Does belief quantization explain (some amount of) polarization?
Suppose people generally do Bayesian updating on beliefs. It seems plausible that most people (unless trained to do otherwise) subconsciosuly quantize their beliefs -- let's say, for the sake of argument, by rounding to the nearest 1%. In other words, if someone's posterior on a statement is 75.2%, it will be rounded to 75%.
Consider questions that exhibit group-level polarization (e.g. on climate change, or the morality of abortion, or whatnot) and imagine that there is a series of "facts" that are floating around that someone uninformed doesn't know about.
If one is exposed to facts in a randomly chosen order, then one will arrive at some reasonable posterior after all facts have been processed -- in fact we can use this as a computational definition of the what it would be rational to conclude.
However, suppose that you are exposed to the facts that support the in-group position first (e.g. when coming of age in your own tribe) and the ones that contradict it later (e.g. when you leave the nest.) If your in-group is chronologically your first source of intel, this is plausible. In this case, if you update on sufficiently many supportive facts of the in-group stance, and you quantize, you'll end up with a 100% belief on the in-group stance (or, conversely, a 0% belief on the out-group stance), after which point you will basically be unmoved by any contradictory facts you may later be exposed to (since you're locked into full and unshakeable conviction by quantization).
One way to resist this is to refuse to ever be fully convinced of anything. However, this comes at a cost, since it's cognitively expensive to hold onto very small numbers, and to intuitively update them well.
Causality is rare! The usual statement that "correlation does not imply causation" puts them, I think, on deceptively equal footing. It's really more like correlation is almost always not causation absent something strong like an RCT or a robust study set-up.
Over the past few years I'd gradually become increasingly skeptical of claims of causality just by updating on empirical observations, but it just struck me that there's a good first principles reason for this.
For each true cause of some outcome we care to influence, there are many other "measurables" that correlate to the true cause but, by default, have no impact on our outcome of interest. Many of these measures will (weakly) correlate to the outcome though, via their correlation to the true cause. So there's a one-to-many relationship between the true cause and the non-causal correlates. Therefore, if all you know is that something correlates with a particular outcome, you should have a strong prior against that correlation being causal.
My thinking previously was along the lines of p-hacking: if there are many things you can test, some of them will cross a given significance threshold by chance alone. But I'm claiming something more specific than that: any true cause is bound to be correlated to a bunch of stuff, which will therefore probably correlate with our outcome of interest (though more weakly, and not guaranteed since correlation is not necessarily transitive).
The obvious idea of requiring a plausible hypothesis for the causation helps somewhat here, since it rules out some of the non-causal correlates. But it may still leave many of them untouched, especially the more creative our hypothesis formation process is! Another (sensible and obvious, that maybe doesn't even require agreement with the above) heuristic is to distrust small (magnitude) effects, since the true cause is likely to be more strongly correlated with the outcome of interest than any particular correlate of the true cause.
Perhaps that can work depending on the circumstances. In the specific case of a toddler, at the risk of not giving him enough credit, I think that type of distinction is too nuanced. I suspect that in practice this will simply make him litigate every particular application of any given rule (since it gives him hope that it might work) which raises the cost of enforcement dramatically. Potentially it might also make him more stressed, as I think there's something very mentally soothing / non-taxing about bright line rules.
I think with older kids though, it's obviously a really important learning to understand that the letter of the law and the spirit of the law do not always coincide. There's a bit of a blackpill that comes with that though, once you understand that people can get away with violating the spirit as long as they comply with the letter, or that complying with the spirit (which you can grok more easily) does not always guarantee compliance with the letter, which puts you at risk of getting in trouble.
Pretending not to see when a rule you've set is being violated can be optimal policy in parenting sometimes (and I bet it generalizes).
Example: suppose you have a toddler and a "rule" that food only stays in the kitchen. The motivation is that each time food is brough into the living room there is a small chance of an accident resulting in a permanent stain. There's cost to enforcing the rule as the toddler will put up a fight. Suppose that one night you feel really tired and the cost feels particularly high. If you enforce the rule, it will be much more painful than it's worth in that moment (meaning, fully discounting future consequences). If you fail to enforce the rule, you undermine your authority which results in your toddler fighting future enforcement (of this and possibly all other rules!) much harder, as he realizes that the rule is in fact negotiable / flexible.
