Abstract
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 μ, if the latter is computable. Hence, M is eligible as a universal sequence predictor in case of unknown μ. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Löf) 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 μ on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role.
This work was partially supported by the Swiss National Science Foundation (SNF grant 2100-67712.02) and the Russian Foundation for Basic Research (RFBR grants N04-01-00427 and N02-01-22001).
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Hutter, M., Muchnik, A. (2004). Universal Convergence of Semimeasures on Individual Random Sequences. In: Ben-David, S., Case, J., Maruoka, A. (eds) Algorithmic Learning Theory. ALT 2004. Lecture Notes in Computer Science(), vol 3244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30215-5_19
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DOI: https://doi.org/10.1007/978-3-540-30215-5_19
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