I’m pointing out that as far as we know, it’s not so.
Also, “then our predictions will only be wrong a limited number of times” is trivially true if the observation string is finite, but ” and after that we’ll predict the correctly every time.” is as far as we know, false (we run out of input string before we gather enough data, even if everything is finite).
We know that the observable universe has a finite size. We can predict arbitrarily (but not infinitely) into the future and with an arbitrary (but not infinite) resolution with a finite model.
A true random number generator is equivalent to a lookup table built into the laws of physics, so long as you only pick a random number finitely many times.
In this case, that frequentist prediction you just quoted is worthless. Our predictions will be wrong a limited number of times only because we’re dealing with a model that extends a limited distance into the future, and random numbers will only occur a limited number of times.
However, it works fine from a bayesian perspective. Once you figure out the main part of the program and just start building up the random number table, it will act the same as if you just told it the results would be random and to give probability distributions rather than just sets of possibilities.
The observable universe being a finite-sized deterministic classical computer is a stronger assumption than the universe being something which only produces observable strings according to computable probability distributions (which is a way to formalize the Church–Turing thesis). Solomonoff induction only requires the latter assumption.
Did you deliberately ignore the qualification “if the observable universe is a big but finite-sized computer”?
I’m pointing out that as far as we know, it’s not so.
Also, “then our predictions will only be wrong a limited number of times” is trivially true if the observation string is finite, but ” and after that we’ll predict the correctly every time.” is as far as we know, false (we run out of input string before we gather enough data, even if everything is finite).
We know that the observable universe has a finite size. We can predict arbitrarily (but not infinitely) into the future and with an arbitrary (but not infinite) resolution with a finite model.
A true random number generator is equivalent to a lookup table built into the laws of physics, so long as you only pick a random number finitely many times.
In this case, that frequentist prediction you just quoted is worthless. Our predictions will be wrong a limited number of times only because we’re dealing with a model that extends a limited distance into the future, and random numbers will only occur a limited number of times.
However, it works fine from a bayesian perspective. Once you figure out the main part of the program and just start building up the random number table, it will act the same as if you just told it the results would be random and to give probability distributions rather than just sets of possibilities.
The observable universe being a finite-sized deterministic classical computer is a stronger assumption than the universe being something which only produces observable strings according to computable probability distributions (which is a way to formalize the Church–Turing thesis).
Solomonoff induction only requires the latter assumption.