On the Existence and Convergence Computable Universal Priors

TL;DR

This paper explores the existence and convergence properties of universal semimeasures across various computability classes, extending Solomonoff's theory and analyzing their implications for inductive inference.

Contribution

It classifies and analyzes the hierarchy of computable universal (semi)measures, revealing new cases and convergence behaviors within this framework.

Findings

M is enumerable but not finitely computable.

Universal semimeasures dominate all in their class.

Convergence types vary across classes and sequence types.

Abstract

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 mu, if the latter is computable. Hence, M is eligible as a universal predictor in case of unknown mu. 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. The results hold for discrete and continuous semimeasures. 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-Loef random sequences. We introduce a generalized concept of randomness for individual sequences and use it to exhibit difficulties regarding these issues.