The problem being discussed is the relativity of complexity. So long as anything can be made out to be more complicated than anything else by an appropriate choice of language, it seems that Solomonoff Induction will be arbitrary, and we won’t be justified in thinking that it is accurate.
Yes, one universal prior will differ from another by just a finite number of terms. But there is no upper bound on how large this finite number can be. So we can’t make any claims about how likely specific predictions are, without arbitrarily ruling out infinite sets of languages/models. So the problem remains.
As you say, A is to be preferred to programs longer than Z. But there is no upper bound on how long Z might be. So any particular program—for example, B—is such that we have no reason to say that it is more or less likely than A. So it seems we have failed to find a full justification for why we should prefer A to B.
Unless, as I said, we start talking about the space of all possible languages/models. But as I said, this threatens to just push the problem up a level.
The problem being discussed is the relativity of complexity. So long as anything can be made out to be more complicated than anything else by an appropriate choice of language, it seems that Solomonoff Induction will be arbitrary, and we won’t be justified in thinking that it is accurate.
Yes, one universal prior will differ from another by just a finite number of terms. But there is no upper bound on how large this finite number can be. So we can’t make any claims about how likely specific predictions are, without arbitrarily ruling out infinite sets of languages/models. So the problem remains.
As you say, A is to be preferred to programs longer than Z. But there is no upper bound on how long Z might be. So any particular program—for example, B—is such that we have no reason to say that it is more or less likely than A. So it seems we have failed to find a full justification for why we should prefer A to B.
Unless, as I said, we start talking about the space of all possible languages/models. But as I said, this threatens to just push the problem up a level.