Recall that logical non-omniscience is an open problem. That is, often we get ‘evidence’ in the form of someone pointing out some feature of the hypothesis that, while deducible from it, we were not aware of. For example, if H = “3542423580 is composite” someone might be stumped until they are reminded that integers ending in the digit 0 are all composite. Of course, this fact is deducible from the definition of composite, we just had forgotten it. P(H) now approaches 1, but we don’t have a Bayesian way of talking about what just happened.
Hypothesis specification is just a special case of this problem. The only difference is that instead of pointing out something that is deducible by assuming the hypothesis (think: lines toward the bottom of a proof) we’re stipulating what it means to assume the hypothesis (like reading off the assumptions at the top of a proof). The reason why “any statement starts out with a 50% probability of being true” sounds silly and is confusing people is that for any particular hypothesis the prior will be set, in part, by stipulating the content of the hypothesis—which is a deductive process. And we don’t know how to handle that with Bayesian math.
Recall that logical non-omniscience is an open problem. That is, often we get ‘evidence’ in the form of someone pointing out some feature of the hypothesis that, while deducible from it, we were not aware of. For example, if H = “3542423580 is composite” someone might be stumped until they are reminded that integers ending in the digit 0 are all composite. Of course, this fact is deducible from the definition of composite, we just had forgotten it. P(H) now approaches 1, but we don’t have a Bayesian way of talking about what just happened.
Hypothesis specification is just a special case of this problem. The only difference is that instead of pointing out something that is deducible by assuming the hypothesis (think: lines toward the bottom of a proof) we’re stipulating what it means to assume the hypothesis (like reading off the assumptions at the top of a proof). The reason why “any statement starts out with a 50% probability of being true” sounds silly and is confusing people is that for any particular hypothesis the prior will be set, in part, by stipulating the content of the hypothesis—which is a deductive process. And we don’t know how to handle that with Bayesian math.