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Notes -
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.
This also ties to the longstanding discussion regarding calibration confidence of 50/50 predictions. One problem with 50/50 prediction of binary event (as in the post) is that it is equivalent language. If you say that you predict "50% chance of tails" is literally the same thing as you saying "I predict 50% of heads" because it is literally the part of the same observation of "I predict 50% chance of heads and 50% chance of tails" that accounts for everything.
This is also well known weakness, you can really pad your prediction capabilities by adding many 50/50 predictions which you phrase as binary - such as that bitcoin will have value greater than X before January 1st 2025 (Yes/No) or that you will get married etc. Just formulate 1,000 such independent scenarios and literally flip a coin to assign yes/no answers and you should do well.
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