is it true that for every Solomonoff-like priors, shorter program have almost always higher probability than longer ones? Almost always in this case means “all but a finite number”.
No, that’s not true. You can take any infinite set of pairs of programs and swap the weights in each pair.
I guess that one can prove this is not the case for computable priors.
Yeah, all universal priors are uncomputable. For any computable prior, you can build a computable sequence whose probability under that prior is 0. For example, you can ask the prior “which choice of next bit would have probability below 0.6?” at each step.
No, that’s not true. You can take any infinite set of pairs of programs and swap the weights in each pair.
Yeah, all universal priors are uncomputable. For any computable prior, you can build a computable sequence whose probability under that prior is 0. For example, you can ask the prior “which choice of next bit would have probability below 0.6?” at each step.