Universal Convergence of Semimeasures on Individual Random Sequences

TL;DR

This paper investigates the convergence properties of universal semimeasures on individual random sequences, revealing limitations and conditions under which convergence occurs or fails.

Contribution

It demonstrates that some universal semimeasures do not converge on all random sequences, and introduces non-universal measures that do converge, advancing understanding of sequence prediction.

Findings

W converges to D on all random sequences

D converges to the true measure mu on all random sequences

Some universal semimeasures do not converge for all random sequences

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 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.