To me, the question is whether an approximation to Solomonoff induction has approximately the same behavior as Solomonoff induction. I think it can’t, for instance because no approximation of the busy beaver function (even the “best compressor you have”) behaves anything like the busy beaver function. If you think this is a misleading way of looking at it please tell me.
Solomonoff induction can’t handle the Busy Beaver function because Busy Beaver is non-computable. So it isn’t an issue for approximations of Solomonoff (except in so far as they can’t handle it either).
I am not saying that “Solomonoff can’t handle Busy Beaver.” (I’m not even sure I know what you mean.) I am saying that Solomonoff is analogous to Busy Beaver, for instance because they are both noncomputable functions. Busy Beaver is non-approximatable in a strong sense, and so I think that Solomonoff might also be non-approximatable in an equally strong sense.
Kolmogorov complexity is uncomputable, but you can usefully approximate Kolmogorov complexity for many applications using PKZIP. The same goes for Solomonoff induction. Its prior is based on Kolmogorov complexity.
Solomonoff induction can’t handle the Busy Beaver function because Busy Beaver is non-computable. So it isn’t an issue for approximations of Solomonoff (except in so far as they can’t handle it either).
I am not saying that “Solomonoff can’t handle Busy Beaver.” (I’m not even sure I know what you mean.) I am saying that Solomonoff is analogous to Busy Beaver, for instance because they are both noncomputable functions. Busy Beaver is non-approximatable in a strong sense, and so I think that Solomonoff might also be non-approximatable in an equally strong sense.
Kolmogorov complexity is uncomputable, but you can usefully approximate Kolmogorov complexity for many applications using PKZIP. The same goes for Solomonoff induction. Its prior is based on Kolmogorov complexity.