(Non-)Equivalence of Universal Priors

Ray Solomonoff invented the notion of universal induction featuring an aptly termed universal prior probability function over all possible computable environments [9]. 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[12]. 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 (Solomonoffs, 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.