A frequentist might say, "P1 = 0.5. P2 is either epsilon or 1-epsilon, we don't know which. P3 is either 0 or 1, we don't know which."
A Bayesian might reply, "P1 = P2 = P3 = 0.5. By the way, there's no such thing as a probability of exactly 0 or 1."
Which is right? As with many such long-unresolved debates, the problem is that two different concepts are being labeled with the word 'probability'. Let's separate them and replace P with:
F = the fraction of possible worlds in which a statement is true. F can be exactly 0 or 1.
B = the Bayesian probability that a statement is true. B cannot be exactly 0 or 1.
Clearly there must be a relationship between the two concepts, or the confusion wouldn't have arisen in the first place, and there is: apart from both obeying various laws of probability, in the case where we know F but don't know which world we are in, B = F. That's what's going on in case 1. In the other cases, we know F != 0.5, but our ignorance of its actual value makes it reasonable to assign B = 0.5.
When does the difference matter?
Suppose I offer to bet my $200 the millionth digit of pi is odd, versus your $100 that it's even. With B3 = 0.5, that looks like a good bet from your viewpoint. But you also know F3 = either 0 or 1. You can also infer that I wouldn't have offered that bet unless I knew F3 = 1, from which inference you are likely to update your B3 to more than 2/3, and decline.
On a larger scale, suppose we search Mars thoroughly enough to be confident there is no life there. Now we know F2 = epsilon. Our Bayesian estimate of the probability of life on Europa will also decline toward 0.
Once we understand F and B are different functions, there is no contradiction.
Do you have any counterexamples in mind where the two approaches give different answers and the difference can't be resolved by noting that they aren't answering the same question?
Consider the following statements:
1. The result of this coin flip is heads.
2. There is life on Mars.
3. The millionth digit of pi is odd.
What is the probability of each statement?
A frequentist might say, "P1 = 0.5. P2 is either epsilon or 1-epsilon, we don't know which. P3 is either 0 or 1, we don't know which."
A Bayesian might reply, "P1 = P2 = P3 = 0.5. By the way, there's no such thing as a probability of exactly 0 or 1."
Which is right? As with many such long-unresolved debates, the problem is that two different concepts are being labeled with the word 'probability'. Let's separate them and replace P with:
F = the fraction of possible worlds in which a statement is true. F can be exactly 0 or 1.
B = the Bayesian probability that a statement is true. B cannot be exactly 0 or 1.
Clearly there must be a relationship between the two concepts, or the confusion wouldn't have arisen in the first place, and there is: apart from both obeying various laws of probability, in the case where we know F but don't know which world we are in, B = F. That's what's going on in case 1. In the other cases, we know F != 0.5, but our ignorance of its actual value makes it reasonable to assign B = 0.5.
When does the difference matter?
Suppose I offer to bet my $200 the millionth digit of pi is odd, versus your $100 that it's even. With B3 = 0.5, that looks like a good bet from your viewpoint. But you also know F3 = either 0 or 1. You can also infer that I wouldn't have offered that bet unless I knew F3 = 1, from which inference you are likely to update your B3 to more than 2/3, and decline.
On a larger scale, suppose we search Mars thoroughly enough to be confident there is no life there. Now we know F2 = epsilon. Our Bayesian estimate of the probability of life on Europa will also decline toward 0.
Once we understand F and B are different functions, there is no contradiction.