A generalized characterization of algorithmic probability

An a priori semimeasure (also known as "algorithmic probability" or "the Solomonoff prior" in the context of inductive inference) is defined as the transformation, by a given universal monotone Turing machine, of the uniform measure on the infinite strings. It is shown in this paper that the class of a priori semimeasures can equivalently be defined as the class of transformations, by all compatible universal monotone Turing machines, of any continuous computable measure in place of the uniform measure. Some consideration is given to possible implications for the prevalent association of algorithmic probability with certain foundational statistical principles

The Fundamental Nature of the Log Loss Function

The standard loss functions used in the literature on probabilistic prediction are the log loss function, the Brier loss function, and the spherical loss function; however, any computable proper loss function can be used for comparison of prediction algorithms. This note shows that the log loss function is most selective in that any prediction algorithm that is optimal for a given data sequence (in the sense of the algorithmic theory of randomness) under the log loss function will be optimal under any computable proper mixable loss function; on the other hand, there is a data sequence and a prediction algorithm that is optimal for that sequence under either of the two other standard loss functions but not under the log loss function.Comment: 12 page

Algorithmic information theory

This article is a brief guide to the field of algorithmic information theory (AIT), its underlying philosophy, and the most important concepts. AIT arises by mixing information theory and computation theory to obtain an objective and absolute notion of information in an individual object, and in so doing gives rise to an objective and robust notion of randomness of individual objects. This is in contrast to classical information theory that is based on random variables and communication, and has no bearing on information and randomness of individual objects. After a brief overview, the major subfields, applications, history, and a map of the field are presented