Predictive complexity and information

Abstract

A new notion of predictive complexity and corresponding amount of information are considered. Predictive complexity is a generalization of Kolmogorov complexity which bounds the ability of any algorithm to predict elements of a sequence of outcomes. We consider predictive complexity for a wide class of bounded loss functions which are generalizations of square-loss function. Relations between unconditional KG(x) and conditional KG(x|y) predictive complexities are studied. We define an algorithm which has some "expanding property". It transforms with positive probability sequences of given predictive complexity into sequences of essentially bigger predictive complexity. A concept of amount of predictive information IG(y : x) is studied. We show that this information is non-commutative in a very strong sense and present asymptotic relations between values IG(y : x), IG(x : y), KG(x) and KG(y).