Entropy Measures vs. Algorithmic Information

TL;DR

This paper investigates the relationships between algorithmic and Shannon entropy, extending the analysis to Rényi and Tsallis entropies, and explores the computational aspects of entropy measures within time-bounded frameworks.

Contribution

It demonstrates that only Shannon entropy aligns with algorithmic entropy for recursive distributions and analyzes the convergence of Rényi and Tsallis entropies in time-bounded settings.

Findings

Expected algorithmic and Shannon entropy are equivalent for recursive distributions.

Rényi and Tsallis entropies only match Shannon entropy at order 1.

Time-bounded algorithmic entropy aligns with unbounded entropy for certain distributions.

Abstract

Algorithmic entropy and Shannon entropy are two conceptually different information measures, as the former is based on size of programs and the later in probability distributions. However, it is known that, for any recursive probability distribution, the expected value of algorithmic entropy equals its Shannon entropy, up to a constant that depends only on the distribution. We study if a similar relationship holds for R\'{e}nyi and Tsallis entropies of order $\alpha$, showing that it only holds for R\'{e}nyi and Tsallis entropies of order 1 (i.e., for Shannon entropy). Regarding a time bounded analogue relationship, we show that, for distributions such that the cumulative probability distribution is computable in time $t(n)$, the expected value of time-bounded algorithmic entropy (where the alloted time is $nt(n)\log (nt(n))$) is in the same range as the unbounded version. So, for these distributions, Shannon entropy captures the notion of computationally accessible information. We prove that, for universal time-bounded distribution $\m^t(x)$, Tsallis and R\'{e}nyi entropies converge if and only if $\alpha$ is greater than 1.