Ubiquity of Benfords law and emergence of the reciprocal distribution

TL;DR

This paper demonstrates that the reciprocal distribution emerges as a universal, scale-invariant distribution through the application of the Law of Total Probability and maximum entropy principles, highlighting its fundamental role in various contexts.

Contribution

It introduces a novel derivation of the reciprocal distribution as a universal scale-invariant distribution using probability theory and entropy maximization.

Findings

Reciprocal distribution is uniquely scale-invariant.

Repeated application of the construction yields the same distribution.

Maximum entropy principle also leads to the reciprocal distribution.

Abstract

We apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdfs), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdfs results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. We separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.