However, you have a third choice, which is to credibly pretend to not see that he's doing it. It's true that this will undermine your perceived competence, as an authority, somewhat. However, it does not undermine the perception that the rule is to be fully enforced if only you noticed the violation. You get to "skip" a particularly costly enforcement, without taking steps back that compromise future enforcement much.
I bet this happens sometimes in classrooms (re: disruptive students) and prisons (re: troublesome prisoners) and regulation (re: companies that operate in legally aggressive ways).
Of course, this stops working and becomes a farce once the pretense is clearly visible. Once your toddler knows that sometimes you pretend not to see things to avoid a fight, the benefit totally goes away. So it must be used judiciously and artfully.
Agreed with your example, and I think that just means that L2 norm is not a pure implementation of what we mean by "simple", in that it also induces some other preferences. In other words, it does other work too. Nevertheless, it would point us in the right direction frequently e.g. it will dislike networks whose parameters perform large offsetting operations, akin to mental frameworks or beliefs that require unecessarily and reducible artifice or intermediate steps.
Worth keeping in mind that "simple" is not clearly defined in the general case (forget about machine learning). I'm sure lots has been written about this idea, including here.
Regularization implements Occam's Razor for machine learning systems.
When we have multiple hypotheses consistent with the same data (an overdetermined problem) Occam's Razor says that the "simplest" one is more likely true.
When an overparameterized LLM is traversing the subspace of parameters that solve the training set seeking the smallest l2-norm say, it's also effectively choosing the "simplest" solution from the solution set, where "simple" is defined as lower parameter norm i.e. more "concisely" expressed.
In early 2024 I think it's worth noting that deep-learning based generative models (presently, LLMs) have the property of generating many plausible hypotheses, not all of which are true. In a sense, they are creative and inaccurate.
An increasingly popular automated problem-solving paradigm seems to be bolting a slow & precise-but-uncreative verifier onto a fast & creative-but-imprecise (deep learning based) idea fountain, a la AlphaGeometry and FunSearch.
Today, in a paper published in Nature, we introduce FunSearch, a method to search for new solutions in mathematics and computer science. FunSearch works by pairing a pre-trained LLM, whose goal is to provide creative solutions in the form of computer code, with an automated “evaluator”, which guards against hallucinations and incorrect ideas. By iterating back-and-forth between these two components, initial solutions “evolve” into new knowledge. The system searches for “functions” written in computer code; hence the name FunSearch.
Perhaps we're getting close to making the valuable box you hypothesize.
Upon reflection, the only way this would work is if verification were easier than deception, so to speak. It's not obvious that this is the case. Among humans, for instance, it seems very difficult for a more intelligent person to tell, in the general case, whether a less intelligent person is lying or telling the truth (unless the verifier is equipped with more resources and can collect evidence and so on, which is very difficult to do about some topics such as the verified's internal state) so, in the case of humans, in general, deception seems easier than verification.
So perhapst the daisy-chain only travels down the intelligence scale, not up.
To be sure, let's say we're talking about something like "the entirety of published material" rather than the subset of it that comes from academia. This is meant to very much include the open source community.
Very curious, in what way are most CS experiments not replicable? From what I've seen in deep learning, for instance, it's standard practice to include a working github repo along with the paper (I'm sure you know lots more about this than I do). This is not the case in economics, for instance, just to pick a field I'm familiar with.
Thanks for these references! I'm a big fan, but for some reason your writing sits in the silly under-exploited part of my 2-by-2 box of "how much I enjoy reading this" and "how much of this do I actually read", so I'd missed all of your posts on this topic! I caught up with some of it, and it's far further along than my thinking. On a basic level, it matches my intuitive model of a sparse-ish network of causality which generates a much much denser network of correlation on top of it. I too would have guessed that the error rate on "good" studies would be lower!