Statistical Criticality arises in Most Informative Representations

TL;DR

This paper demonstrates that statistical criticality, such as Zipf's law, naturally emerges in samples that are maximally informative about the underlying data-generating process, linking information theory and critical phenomena.

Contribution

It introduces the concept of relevance as a measure of informative frequency distributions and shows that maximally informative samples exhibit criticality and Zipf's law without fine tuning.

Findings

Maximally informative samples maximize relevance at a given resolution.

Zipf's law arises at the optimal trade-off between resolution and relevance.

A bound on the maximum number of estimable parameters is derived.

Abstract

We show that statistical criticality, i.e. the occurrence of power law frequency distributions, arises in samples that are maximally informative about the underlying generating process. In order to reach this conclusion, we first identify the frequency with which different outcomes occur in a sample, as the variable carrying useful information on the generative process. The entropy of the frequency, that we call relevance, provides an upper bound to the number of informative bits. This differs from the entropy of the data, that we take as a measure of resolution. Samples that maximise relevance at a given resolution - that we call maximally informative samples - exhibit statistical criticality. In particular, Zipf's law arises at the optimal trade-off between resolution (i.e. compression) and relevance. As a byproduct, we derive a bound of the maximal number of parameters that can be estimated from a dataset, in the absence of prior knowledge on the generative model. Furthermore, we relate criticality to the statistical properties of the representation of the data generating process. We show that, as a consequence of the concentration property of the Asymptotic Equipartition Property, representations that are maximally informative about the data generating process are characterised by an exponential distribution of energy levels. This arises from a principle of minimal entropy, that is conjugate of the maximum entropy principle in statistical mechanics. This explains why statistical criticality requires no parameter fine tuning in maximally informative samples.