But I have no idea how big the quantum effects are on the weather tomorrow, and when I say I give a 10% chance for rain, I’m clearly not referring to the true quantum probabilities.
After reading this, I was confused by you not raising a very similar objection to grounding probabilities in what Solomonoff would say. Like, it similarly seems clear that you’re not referring to the true Solomonoff probabilities either? In many situations, a very good predictor would already 99.99%-know the answer to a question you’re uncertain about. Good probabilities needn’t have much to do with the probabilities of an ideal predictor. In particular, in the following later paragraph, I think you’re making sth close to the mistake you critiqued in the quantum proposal:
It’s tempting to say that one should define probabilities as the result of Solomonoff induction. Probabilities would be still subjective in the sense that no one can actually run the full Solomonoff induction, so we are all just giving our best guesses. But I can at least still say that the guy who gives 50% probability to Bigfoot standing next door is wrong in the sense that I’m confident that’s not close to what the Solomonoff induction says.
Say that we have a data sequence which has been the digits of pi in binary for the first 1000 items, and we’re predicting the th item. I say it’s 50:50; you say “that’s really wrong! that’s clearly far from what solomonoff induction thinks, because it already basically knows the answer!”. Or if you say it’s 99.9:0.1 and you turn out to be right, then you were being reasonable with your probability because that’s similar to what Solomonoff would have said (I’m certainly not confident that is really wrong; indeed, I have close to 50% that solomonoff thinks something close to it)? Or if we have a UTM such that with probability the first item in an empty sequence is 1, then I’m unreasonable to guess ? One could say something about better and worse strategies for guessing Solomonoff’s probabilities, or maybe something about how predictions are supposed to be eventually graded with a proper scoring rule, or something, but I think one can approximately equally try to save the quantum definition this way, and at that point talking about Solomonoff or quantum amplitudes isn’t adding any clarity. Even if we were guessing Solomonoff’s probabilities, one would want to give some account of what we are doing when we are doing this guessing; probably one would end up wanting to say that this guessing would itself be done in probabilistic terms, but then one would still need to explain that sort of probabilistic reasoning; and it presumably wouldn’t be explained as “we are guessing Solomonoff’s probabilities about Solomonoff’s probabilities” (where the “guessing” again gets unfolded the same way, repeated arbitrarily many times, I guess?). So this looks circular and it looks like one would want to give some other account of probabilistic reasoning.
I think a much better picture is that we’re not guessing what an ideal predictor would say about whether Bigfoot is in the room, we’re guessing whether Bigfoot is in fact in the room. And it would just be silly to think that Bigfoot is in the room with probability; from inside our thinking community, this looks like an objective mistake, and one doesn’t need to reference Solomonoff to make this judgment. This is maybe like how a pretrained LLM is not registering its guesses for what solomonoff would say next, it’s just guessing next tokens.
This is a bit similar to how truth is not proVability. Probabilities aren’t defined as the outputs of some ideal thing. We reason probabilistically and this is a successful activity, and we can make some sense of the success of this sort of activity with eg coherence theorems or theorems saying solomonoff induction has some nice properties. (I think it makes sense to say solomonoff induction is an ideal thing that’s somewhat analogous to good probabilistic reasoning; I just think it doesn’t make sense to try to translate probabilistic statements into statements about solomonoff.) This doesn’t require giving any definition to “the probability of P is p”, just like one doesn’t need to define “P is true”[1].
In conclusion, I think it makes sense to use solomonoff induction as an analogy to what one is doing when one reasons probabilistically, but I don’t think it makes sense to try to rewrite probabilistic statements into some statements about solomonoff induction. (To clarify, I don’t think this is a serious criticism of the broader philosophical thesis in the sequence, I just think you’re confused/wrong about a subtle philosophical point about probabilities which doesn’t sink the overall framework.)
After reading this, I was confused by you not raising a very similar objection to grounding probabilities in what Solomonoff would say. Like, it similarly seems clear that you’re not referring to the true Solomonoff probabilities either? In many situations, a very good predictor would already 99.99%-know the answer to a question you’re uncertain about. Good probabilities needn’t have much to do with the probabilities of an ideal predictor. In particular, in the following later paragraph, I think you’re making sth close to the mistake you critiqued in the quantum proposal:
Say that we have a data sequence which has been the digits of pi in binary for the first 1000 items, and we’re predicting the
I think a much better picture is that we’re not guessing what an ideal predictor would say about whether Bigfoot is in the room, we’re guessing whether Bigfoot is in fact in the room. And it would just be silly to think that Bigfoot is in the room with
This is a bit similar to how truth is not proVability. Probabilities aren’t defined as the outputs of some ideal thing. We reason probabilistically and this is a successful activity, and we can make some sense of the success of this sort of activity with eg coherence theorems or theorems saying solomonoff induction has some nice properties. (I think it makes sense to say solomonoff induction is an ideal thing that’s somewhat analogous to good probabilistic reasoning; I just think it doesn’t make sense to try to translate probabilistic statements into statements about solomonoff.) This doesn’t require giving any definition to “the probability of P is p”, just like one doesn’t need to define “P is true”[1].
In conclusion, I think it makes sense to use solomonoff induction as an analogy to what one is doing when one reasons probabilistically, but I don’t think it makes sense to try to rewrite probabilistic statements into some statements about solomonoff induction. (To clarify, I don’t think this is a serious criticism of the broader philosophical thesis in the sequence, I just think you’re confused/wrong about a subtle philosophical point about probabilities which doesn’t sink the overall framework.)
and in fact in a certain precise sense cannot define “P is true”