of optimization. On the other hand, it's not just multiplicative, because our xor example at the top of this post showed a jump from 0 bits of optimization to 1 bit from observing 1 bit.