On the existence and convergence of computable universal priors

Solomonoff unified Occam’s razor and Epicurus’ principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the posterior of his universal semimeasure M converges rapidly to the true sequence generating posterior μ, if the latter is computable. Hence, M is eligible as a universal predictor in case of unknown μ. We investigate the existence and convergence of computable universal (semi)measures for a hierarchy of computability classes: finitely computable, estimable, enumerable, and approximable. For instance, M is known to be enumerable, but not finitely computable, and to dominate all enumerable semimeasures. We define seven classes of (semi)measures based on these four computability concepts. Each class may or may not contain a (semi)measure which dominates all elements of another class. The analysis of these 49 cases can be reduced to four basic cases, two of them being new. We also investigate more closely the types of convergence, possibly implied by universality: in difference and in ratio, with probability 1, in mean sum, and for Martin-Löf random sequences. We introduce a generalized concept of randomness for individual sequences and use it to exhibit difficulties regarding these issues.