(Non-)Equivalence of Universal Priors

TL;DR

This paper clarifies the relationships between different notions of universal priors in algorithmic information theory, showing that Solomonoff's and Levin's priors are equivalent, while universally dominant priors form a larger class.

Contribution

The paper characterizes the differences and relationships between Solomonoff's, Levin's, and universally dominant priors, clarifying their equivalences and distinctions.

Findings

Solomonoff's and Levin's priors are equivalent.

Universally dominant priors form a strictly larger class.

The paper provides a characterization of the discrepancy.

Abstract

Ray Solomonoff invented the notion of universal induction featuring an aptly termed "universal" prior probability function over all possible computable environments. The essential property of this prior was its ability to dominate all other such priors. Later, Levin introduced another construction --- a mixture of all possible priors or `universal mixture'. These priors are well known to be equivalent up to multiplicative constants. Here, we seek to clarify further the relationships between these three characterisations of a universal prior (Solomonoff's, universal mixtures, and universally dominant priors). We see that the the constructions of Solomonoff and Levin define an identical class of priors, while the class of universally dominant priors is strictly larger. We provide some characterisation of the discrepancy.