I wrote "your brain can wind up settling on either of [the two generative models]", not both at once.

Ah that makes sense. So the picture I should have is: whatever local algorithm oscillates between multiple local MAP solutions over time that correspond to qualitatively different high-level information (e.g., clockwise vs counterclockwise). Concretely, something like the metastable states of a Hopfield network, or the update steps of predictive coding (literally gradient update to find MAP solution for perception!!) oscillating between multiple local minima?

Curious about the claim regarding bistable perception as the brain "settling" differently on two distinct but roughly equally plausible generative model parameters behind an observation. In standard statistical terms, should I think of it as: two parameters having similarly high Bayesian posterior probability, but the brain not explicitly representing this posterior, instead using something like local hill climbing to find a local MAP solution—bistable perception corresponding to the two different solutions this process converges to?

If correct, to what extent should I interpret the brain as finding a single solution (MLE/MAP) versus representing a superposition or distribution over multiple solutions (fully Bayesian)? Specifically, in which context should I interpret the phrase "the brain settling on two different generative models"?

I just read your koan and wow it's a great post, thank you for writing it. It also gave me some new insights as to how to think about my confusions and some answers. Here's my chain of thought:

• if I want my information theoretic quantities to not degenerate, then I need some distribution over the weights. What is the natural distribution to consider?
• Well, there's the Bayesian posterior.
• But I feel like there is a sense in which an individual neural network with its weight should be considered as a deterministic information processing system on its own, without reference to an ensemble.
• Using the Bayesian posterior won't let me do this:
  • If I have a fixed neural network that contains a circuit $C$ that takes activation $X$ (at a particular location in the network) to produce activation $Y$ (at a different location), it would make sense to ask questions about the nature of information processing that $C$ does, like $I(X;Y)$.
  • But intuitively, taking the weight as an unknown averages everything out - even if my original fixed network had a relatively high probability density in the Bayesian posterior, it is unlikely that $X$ and $Y$ would be related by similar circuit mechanisms given another random sample weight from the posterior.
  • Same with sampling from the post-SGD distribution.
• So it would be nice to find a way to interpolate the two. And I think the idea of a tempered local Bayesian posterior from your koan post basically is the right way to do this! (and all of this makes me think papers that measure mutual information between activations in different layers via introducing a noise distribution over the parameters of $f$ are a lot more reasonable than I originally thought)

I like the definition, it's the minimum expected code length for a distribution under constraints on the code (namely, constraints on the kind of beliefs you're allowed to have - after having that belief, the optimal code is as always the negative log prob).

Also the examples in Proposition 1 were pretty cool in that it gave new characterizations of some well-known quantities - log determinant of the covariance matrix does indeed intuitively measure the uncertainty of a random variable, but it is very cool to see that it in fact has entropy interpretations!

It's kinda sad because after a brief search it seems like none of the original authors are interested in extending this framework.

That makes sense. I've updated towards thinking this is reasonable (albeit binning and discretization is still ad hoc) and captures something real.

We could formalize it like $I_{\sigma }(X;f(X\left)\right)$ where $I_{\sigma }(X;f(X\left)\right)=I(X;f(X)+{ϵ}_{\sigma })$ with ${ϵ}_{\sigma }$ being some independent noise parameterized by \sigma. Then $I_{\sigma }(X;f(X\left)\right)$ would become finite. We could think of binning the output of a layer to make it stochastic in a similar way.

Ideally we'd like the new measure to be finite even for deterministic maps (this is the case for above) and some strict data processing inequality like $I_{\sigma }(X;g(f\left(X\right)\left)\right)<I_{\sigma }\left(f\right(X);g(f\left(X\right)\left)\right)$ to hold, intuition being that each step of the map adds more noise.

But $I_{\sigma }(X;f(X\left)\right)$ is just $h\left(f\right(X)+{ϵ}_{\sigma })$ up to a constant that depends on the noise statistic, so the above is an equality.

Issue is that the above intuition is based on each application of $f$ and $g$ adding additional noise to the input (just like how discretization lets us do this: each layer further discretizes and bins its input, leading in gradual loss of information hence letting mutual information capture something real in the sense of amount of bits needed to recover information up to certain precision across layers), but I_\sigma just adds an independent noise. So any relaxation if $I(X;f(X\left)\right)$ will have to depend on the functional structure of $f$.

With that (+ Dmitry's comment on precision scale), I think the papers that measure mutual information between activations in different layers with a noise distribution over the parameters of $f$ sound a lot more reasonable than I originally thought.

Ah you're right. I was thinking about the deterministic case.

Your explanation of the jacobian term accounting for features "squeezing together" makes me update towards thinking maybe the quantizing done to turn neural networks from continuous & deterministic to discrete & stochastic, while ad hoc, isn't as unreasonable as I originally thought it was. This paper is where I got the idea that discretization is bad because it "conflates 'information theoretic stuff' with 'geometric stuff', like clustering" - but perhaps this is in fact capturing something real.

Thank you, first three examples make sense and seem like an appropriate use of mutual information. I'd like to ask about the fourth example though, where you take the weights as unknown:

• What's the operational meaning of $I(X;Z)$ under some $p\left(W\right)$? More importantly: to what kind of theoretical questions we could ask are these quantities an answer to? (I'm curious of examples in which such quantities came out as a natural answer to the research questions you were asking in practice.)
  • I would guess the choice of $p\left(W\right)$ (maybe the bayesian posterior, maybe the post-training SGD distribution) and the operational meaning they have will depend on what kind of questions we're interested in answering, but I don't yet have a good idea as to in what contexts these quantities come up as answers to natural research questions.

