Can algorithmic probability tell me what mathematical structure generated a string, when some of the possible mathematical structures are not computable?
Presumably you’ll only ever attempt to infer from a finite prefix of such a string, which is guaranteed to have a computable description. (Worst case scenario: “the string whose first character is L, whose second character is e, …”).
But, we’re trying to infer the actual generator, right? If, big-worldily, the actual generator is in some set of incomputible generators, it doesn’t help us at all to come up with an n-parameter generating program for the first n bits—although no computable predictor can do better. If the set of possible generators is, itself, incomputable, how do we set a probability distribution over it?
Presumably you’ll only ever attempt to infer from a finite prefix of such a string, which is guaranteed to have a computable description. (Worst case scenario: “the string whose first character is L, whose second character is e, …”).
But, we’re trying to infer the actual generator, right? If, big-worldily, the actual generator is in some set of incomputible generators, it doesn’t help us at all to come up with an n-parameter generating program for the first n bits—although no computable predictor can do better. If the set of possible generators is, itself, incomputable, how do we set a probability distribution over it?
Ah, good point. Yes, that’s probably a safe assumption for practical purposes.