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Yes, good point, but if the prior is positive it drops out of the asymptotic as it doesn't contribute to the order of vanishing, so you can just ignore it from the start.

There was a sign error somewhere, you should be getting + lambda and - (m-1). Regarding the integral from 0 to 1, since the powers involved are even you can do that and double it rather than -1 to 1 (sorry if this doesn't map exactly onto your calculation, I didn't read all the details).

There is some preliminary evidence in favour of the view that transformers approximate a kind of Bayesian inference in-context (by which I mean something like, they look at in-context examples and process them to represent in their activations something like a Bayesian posterior for some "inner" model based on those examples as samples, and then predict using the predictive distribution for that Bayesian posterior). I'll call the hypothesis that this is taking place "virtual Bayesianism".

I'm not saying you should necessarily believe that, for current generation transformers. But fwiw I put some probability on it, and if I had to predict one significant capability advance in the next generation of LLMs it would be to predict that virtual Bayesianism becomes much stronger (in-context learning being a kind of primitive pre-cursor).

Re: the points in your strategic upshots. Given the above, the following question seems quite important to me: putting aside transformers or neural networks, and just working in some abstract context where we consider Bayesian inference on a data distribution that includes sequences of various lengths (i.e. the kinds of distribution that elicits in-context learning), is there a general principle of Bayesian statistics according to which general-purpose search algorithms tend to dominate the Bayesian posterior?

In mathematical terms, what separates agents that could arise from natural selection from a generic agent?

To ask a more concrete question, suppose we consider the framework of DeepMind's Population Based Training (PBT), chosen just because I happen to be familiar with it (it's old at this point, not sure what the current thing is in that direction). This method will tend to produce a certain distribution over parametrised agents, different from the distribution you might get by training a single agent in traditional deep RL style. What are the qualitative differences in these inductive biases?

This is an open question. In practice it seems to work fine even at strict saddles (i.e. things where there are no negative eigenvalues in the Hessian but there are still negative directions, i.e. they show up at higher than second order in the Taylor series), in the sense that you can get sensible estimates and they indicate something about the way structure is developing, but the theory hasn't caught up yet.

I think there's no such thing as parameters, just processes that produce better and better approximations to parameters, and the only "real" measures of complexity have to do with the invariants that determine the costs of those processes, which in statistical learning theory are primarily geometric (somewhat tautologically, since the process of approximation is essentially a process of probing the geometry of the governing potential near the parameter).

From that point of view trying to conflate parameters  such that  is naive, because  aren't real, only processes that produce better approximations to them are real, and so the  derivatives of  which control such processes are deeply important, and those could be quite different despite  being quite similar.

So I view "local geometry matters" and "the real thing are processes approximating parameters, not parameters" as basically synonymous.

You might reconstruct your sacred Jeffries prior with a more refined notion of model identity, which incorporates derivatives (jets on the geometric/statistical side and more of the algorithm behind the model on the logical side).

Except nobody wants to hear about it at parties.


You seem to do OK... 

If they only would take the time to explain things simply you would understand. 

This is an interesting one. I field this comment quite often from undergraduates, and it's hard to carve out enough quiet space in a conversation to explain what they're doing wrong. In a way the proliferation of math on YouTube might be exacerbating this hard step from tourist to troubadour.

As a supervisor of numerous MSc and PhD students in mathematics, when someone finishes a math degree and considers a job, the tradeoffs are usually between meaning, income, freedom, evil, etc., with some of the obvious choices being high/low along (relatively?) obvious axes. It's extremely striking to see young talented people with math or physics (or CS) backgrounds going into technical AI alignment roles in big labs, apparently maximising along many (or all) of these axes!

Especially in light of recent events I suspect that this phenomenon, which appears too good to be true, actually is.

Please develop this question as a documentary special, for lapsed-Starcraft player homeschooling dads everywhere.

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