Then I’m confused, because the two would seem to produce two very different answers on the same string. Since a string with very high Kolmogorov complexity can be clearly produced by a uniform distribution, the Solomonoff prior would converge to a very high complexity hypothesis, while the Levin mixture would just assign 0.5 to 0 and 0.5 to 1. What am I missing here?
The Solomonoff prior would have many surviving hypotheses at each step, and the total weight of those that predict a 0 for the next bit would be about equal to the total weight of those that predict a 1. If the input distribution is biased, e.g. 0 with probability 5⁄6 and 1 with probability 1⁄6, then the Solomonoff prior will converge on that as well. That works for any computable input distribution, with probability 1 according to the input distribution.
Then I’m confused, because the two would seem to produce two very different answers on the same string.
Since a string with very high Kolmogorov complexity can be clearly produced by a uniform distribution, the Solomonoff prior would converge to a very high complexity hypothesis, while the Levin mixture would just assign 0.5 to 0 and 0.5 to 1.
What am I missing here?
The Solomonoff prior would have many surviving hypotheses at each step, and the total weight of those that predict a 0 for the next bit would be about equal to the total weight of those that predict a 1. If the input distribution is biased, e.g. 0 with probability 5⁄6 and 1 with probability 1⁄6, then the Solomonoff prior will converge on that as well. That works for any computable input distribution, with probability 1 according to the input distribution.
nitpick: the prior does not converge, the prior is what you have before you start observing data, then it is a posterior.
Many thanks, I get it now.
What matters is the probability that they assign to the next bit being equal to one.