On Semimeasures Predicting Martin-Loef Random Sequences

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dc.creatorHutter, Marcus
dc.creatorMuchnik, Andrej
dc.date2007-08-17
dc.date.accessioned2026-07-07T08:24:05Z
dc.date.available2026-07-07T08:24:05Z
dc.descriptionSolomonoff'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.
dc.description21 LaTeX pages
dc.identifierhttps://arxiv.org/abs/0708.2319
dc.identifierhttp://arxiv.org/abs/0708.2319
dc.identifierTheoretical Computer Science, 382 (2007) 247-261
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/136221
dc.subjectInformation Theory
dc.subjectMachine Learning
dc.subjectProbability
dc.titleOn Semimeasures Predicting Martin-Loef Random Sequences
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