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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.
There's also the problem that 50-50 is not actually a neutral probability, if you're a coherent Bayesian and you don't have an ultra-simple sample space. For example, if I think that the probability of each possible bloxor being greeblic is 50%, then I am committed to thinking that the probability that 70/100 bloxors being greeblic is 0.004%. So my "neutral" prior commits me to extremely strong confidence that the distribution of greeblic among those 100 bloxors is not 70!
If I set my prior for each bloxor being greeblic to 69.5%, then it is approximately neutral with respect to 70/100 bloxors being greeblic. But now I'm obviously far from neutral with respect to any individual bloxor being greeblic.
This is one of the limitations of Bayesianism as a formalism: it can model neutral belief with respect to any individual partition of the sample space, but not all partitions of the sample space. So, Scott is just wrong and frankly hasn't understood the mathematics, given his statement "If you have total uncertainty about a statement (“are bloxors greeblic?”), you should assign it a probability of 50%," since this norm implies incoherence, but coherence is a fundamental Bayesian norm.
Put briefly, what Scott is saying requires that you reject Bayesian epistemology/decision theory. I haven't read the whole post yet, but I would be surprised if he realised that.
A different model solves this. If you treat the proportion of greeblic bloxors as an unknown parameter, then assign a prior to that parameter, you can have both
a single bloxor has a 50% chance of being greeblic
the chance of 70/100 bloxors being greeblic is not negligible
This works because the bloxors are no longer independent; they are related through the proportion parameter. Observing one bloxor would change your belief about the parameter, and thus about the other bloxors.
A sufficiently large number of conjunctions of single-case hypotheses of the "bloxor x is greeblic" regenerates the problem. I put it in terms of proportions for familiarity's sake, but formally it's easier to understand the point if you consider Boolean operations on the elements of partitions, and note that in Bayesian epistemology the sample space is assumed to be closed under Boolean operations.
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