Unlike my previous posts, this one isn't an announcement of some finished result. I just want to get some ideas out for public discussion. A big part of the credit goes to Wei Dai and Vladimir Nesov, though the specific formulations are mine.
Wei Dai wonders: what are probabilities, anyway? Eliezer wonders: what are the Born probabilities of? I cannot claim to know the answers, but I strongly hold that these questions are, in fact, answerable. And as evidence, I'll try to show how normal the answers might plausibly turn out to be.
Perhaps counterintuitively, the easiest way for probabilities to arise is not by postulating "different worlds" that you could "end up" in starting from now. No, the easiest setting is a single, purely deterministic world with only one possible future.
The first thought experiment goes like this. Imagine a coarse-grained classical universe whose physical laws are allowed to consume symbols from a "Tape", located outside the Matrix, which contains a sequence of ones and zeroes coming from a pseudorandom number generator. If we flip a coin in that world, the result of the flip will depend on several consecutive ones and zeroes read from the Tape. If the coin is "fair", it will come up heads 50% of the time, on average, as we advance along the Tape. (Yes, Virginia, in this setting the fairness would be a mathematical property of the coin, not of our own ignorance.) In such a world, creatures who are too computationally weak to predict the Tape will probably find useful a concept of "probability", and will indeed find "probabilistic" events happening with the limiting frequencies, standard deviations, etc. predicted by our familiar probability theory - even though the world has only one timeline that never branches.
But we know that in our world, observations are not completely deterministic: they can be influenced by the mysterious Born probabilities. How could that kind of law arise from a Nature that doesn't contain it already?
To understand the second thought experiment, you need to be able to imagine the MWI without the Born probabilities: just a big wavefunction evolving according to the usual laws, without any observers who could collapse it or experience probabilities (whatever that means). And imagine an outside observer that samples it according to different probability measures, and sees different worlds. An observer using the Born rule will see our familiar "2-world". But other observers looking at the same wavefunction can see the "3-world" and many, many other worlds. What do they look like? That is an empirical question, completely answerable by modern physics, that I don't know enough math to answer; but intuitively it seems that most rules different from the 2-rule should either reward or penalize interactions that lead to branching, so the other worlds look either like huge neverending explosions, or static crystals at close to absolute zero. It's entirely possible that only 2-sampling is "stable" enough to contain stars, planets, proteins and biological evolution - which, if you think about it, "explains" the "existence" of the 2-world without assuming the Born rule.
The above does sound like it could lead to an explanation of some statement vaguely similar to the Born rule, but to clarify matters even further - or confuse you even deeper - let us go on to...
Imagine an algorithm running on a classical physical computer, sitting on a table in our quantum universe. The computer has the above-explained property of being "stable" under the Born rule: a weighted-majority of near futures ranked by the 2-norm have the computer correctly executing the next few steps, but for the 1-norm this isn't necessarily the case - the computer will likely glitch or self-destruct. (All computers built by humans probably have this property. Also note that it can be defined in terms of the wavefunction alone, without assuming weights a priori.) Then the algorithm will have "subjective anticipation" of an extremely weird kind: conditioned on the algorithm itself running faithfully in the future, it can conclude that some future histories with higher Born-weight are more likely. So if I'm some kind of Platonic mathematical algorithm, this says something about what I should expect to happen in the world.
This "explanation" has the big drawback that it doesn't explain experimental observations. Why should an apparatus measuring a property of one individual particle (say) give rise to observed probabilities predicted by quantum theory? I don't have an answer to that.
Moreover, I don't think the above thought experiments should be taken as a final answer to anything at all. The intent of this post was to show that confusing questions can and should be approached empirically, and that we can and should strive to achieve perfectly normal answers.