I want to go a bit deep here on "maximum entropy" and misunderstandings thereof by the straw-man Humbali character, mostly to clarify things for myself, but also in the hopes that others might find it useful. I make no claim to novelty here—I think all this ground was covered by Jaynes (1968)—but I do have a sense that this perspective (and the measure-theoretic intuition behind it) is not pervasive around here, the way Bayesian updating is.
First, I want to point out that entropy of a probability measure p is only definable relative to a base measure μ, as follows:
(The derivatives notated here denote Radon-Nikodym derivatives; the integral is Lebesgue.) Shannon's formulae, the discrete H(p)=−∑ip(xi)logp(xi) and the continuous H(p)=−∫Xp(x)logp(x)dx, are the special cases of this where μ is assumed to be counting measure or Lebesgue measure, respectively. These formulae actually treat p as having a subtly different type than "probability measure": namely, they treat it as a density with respect to counting measure (a "probability mass function") or a density with respect to Lebesgue measure (a "probability density function"), and implicitly supply the corresponding μ.
If you're familiar with Kullback–Leibler divergence (DKL), and especially if you've heard DKL called "relative entropy," you may have already surmised that Hμ(p)=−DKL(p||μ). Usually, KL divergence is defined with both arguments being probability measures (measures that add up to 1), but that's not required for it to be well-defined (what is required is absolute continuity, which is sort of orthogonal). The principle of "maximum entropy," or argmaxpHμ(p), is equivalent to argminpDKL(p||μ). In the absence of additional constraints on p, the solution of this is p=μ, so maximum entropy makes sense as a rule for minimum confidence to exactly the same extent that the implicit base measure μ makes sense as a prior. The principle of maximum entropy should really be called "the principle of minimum updating", i.e., making a minimum-KL-divergence move from your prior μ to your posterior p when the posterior is constrained to exactly agree with observed facts. (Standard Bayesian updating can be derived as a special case of this.)
Sometimes, the structure of a situation has some symmetry group with respect to which the situation of uncertainty seems to be invariant, with classic examples being relabeling heads/tails on a coin, or arbitrarily permuting a shuffled deck of cards. In these examples, the requirement that a prior be invariant with respect to those symmetries (in Jaynes' terms, the principle of transformation groups) uniquely characterizes counting measure as the only consistent prior (the classical principle of indifference, which still lies at the core of grade-school probability theory). In other cases, like a continuous roulette wheel, other Haar measures (which generalize both counting and Lebesgue measure) are justified. But taking "indifference" or "insufficient reason" to justify using an invariant measure as a prior in an arbitrary situation (as Laplace apparently did) is fraught with difficulties:
I think Humbali's confusion can be partially explained as conflating an invariant measure and a prior—in both directions:
There is a bit of motte-and-bailey uncovered by the bad-faith in position 2. Humbali all along primarily wants to defend his prior as unquestionably reasonable (the bailey), and when he brings up "maximum entropy" in the first place, he's retreating to the motte of Lebesgue measure, which seems to have a formidable air of mathematical objectivity about it. Indeed, by its lights, Humbali's own prior does happen to have more entropy than Eliezer's, though Lebesgue measure fails to support the full bailey of Humbali's actual prior. However, in this case even the motte is not defensible, since Lebesgue measure is an improper prior and the translation-invariance that might justify it simply has no relevance in this context.Meta: any feedback about how best to make use of the channels here (commenting, shortform, posting, perhaps others I'm not aware of) is very welcome; I'm new to actually contributing content on AF.
Ha, I was just about to write this post.
To add something, I think you can justify the uniform measure on bounded intervals of reals (for illustration purposes, say [0,1]) by the following argument: "Measuring a real number x∈[0,1]" is obviously simply impossible if interpreted literally, containing an infinite amount of data. Instead this is supposed to be some sort of idealization of a situation where you can observe "as many bits as you want" of the binary expansion of the number (choosing another base gives the same measure). If you now apply the principle of indifference to each measured bit, you're left with Lebesgue measure.
