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brian_jaress comments on Bayesian Flame - Less Wrong
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I can't tell without more information whether that's an example of what I mean by "changing your mind." Here's one that I think definitely qualifies:
Let's say you're going to bet on a coin toss. You only have a small amount of information on the coin, and you decide for whatever reason that there's a 51% chance of getting heads. So you're going to bet on heads. But then you realize that there's a way to get more data.
At this point, I'm thinking, "Gee, I hardly know anything about this coin. Maybe I'm better off betting on tails and I just don't know it. I should get that data."
What I think you're saying about bayesians is that a bayesian would say, "Gee, 51% isn't very high. I'd like to be at least 80% sure. Since I don't know very much yet, it wouldn't take much more to get to 80%. I should get that data so I can bet on heads with confidence."
Which sort of makes sense but is also a little strange.
Technical stuff: under the standard assumption of infinite exchangeability of coin tosses, there exists some limiting relative frequency for coin toss results. (This is de Finetti's theorem.)
Key point: I have a probability distribution for this relative frequency (call it f) -- not a probability of a probability.
Here you've said that my probability density for f is dispersed, but slightly asymmetric. I too can say, "Well, I have an awful lot of probability mass on values of f less than 0.5. I should collect more information to tighten this up."
This mixes up f on the one hand with my distribution for f on the other. I can certainly collect data until I'm 80% sure that f is bigger than 0.5 (provided that f really is bigger than 0.5). This is distinct from being 80% sure of getting heads on the next toss.
I guess I just don't understand the difference between bayesianism and frequentism. If I had seen your discussion of limiting relative frequency somewhere else, I would have called it frequentist.
I think I'll go back to borrowing bits and pieces. (Thank you for some nice ones.)
The key difference is that a frequentist would not admit the legitimacy of a distribution for f -- the data are random, so they get a distribution, but f is fixed, although unknown. Bayesians say that quantities that are fixed but unknown get probability distributions that encode the information we have about them.