Universal Convergence of Semimeasures on Individual Random Sequences
- Cite this paper as:
- 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
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.
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