Open Problems 1 When Is Small Beautiful ?

Is the textbook result that Solomonoff’s universal posterior convergesto the true posterior for all Martin-Löf random sequences true?Universal Induction Induction problems can be phrased as sequence predic-tion tasks. This is, for instance, obvious for time series prediction, but alsoincludes classification tasks. Having observed data xt at times t < n, the task isto predict the t-th symbol xt from sequence x = x1...xt−1. The key concept toattack general induction problems is Occam’s razor and to a less extent Epicu-rus’ principle of multiple explanations. The former/latter may be interpreted asto keep the simplest/all theories consistent with the observations x1...xt−1 andto use these theories to predict xt. Solomonoff [4, 5] formalized and combinedboth principles in his universal prior M(x) which assigns high/low probability tosimple/complex environments, hence implementing Occam and Epicurus. M(x)is defined as the probability that a universal Turing machine U outputs a stringstarting with x, when provided with fair coin flips on the input.Posterior Convergence Solomonoff’s [5] central result is that if the probabil-ityμ(xt|x1...xt−1) of observing xt at time t, given past observations x1...xt−1is a computable function, then the universal posterior Mt :=M(xt|x1...xt−1)converges (rapidly!) with μ-probability 1 (w.p.1) for t → ∞ to the true poste-rior μt :=μ(xt|x1...xt−1), hence M represents a universal predictor in case ofunknown μ. Convergence of Mt to μt w.p.1 tells us that Mt is close to μt for suf-ficiently large t for “almost all” sequences x1:∞ (we abbreviate x1:n := x1...xn).It says nothing about whether convergence is true for any particular sequence(of measure 0).Martin-Löf Randomness is the standard notion to capture convergence forindividual sequences and is closely related to Solomonoff’s universal prior. Levingave a characterization equivalent to Martin-Löf’s (M.L.) original definition [2]:A sequence x1:∞ is μ-random (in the sense of M.L.) iff there is a constant csuch thatM(x1:n) ≤ c · μ(x1:n) for all n.One can show that a μ-random sequence x1:∞ passes all thinkable effectiverandomness tests, e.g. the law of large numbers, the law of the iterated logarithm,etc. In particular, the set of all μ-random sequences has μ-measure 1.Open Problem An interesting open question is whether Mt converges to μt(in difference or ratio) individually for all Martin-Löf random sequences. Clearly,Solomonoff’s result shows that convergence may at most fail for a set of sequenceswith μ-measure zero. A convergence result for μ-random sequences is particularlyinteresting and natural in this context, since μ-randomness can be defined interms of M itself (see above). 11 IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland [email protected] http://www.idsia.ch/~marcus12 A prize of 128 Euro for a solution of this problem is offered. Proof Attempts Attempts to convert the convergence results w.p.1 to effectiveμ-randomness tests fail, since Mt is not lower semi-computable. In [3, Th.5.2.2]and [6, Th.10] the following Theorem is stated:“Let μ be a positive recursive measure. If the length of y is fixed and thelength of x grows to infinity, then M(y|x)/μ(y|x) → 1 with μ-probability one.The infinite sequences ω with prefixes x satisfying the displayed asymptotics areprecisely [‘⇒’ and ‘⇐’] the μ-random sequences.”While convergence w.p.1 is correct if appropriately interpreted, the proofthat convergence holds for μ-random sequences is incomplete:“M(x1:n) ≤ c·μ(x1:n)∀n ⇒ limn→∞ M(x1:n)/μ(x1:n) exists” has been used, butnot proven, and may indeed be wrong. Vovk [7] shows that for two finitely com-putable semi-measures μ and ρ, and x1:∞ being μand ρ-random that ρt/μt → 1.If M were recursive, then this would imply Mt/μt → 1 for every μ-random se-quencex1:∞, since every sequence is M -random. Since M is not recursive Vovk’stheorem cannot be applied and it is not obvious how to generalize it. So thequestion of individual convergence remains open.Conclusions Contrary to what was believed before, the question of posteriorconvergence Mt/μt → 1 (also Mt → μt) for all μ-random sequences is still open.In [1] we introduce a new flexible notion of randomness which contains Martin-Löf randomness as a special case. This notion is used to show that standardproof attempts of Mt/μtM.L.−→ 1 based on so called dominance only must fail,indicating that this problem may be a hard one.