It's not clear that there's a "right" way to apply this type of thinking to produce "the correct" prior on N (or R or any other non-compact space.
Given any particular admissible representation of a topological space, I do agree you can generate a Borel probability measure by pushing forward the Haar measure of the digit-string space ΣN (considered as a countable product of ω copies of Σ, considered as a group with the modular-arithmetic structure of Z/|Σ|) along the representation. This construction is studied in detail in (Mislove, 2015).
But, actually, the representation itself (in this case, the Cantor map) smuggles in Lebesgue measure, because each digit happens to cut the target space "in half" according to Lebesgue measure. If I postcompose, say, x↦√x after the Cantor map, that is also an admissible representation of [0,1], but it no longer induces Lebesgue measure. This works for any continuous bijection, so any absolutely continuous probability measure on [0,1] can be induced by such a representation. In fact, this is why the inverse-CDF algorithm for drawing samples from arbitrary distributions, given only uniform random bits, works.
That being said, you can apply this to non-compact spaces. I could get a probability measure on R via a decimal representation, where, say, the number of leading zeros encodes the exponent in unary and the rest is the mantissa. [Edit: I didn't think this through all the way, and it can only represent real numbers ≥1. No big obstacle; post-compose x↦log(x−1).] The reason there doesn't seem to be a "correct" way to do so is that, because there's no Haar probability measure on non-compact spaces (at least, usually?), there's no digit representation that happens to cut up the space "evenly" according to such a canonical probability measure.
Out of curiosity, this morning I did a literature search about "hard-coded optimization" in the gradient-based learning space—that is, people deliberately setting up "inner" optimizers in their neural networks because it seems like a good way to solve tasks. (To clarify, I don't mean deliberately trying to make a general-purpose architecture learn an optimization algorithm, but rather, baking an optimization algorithm into an architecture and letting the architecture learn what to do with it.)
Why is this interesting?
Anyway, here's (some of) what I found:
Here are some other potentially relevant papers I haven't processed yet:
Among computational constraints, I think the most significant/fundamental are, in order,
Useful primitives for incentivizing alignment-relevant metrics without compromising on task performance might include methods like Orthogonal Gradient Descent or Averaged Gradient Episodic Memory, evaluated and published in the setting of continual learning or multi-task learning. Something like “answer questions honestly” could mathematically be thought of as an additional task to learn, rather than as an inductive bias or regularization to incorporate. And I think these two training modifications are quite natural (I just came to essentially the same ideas independently and then thought “if either of these would work then surely the multi-task learning folks would be doing them?” and then I checked and indeed they are). Just some more nifty widgets to add to my/our toolbox.
Re: alignment tasks in multi-task settings. I think this makes a lot of sense. Especially in worlds where we have a lot of ML/AI systems doing a bunch of different things, even if they have very different specific tasks, the "library" of alignment objectives is probably widely shared.
“Concern, Respect, and Cooperation” is a contemporary moral-philosophy book by Garrett Cullity which advocates for a pluralistic foundation of morality, based on three distinct principles:
What I recently noticed here and want to write down is a loose correspondence between these different foundations for morality and some approaches to safe superintelligence:
Cullity argues that none of his principles is individually a satisfying foundation for morality, but that all four together (elaborated in certain ways with many caveats) seem adequate (and maybe just the first three). I have a similar intuition about AI safety approaches. I can’t yet make the analogy precise, but I feel worried when I imagine corrigibility alone, CEV alone, bargaining alone (whether causal or acausal), or Earth-as-wildlife-preserve; whereas I feel pretty good imagining a superintelligence that somehow balances all four. I can imagine that one of them might suffice as a foundation for the others, but I think this would be path-dependent at best. I would be excited about work that tries to do for Cullity’s entire framework what CEV does for pure single-agent utilitarianism (namely, make it more coherent and robust and closer to something that could be formally specified).