The Redundancy of a Computable Code on a Noncomputable Distribution

TL;DR

This paper explores the limits of computable codes in approximating Kolmogorov complexity for noncomputable distributions, introducing concepts like superuniversal codes and catch-up time, with implications for statistical coding efficiency.

Contribution

It defines new universal coding concepts, analyzes their properties under Bayesian measures, and introduces catch-up time as a measure of coding efficiency for noncomputable distributions.

Findings

Bayesian codes are superuniversal for almost all sequences under certain measures.

No computable code can significantly outperform Bayesian codes in redundancy.

Catch-up time varies among codes, affecting practical coding performance.

Abstract

We introduce new definitions of universal and superuniversal computable codes, which are based on a code's ability to approximate Kolmogorov complexity within the prescribed margin for all individual sequences from a given set. Such sets of sequences may be singled out almost surely with respect to certain probability measures. Consider a measure parameterized with a real parameter and put an arbitrary prior on the parameter. The Bayesian measure is the expectation of the parameterized measure with respect to the prior. It appears that a modified Shannon-Fano code for any computable Bayesian measure, which we call the Bayesian code, is superuniversal on a set of parameterized measure-almost all sequences for prior-almost every parameter. According to this result, in the typical setting of mathematical statistics no computable code enjoys redundancy which is ultimately much less than that of the Bayesian code. Thus we introduce another characteristic of computable codes: The catch-up time is the length of data for which the code length drops below the Kolmogorov complexity plus the prescribed margin. Some codes may have smaller catch-up times than Bayesian codes.