after taking account of all available evidence—is SI then well-calibrated?
Yes. The prediction error theorem states that as long as the true distribution is computable, the estimate will converge quickly to the true distribution.
However, almost all the work done here, comes from the conditioning. The proof uses that for any computable mu, M(x) > 2^(-K(mu)) mu(x). That is, M does not assign a “very” small probablility to any possible observation.
The exact prior you pick does not matter very much, as long as it dominates the set of all possible distributions mu in this sense. If you have some other distribution P, such that for every mu there is a C with P(x) > C mu(x), you get a similar theorem, differing by just the constant in the inequality.
So I disagree with this:
Essentially the theory seems to predict that possible (evidence-compatible) events or states in the universe will occur in exact or fairly exact proportion to their relative complexities as measured in bits
It’s ok if the prior is not very exact. As long as we don’t overlook any possibilities as a priori super-unlikely when they are not, we can use observations to pin down the exact proportions later.
Yes. The prediction error theorem states that as long as the true distribution is computable, the estimate will converge quickly to the true distribution.
However, almost all the work done here, comes from the conditioning. The proof uses that for any computable mu, M(x) > 2^(-K(mu)) mu(x). That is, M does not assign a “very” small probablility to any possible observation.
The exact prior you pick does not matter very much, as long as it dominates the set of all possible distributions mu in this sense. If you have some other distribution P, such that for every mu there is a C with P(x) > C mu(x), you get a similar theorem, differing by just the constant in the inequality.
So I disagree with this:
It’s ok if the prior is not very exact. As long as we don’t overlook any possibilities as a priori super-unlikely when they are not, we can use observations to pin down the exact proportions later.