I interpreted the story as EY says “20% odds it’s mine” and NB says “15% odds it’s mine”, so the first approximation is to renormalize the odds to add to 100% and split the bill as 20/(20+15)= and 15/(20+15) respectively. Anything more involved requires extra information and progressively involved assumptions. For example, what can you conclude about the calibration level of each one? Did NB actually mean 1⁄7 when he said 15%? Is EY prone to over/under-estimating the reliability of his memory? And practical questions: how cumbersome would it be to split the $20 given the change available?
So the real exchange probably went something like this: “Here, I’ll take the twenty and give you a ten and a dollar bill… Oh, here is a couple of quarters, too”. “Keep the quarters, I hate coins.” ’Done.”
Your concluding question does not seem to be relevant, and the calculation depends heavily on how you assigned p and q to begin with. Was there shared evidence? What other alternatives were considered? I can easily imagine that in some circumstances the combined probability could be lower than both p and q, because both heavily rely on the same piece of evidence and not seeing that the two hypotheses were equivalent weakens the value of this evidence (what else did you miss?).
I fully agree that you could have more information that tells you how to combine the probabilities, but we don’t always have that information, and we need to make a decision anyway. Maybe this means the problem does not have a definitive answer, but I am still trying to decide what I would do.
This is valid. However, for the reasons I am interested in the problem (which I am not going to describe now) I don’t get any to consider those probabilities. Pretend that I am stubborn, and refuse to consider anything other than p and q, but still would like advice on how to combine them.
I interpreted the story as EY says “20% odds it’s mine” and NB says “15% odds it’s mine”, so the first approximation is to renormalize the odds to add to 100% and split the bill as 20/(20+15)= and 15/(20+15) respectively. Anything more involved requires extra information and progressively involved assumptions. For example, what can you conclude about the calibration level of each one? Did NB actually mean 1⁄7 when he said 15%? Is EY prone to over/under-estimating the reliability of his memory? And practical questions: how cumbersome would it be to split the $20 given the change available?
So the real exchange probably went something like this: “Here, I’ll take the twenty and give you a ten and a dollar bill… Oh, here is a couple of quarters, too”. “Keep the quarters, I hate coins.” ’Done.”
Your concluding question does not seem to be relevant, and the calculation depends heavily on how you assigned p and q to begin with. Was there shared evidence? What other alternatives were considered? I can easily imagine that in some circumstances the combined probability could be lower than both p and q, because both heavily rely on the same piece of evidence and not seeing that the two hypotheses were equivalent weakens the value of this evidence (what else did you miss?).
I fully agree that you could have more information that tells you how to combine the probabilities, but we don’t always have that information, and we need to make a decision anyway. Maybe this means the problem does not have a definitive answer, but I am still trying to decide what I would do.
If I’m the person who came up with probabilities p and q in the first place, surely I know how I came up with those probabilities.
This is valid. However, for the reasons I am interested in the problem (which I am not going to describe now) I don’t get any to consider those probabilities. Pretend that I am stubborn, and refuse to consider anything other than p and q, but still would like advice on how to combine them.