And more generally, do you think shannon information measures (at least in your research experience) basically work for most theoretical purposes in saying interesting stuff about deterministic neural networks, or perhaps do you think we need something else?

• Reason being (sorry this is more vibes, I am confused and can't seem to state something more precise): Neural networks seem like they could (and should perhaps) be analyzed as individual deterministic information processing systems without reference to ensembles, since each individual weights are individual algorithms that on its own does some "information processing" and we want to understand its nature.
• Meanwhile shannon information measures seem bad at this, since all they care about is the abstract partition that functions induce on their domain by the preimage map. Reversible networks (even when trained) for example have the same partition induced even if you keep stacking more layers, so from the perspective of information theory, everything looks the same - even though as individual weights without considering the whole ensemble, the networks are changing in its information processing nature in some qualitative way, like representation learning - changing the amount of easily accessible / usable information - hence why I thought something in the line of V-information might be useful.

I would be surprised (but happy!) if such notions could be captured by cleverly using shannon information measures. I think that's what the papers attempting to put a distribution over weights were doing, using languages like (in my words) "$P\left(w\right)$ is an arbitrary choice and that's fine, since it is used as means of probing information" or smth, but the justifications feel hand-wavy. I have yet to see a convincing argument for why this works, or more generally how shannon measures could capture these aspects of information processing (like usable information).

The 3-4 Chasm of Theoretical Progress

epistemic status: unoriginal. trying to spread a useful framing of theoretical progress introduced from an old post.

Tl;dr, often the greatest theoretical challenge comes from the step of crossing the chasm from [developing an impractical solution to a problem] to [developing some sort of a polytime solution to a problem], because the nature of their solutions can be opposites.

Summarizing Diffractor's post on Program Search and Incomplete Understanding:

Solving a foundational problem to its implementation often takes the following steps (some may be skipped):

1. developing a philosophical problem
2. developing a solution given infinite computing power
3. developing an impractical solution
4. developing some sort of polytime solution
5. developing a practical solution

and he says that it is often during the 3 -> 4 step in which understanding gets stuck and the most technical and brute-force math (and i would add sometimes philosophical) work is needed, because:

• a common motif in 3) is that they're able to proving interesting things about their solutions, like asymptotic properties, by e.g., having their algorithms iterate through all turing machines, hence somewhat conferring the properties of the really good turing machine solution that exists somewhere in this massive search space to the overall search algorithm (up to a massive constant, usually).
  • think of Levin's Universal Search, AIXItl, Logical Induction.
  • he says such algorithms are secretly a black box algorithm; there are no real gears.
• Meanwhile, algorithms in 4) have the opposite nature - they are polynomial often because they characterize exploitable patterns that make a particular class of problems easier than most others, which requires Real Understanding. So algorithms of 3) and 4) often look nothing alike.

I liked this post and the idea of the "3-4 chasm," because it explicitly captures the vibes of why I personally felt the vibes that, e.g., AIT, might be less useful for my work: after reading this post, I realized that for example when I refer to the word "structure," I'm usually pointing at the kind of insights required to cross the 3-4 gap, while others might be using the same word to refer to things at a different level. This causes me to get confused as to how some tool X that someone brought up is supposed to help with the 3-4 gap I'm interested in.^[1]

Vanessa Cosoy refers to this post, saying (in my translation of her words) that a lot of the 3-4 gap in computational learning theory has to do with our lack of understanding of deep learning theory, like how the NP-complete barrier is circumvented in practical problems, what are restrictions we can put on out hypothesis class to make them efficiently learnable in the same way our world seems efficiently learnable, etc.

• She mentions that this gap, at least in the context of deep learning theory, isn't too much of a pressing problem because it already has mainstream attention - which explains why a lot of her work seems to lie in the 1-3 regime.

I asked GPT for examples of past crossings of the 3-4 chasm in other domains, and it suggested [Shannon's original technically-constructive-but-highly-infeasible proof for the existence of optimal codes] vs. [recent progress on Turbocodes that actually approach this limit while being very practical], which seems like a perfect example.

1. ^{^}

   AIT in specific seems to be useful primarily in the 1-2 level?

Thank you! I tried it on this post and while the post itself is pretty short, the raw content that i get seems to be extremely long (making it larger than the o1 context window, for example), with a bunch of font-related information inbetween. Is there a way to fix this?

The critical insight is that this is not always the case!

Let's call two graphs I-equivalent if their set of independencies (implied by d-separation) are identical. A theorem of Bayes Nets say that two graphs are I-equivalent if they have the same skeleton and the same set of immoralities.

This last constraint, plus the constraint that the graph must be acyclic, allows some arrow directions to be identified - namely, across all I-equivalent graphs that are the perfect map of a distribution, some of the edges have identical directions assigned to them.

The IC algorithm (Verma & Pearl, 1990) for finding perfect maps (hence temporal direction) is exactly about exploiting these conditions to orient as many of the edges as possible:

More intuitively, (Verma & Pearl, 1992) and (Meek, 1995) together shows that the following four rules are necessary and sufficient operations to maximally orient the graph according to the I-equivalence (+ acyclicity) constraint:

Anyone interested in further detail should consult Pearl's Causality Ch 2. Note that for some reason Ch 2 is the only chapter in the book where Pearl talks about Causal Discovery (i.e. inferring time from observational distribution) and the rest of the book is all about Causal Inference (i.e. inferring causal effect from (partially) known causal structure).