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A limitation of usual Bayesian reasoning.
Scott is doing his annual subscription drive and I was reminded of a (still) private post of his I disagree with: https://www.astralcodexten.com/p/but-seriously-are-bloxors-greeblic
And then he proceeded to list some of the objections and his objections to objections. The objection I'm personally most partial to was not listed, so I assume it's a sort of novel idea, at least in that (and this) community.
Suppose that in your travels you encounter a shady guy who offers you an opportunity to bet on the outcome of a coin flip. Nearby stands a yudkowsky, who tells you that according to his observations the coin is biased and the next flip is about 66% likely to land on heads. You know that yudkowskis are honest and good Bayesians, so you trust his assessment.
The shady guy flips the coin and it lands on tails. Now consider two possible worlds: in one the yudkowsky says that his new estimate is 50% heads, in another he says that he has updated to 65% heads. That's two very different worlds! It turns out that the yudkowsky has an important parameter: how many coinflips he has observed so far, and therefore how much of his estimation comes from the observations and how much from the prior, and for some reason he doesn't tell you its value!
Scott's assertion is correct in a narrow technical sense: in a world where the shady stranger forces you to make a bet at gunpoint, you are forced to use the yudkowsky's estimation and the yudkowsky is forced to use a symmetric prior that gives him a 50% probability of heads when he has not seen any flips at all yet.
However in the real world there's almost always an option to wait and collect more data, and whether you want to exercise it critically depends on the difference between "it's a 50/50 chance based on observing 100 coinflips" and "it's a 50/50 chance based solely on the prior I pulled out of my ass".
So what's going on I think is that people intuitively understand that there's this important difference and suspect that when Scott says that normally they should start with a 50/50 prior, he's trying to swindle them into accepting Bayesians' estimations without asking how sure they are about them. And rightfully so, because that's a valid and important question to ask and honestly Bayesians ought to get a habit of volunteering this information unprompted, instead of making incorrect technical arguments insinuating that the estimated probability alone should be enough for everyone.
I am not a paying subscriber so I cannot access the post in question to check if my objection is addressed. I think there's a simpler problem than what you've articulated here. Consider two statements: "I think X occurs with probability 50%" and "I think X is equally likely to have any probability [0..100]". There is a sense in which both statements are "the same" because the expected probability of a uniform distribution over [0..100] is 50 but the statements (to me) clearly convey different information. Sure, if you're forced to give a particular integer value for a statement's probability you would choose "50" in both cases, but there is clearly a distinction between the subjective states that lead to that same probability. The assertion that you should use 50% feels like it is an attempt to treat these two statements as equivalent when they aren't.
This is basically Bayesian-vs-frequentist. I think the counterargument would be "the statement that X is likely to have a probability isn't even coherent, that's a type error". You can say that a class of events has an objectively true rate of occurrence, ie. if a coin will be thrown 100 times, then there will be a factual number of heads that show up, but you cannot say that any individual cointhrow has a likelihood of having a likelihood - that's just a simple likelihood. In other words, you can assign 10% probability to a model of the coin in which it has a 60% probability of landing on heads, but the word "probability" there carried two different meanings: observational credence (subjective) vs outcome ratio (objective). You can't have a credence over a credence; one is observational, the other is physical.
Not sure if that makes sense.
Rephrase my second statement slightly. "I have no bias towards any number [0..100] as the probability for X." Does that convey the same information as "I think X occurs with probability 50%?"
Yes, but it's near impossible to genuinely have no bias about X; to have absolutely no bias X has to be decoupled from any causal modeling. We have bias for almost anything that happens in the world, so I think this just makes for bad intuition because it's such a cornercase.
Sure. I don't intend to make any particular claim about how often one is actually in the described state. My point is that Scott is wrong when he says you should say something happens with probability 50% if one finds themselves in the described state.
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Why couldn't you just nest them? If I have a lottery ticket that pays off in other lottery tickets which finally pay money, then there are likelihoods of having a likelihood. You could of course calculate the average likelihood but sometimes this information is useful. Another example, if I have a game-theory situation where one player has beliefs over the beliefs of the other player, I have probabilities over probabilities.
AIUI technically speaking you have conditional probabilities, but that's not quite a "likelihood of having a likelihood" but "a likelihood given a precondition event which also has a likelihood".
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I agree with this. I think this relates to another interesting problem in probability. "What's the probability that the 10^10^10 th prime is 3 mod 4?" Its tempting to say 1/2 since we know that the asymptotic density is 1/2 and we have no way of knowing. But this is iconsistent with the axioms of probability theory. Since it's a statement with a definite answer the probability has to be either 1 or 0 to be consistent.
Even this is operating a critical assumption that the probable outcome must be the true outcome. What if it isn't?
I have no idea what you could possibly mean. True statements have probability 1, that's axiomatic.
Yes, and therein lies the fundamental contradiction/weakness of Bayesian reasoning. A cursory examination of the world around us will show that improbable things happen all the time and thus one must conclude that the probability of improbable things occurring is 1.
Improbable events do not happen every time an improbable event could happen, so the probability of something improbable happening in a particular instance is not 1.
The probability that "something improbable will happen today somewhere in the world" is 1-epsilon, but that's correct.
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