> If we state the full nature of our experiment, what we controlled and what we didn't... how can it be a "degree of belief"?
Here's a question for you: what's the millionth digit of Pi? (In the standard decimal expansion, of course.)
This is a question (kind of an "experiment") which literally has only one, constant answer, that is theoretically knowable. But unless you search online, or happen to know the answer, you actually don't know which of the digits 0-9 it is. And I can just as easily ask about the 10^100th digit of Pi, which is, again, a constant - and yet no one knows what it is.
So using the Frequentist approach to statistics doesn't make much sense - there's no repeated experiment with possible different outcomes.
But there is a real sense in which your answer should be "it's one of the digits 0-9 with 1/10 probability each". That answer makes sense, because the answer isn't unknowable, just unknown, and probability reflects your lack of knowledge and degree of belief.
The repeatable experiment is asking for different digits of different normal irrational numbers. If repeated enough times, the answer will be each digit 0-9, each one about 1/10 of the time.
Here's a question for you: what's the millionth digit of Pi? (In the standard decimal expansion, of course.)
This is a question (kind of an "experiment") which literally has only one, constant answer, that is theoretically knowable. But unless you search online, or happen to know the answer, you actually don't know which of the digits 0-9 it is. And I can just as easily ask about the 10^100th digit of Pi, which is, again, a constant - and yet no one knows what it is.
So using the Frequentist approach to statistics doesn't make much sense - there's no repeated experiment with possible different outcomes.
But there is a real sense in which your answer should be "it's one of the digits 0-9 with 1/10 probability each". That answer makes sense, because the answer isn't unknowable, just unknown, and probability reflects your lack of knowledge and degree of belief.