Universal Convergence of Semimeasures on Individual Random Sequences
<|reference_start|>Universal Convergence of Semimeasures on Individual Random Sequences: Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a universal sequence predictor in case of unknown mu. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Loef) random sequences remained open. Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of M itself. We show that there are universal semimeasures M which do not converge for all random sequences, i.e. we give a partial negative answer to the open problem. We also provide a positive answer for some non-universal semimeasures. We define the incomputable measure D as a mixture over all computable measures and the enumerable semimeasure W as a mixture over all enumerable nearly-measures. We show that W converges to D and D to mu on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role.<|reference_end|>
@article{hutter2004universal,
    title={Universal Convergence of Semimeasures on Individual Random Sequences},
    author={Marcus Hutter and Andrej Muchnik},
    journal={Proc. 15th International Conf. on Algorithmic Learning Theory
  (ALT-2004), pages 234-248},
    year={2004},
    number={IDSIA-14-04},
    archivePrefix={arXiv},
    eprint={cs/0407057},
    primaryClass={cs.LG cs.AI cs.CC cs.IT math.IT math.PR}
}