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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.
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.
That was one of the objections listed in the post, Scott's response was that you should only be neutral about elementary propositions, not about compound ones ("bloxors are greeblic AND bloxors are grue").
I personally think that this entire kind of objections can be dismissed by pointing out that Bayesian math works correctly and without contradictions, and when looking at actual priors there's not much disagreement about how to choose them either, in practice. Nobody actually has arguments against assigning a symmetric prior to a coin bias, or even can muster a lot of enthusiasm to argue that you should use a gaussian instead of a uniform prior.
People get hot and bothered when they feel that someone tries to hide how much information they have actually updated on and how much is their prior.
How do I know that "bloxor-1 is greeblic" is elementary, if I am totally uncertain about this proposition, and I don't even understand the terms? Additionally, it's arbitrary to say that one should be neutral about the elementary propositions.
What do you mean "correctly"?
Depends. If you interpret the probabilities as subjective degrees of belief and interpret degrees of belief in terms of idealised betting dispositions, then it's not obvious that people can introspect their own odds. Experimental work from about the Allais paradox onwards doesn't suggest that Bayesianism is a good fit with how humans actually reason under uncertainty, and without some evidence of reliability of personal introspection of priors, "My prior is X" is potentially just hot air.
How many of the arguments in probability theory have you read to come to this judgement? Because I can think of large parts of the literature dedicated to exactly this point.
Skill issue.
That I, doing Bayesian math about some bets against you, will leave you poor and destitute in the long run, unless you're using Bayes too. What do you want to use instead of Bayes for the record?
My point is not that the poors are always instinctively right. My point is that they have well-honed instincts for when someone is trying to take advantage of them, and the usual Bayesian reasoning like the above rightfully triggers it, even if they don't have the concepts or the introspection to communicate to us what was that, that triggered them.
My point is that a Bayesian megamind is entirely justified in asking the yudkowsky what fraction of his prediction came from the data, and basing his bet amount on that, and grumbling about the yudkowsky being useless if he refuses to answer.
Huh?
It's possible to set up some types of games where this is true, as well as some types of games when using Bayesian math can lead to disasters. See this paper for a pretty simple example of how setting up the game in a way that Bayesianism looks good is more complex than you seem to think: https://www.jstor.org/stable/40210799
If you're thinking of conditionalization as part of "Bayesian math" and alluding to diachronic Dutch Book Arguments, the problems here are particularly vexing. See here: https://link.springer.com/article/10.1007/s10670-020-00228-1
Richard Pettigrew, who has a background in both mathematics and philosophy, has done a lot of great work on these issues. Here's a brief and relatively simple introduction: http://m-phi.blogspot.com/2018/10/dutch-books-and-conditionalization.html
Basically, the literature thus far has been a long series of failed attempts to squeeze Bayesian epistemological juice out of pragmatic rocks.
The task is underspecified and hence so is your question. Can you explain more?
I agree.
One strand: Bayesians tend to be subjectivists, so symmetric priors are only a personal decision. Another strand: imprecise probabilists (like set-based Bayesians) tend to deny that any additive prior is mandatory (and perhaps not even permissible). Another strand: frequentists are critical of the whole Bayesian enterprise; note that criticisms of frequentists' positive claims are beside the point here.
Of course, all those criticisms of symmetric priors (as mandatory) might be wrong, but it's not true that symmetric priors are controversial, even among people with apparent expertise in the relevant mathematics and logic.
You might say, "Well, obviously if I asked you what the probability of heads is with this perfectly ordinary coin, you'd say 50%." However, we are both far from lacking any evidence with respect to that coin, and "The probability is 50%" can be interpreted in all sorts of different ways, e.g. a frequentist would want to interpret it in terms of hypothetical frequencies in a mathematical model of the coin tossing; some Bayesians would interpret it in terms of degrees of evidential support; other Bayesians would interpret it in terms of degrees of belief; some Bayesians would interpret it in terms of the degrees of belief that a rational person should have given the evidence...
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