Suppose I get a piece of evidence apparently relevant to a proposition p that can't be handled using the standard Bayesian probabilistic apparatus. For instance, maybe I am trying to figure out how close a dart landed to the center of the target, and I learn that, mirabilis, the dart's landing point had rational numbers as coordinates. That's a case where the likelihoods of the evidence on all the relevant hypotheses are zero and there is no good limiting procedure to get around that. Or maybe I am trying to figure out how old Jones is, and I am told that Jones' age in years happens to be equal to a number that an angel yesterday picked out from among all natural numbers by a procedure that has no biases in favor of any numbers. That's a case where there are no meaningful likelihoods at all, since countable fair lotteries cannot be handled by the probability calculus. Or perhaps I am wondering whether some large number is prime, and I am told that infinitely many people in the multiverse think it is and infinitely many think it's not.
In some cases, the evidence should infect my credences in such a way that I no longer have a probability assigned to p. In other cases, I should just ignore the evidence. How to judge what should be done when? My intuitions say that I can just ignore what the infinitely many people in the multiverse think. But I don't know what to make of the other two pieces of evidence. I have a weak inclination to ignore the rational-number-coordinates fact. But the fact about Jones' age happening to match up with the infinite lottery result, that I don't think should be ignored. Maybe I should no longer have credences about Jones' age?
Can anything in general be said? Maybe, but I don't know how to do it.
3 comments:
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I think the problem may be in the question. If "evidence" is defined in a Bayesian fashion (something like "X is evidence of H iff P(X|H)>> P(X|~H)"), then a fact which cannot be handled probabilistically is not evidence. It may be in some way related to the matter at hand, but it does not count as evidence.
This might seem arbitrary, but it seems to cohere with the commonsense notion that, if X does not increase the likelihood of H then X is not evidence for H. That surely seems intuitively right. And, if X cannot be evaluated in a probabilistic way, then I can't see how it would increase the likelihood of H at all.
So, to go through your examples: I think the multiverse example is utterly untenable. If there is a reason for an infinite number of people to even form an opinion on the matter of whether the number is prime, then that reason should govern the choice of which one they think it is. It just seems utterly implausible that it could be as you stipulated. The example of Jones' age seems unrelated to the matter of "evidence" at all, for the reasons I mention in my first two paragraphs. The angel choosing a number that happens to be the same as Jones' age is not evidence of his age in any meaningful sense. And in the case of the dart, how does knowing that the coordinates are rational numbers give any evidence at all of how close the dart is to the center??
Perhaps I'm misunderstanding the question....
Michael:
The problem is that if P(E|H) is not defined, then we can't check if E is relevant to H or not.
Suppose you find out that a dart hit some nonmeasurable set. That could well be relevant evidence with respect to whether the dart hit near the center, even though the probabilities are undefined.
Huume:
Thank you for the kind words.
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